Quasi-Hopf Algebras: A Categorical Approach [ebook ed.] 1108632653, 9781108632652

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Q UA S I - H O P F A L G E B R A S This is the first book to be dedicated entirely to Drinfeld’s quasi-Hopf algebras. Ideal for graduate students and researchers in mathematics and mathematical physics, this treatment is largely self-contained, taking the reader from the basics, with complete proofs, to much more advanced topics, with almost complete proofs. Many of the proofs are based on general categorical results; the same approach can then be used in the study of other Hopf-type algebras, for example Turaev or Zunino Hopf algebras, Hom-Hopf algebras, Hopfish algebras, and in general any algebra for which the category of representations is monoidal. Newcomers to the subject will appreciate the detailed introduction to (braided) monoidal categories, (co)algebras and the other tools they will need in this area. More advanced readers will benefit from having recent research gathered in one place, with open questions to inspire their own research.

Encyclopedia of Mathematics and Its Applications This series is devoted to significant topics or themes that have wide application in mathematics or mathematical science and for which a detailed development of the abstract theory is less important than a thorough and concrete exploration of the implications and applications. Books in the Encyclopedia of Mathematics and Its Applications cover their subjects comprehensively. Less important results may be summarized as exercises at the ends of chapters. For technicalities, readers can be referred to the bibliography, which is expected to be comprehensive. As a result, volumes are encyclopedic references or manageable guides to major subjects.

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S. Khrushchev Orthogonal Polynomials and Continued Fractions H. Nagamochi and T. Ibaraki Algorithmic Aspects of Graph Connectivity F. W. King Hilbert Transforms I F. W. King Hilbert Transforms II O. Calin and D.-C. Chang Sub-Riemannian Geometry M. Grabisch et al. Aggregation Functions L. W. Beineke and R. J. Wilson (eds.) with J. L. Gross and T. W. Tucker Topics in Topological Graph Theory J. Berstel, D. Perrin and C. Reutenauer Codes and Automata T. G. Faticoni Modules over Endomorphism Rings H. Morimoto Stochastic Control and Mathematical Modeling G. Schmidt Relational Mathematics P. Kornerup and D. W. Matula Finite Precision Number Systems and Arithmetic Y. Crama and P. L. Hammer (eds.) Boolean Models and Methods in Mathematics, Computer Science, and Engineering V. Berth´e and M. Rigo (eds.) Combinatorics, Automata and Number Theory A. Krist´aly, V. D. R˘adulescu and C. Varga Variational Principles in Mathematical Physics, Geometry, and Economics J. Berstel and C. Reutenauer Noncommutative Rational Series with Applications B. Courcelle and J. Engelfriet Graph Structure and Monadic Second-Order Logic M. Fiedler Matrices and Graphs in Geometry N. Vakil Real Analysis through Modern Infinitesimals R. B. Paris Hadamard Expansions and Hyperasymptotic Evaluation Y. Crama and P. L. Hammer Boolean Functions A. Arapostathis, V. S. Borkar and M. K. Ghosh Ergodic Control of Diffusion Processes N. Caspard, B. Leclerc and B. Monjardet Finite Ordered Sets D. Z. Arov and H. Dym Bitangential Direct and Inverse Problems for Systems of Integral and Differential Equations G. Dassios Ellipsoidal Harmonics L. W. Beineke and R. J. Wilson (eds.) with O. R. Oellermann Topics in Structural Graph Theory L. Berlyand, A. G. Kolpakov and A. Novikov Introduction to the Network Approximation Method for Materials Modeling M. Baake and U. Grimm Aperiodic Order I: A Mathematical Invitation J. Borwein et al. Lattice Sums Then and Now R. Schneider Convex Bodies: The Brunn–Minkowski Theory (Second Edition) G. Da Prato and J. Zabczyk Stochastic Equations in Infinite Dimensions (Second Edition) D. Hofmann, G. J. Seal and W. Tholen (eds.) Monoidal Topology ´ Rodr´ıguez Palacios Non-Associative Normed Algebras I: The Vidav–Palmer and M. Cabrera Garc´ıa and A. Gelfand–Naimark Theorems C. F. Dunkl and Y. Xu Orthogonal Polynomials of Several Variables (Second Edition) L. W. Beineke and R. J. Wilson (eds.) with B. Toft Topics in Chromatic Graph Theory T. Mora Solving Polynomial Equation Systems III: Algebraic Solving T. Mora Solving Polynomial Equation Systems IV: Buchberger Theory and Beyond V. Berth´e and M. Rigo (eds.) Combinatorics, Words and Symbolic Dynamics B. Rubin Introduction to Radon Transforms: With Elements of Fractional Calculus and Harmonic Analysis M. Ghergu and S. D. Taliaferro Isolated Singularities in Partial Differential Inequalities G. Molica Bisci, V. D. Radulescu and R. Servadei Variational Methods for Nonlocal Fractional Problems S. Wagon The Banach–Tarski Paradox (Second Edition) K. Broughan Equivalents of the Riemann Hypothesis I: Arithmetic Equivalents K. Broughan Equivalents of the Riemann Hypothesis II: Analytic Equivalents M. Baake and U. Grimm (eds.) Aperiodic Order II: Crystallography and Almost Periodicity ´ Rodr´ıguez Palacios Non-Associative Normed Algebras II: Representation M. Cabrera Garc´ıa and A. Theory and the Zel’manov Approach A. Yu. Khrennikov, S. V. Kozyrev and W. A. Z´un˜ iga-Galindo Ultrametric Pseudodifferential Equations and Applications S. R. Finch Mathematical Constants II J. Kraj´ıcˇ ek Proof Complexity D. Bulacu, S. Caenepeel, F. Panaite and F. Van Oystaeyen Quasi-Hopf Algebras

E n cyc l o p e d i a o f M at h e m at i c s a n d i t s A p p l i c at i o n s

Quasi-Hopf Algebras A Categorical Approach DA N I E L BU L AC U Universitatea din Bucures¸ti, Romania

S T E FA A N C A E N E P E E L Vrije Universiteit Brussel, Belgium

F L O R I N PA NA I T E Institute of Mathematics of the Romanian Academy

F R E D DY VA N OY S TA E Y E N Universiteit Antwerpen, Belgium

University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 79 Anson Road, #06–04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781108427012 DOI: 10.1017/9781108582780 © Daniel Bulacu, Stefaan Caenepeel, Florin Panaite and Freddy Van Oystaeyen 2019 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2019 Printed and bound in Great Britain by Clays Ltd, Elcograf S.p.A. A catalogue record for this publication is available from the British Library Library of Congress Cataloging-in-Publication Data Names: Bulacu, Daniel, 1973– author. | Caenepeel, Stefaan, 1956– author. | Panaite, Florin, 1970– author. | Oystaeyen, F. Van, 1947– author. Title: Quasi-Hopf algebras : a categorical approach / Daniel Bulacu (Universitatea din Bucureti, Romania), Stefaan Caenepeel (Vrije Universiteit, Amsterdam), Florin Panaite (Institute of Mathematics of the Romanian Academy), Freddy van Oystaeyen (Universiteit Antwerpen, Belgium). Description: Cambridge ; New York, NY : Cambridge University Press, [2019] | Series: Encyclopedia of mathematics and its applications ; 171 | Includes bibliographical references and index. Identifiers: LCCN 2018034517 | ISBN 9781108427012 (hardback) Subjects: LCSH: Hopf algebras. | Tensor products. | Tensor algebra. Classification: LCC QA613.8 .B85 2019 | DDC 512/.55–dc23 LC record available at https://lccn.loc.gov/2018034517 ISBN 978-1-108-42701-2 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Dedicated to our wives Adriana, Lieve, Cristina, Danielle.

Contents

Preface

page xi

1

Monoidal and Braided Categories 1.1 Monoidal Categories 1.2 Examples of Monoidal Categories 1.2.1 The Category of Sets 1.2.2 The Category of Vector Spaces 1.2.3 The Category of Bimodules 1.2.4 The Category of G-graded Vector Spaces 1.2.5 The Category of Endo-functors 1.2.6 A Strict Category Associated to a Monoidal Category 1.3 Monoidal Functors 1.4 Mac Lane’s Strictification Theorem for Monoidal Categories 1.5 (Pre-)Braided Monoidal Categories 1.6 Rigid Monoidal Categories 1.7 The Left and Right Dual Functors 1.8 Braided Rigid Monoidal Categories 1.9 Notes

1 1 7 7 7 7 8 13 15 16 25 28 38 43 48 54

2

Algebras and Coalgebras in Monoidal Categories 2.1 Algebras in Monoidal Categories 2.2 Coalgebras in Monoidal Categories 2.3 The Dual Coalgebra/Algebra of an Algebra/Coalgebra 2.4 Categories of Representations 2.5 Categories of Corepresentations 2.6 Braided Bialgebras 2.7 Braided Hopf Algebras 2.8 Notes

55 55 65 70 78 82 87 95 101

3

Quasi-bialgebras and Quasi-Hopf Algebras 3.1 Quasi-bialgebras 3.2 Quasi-Hopf Algebras 3.3 Examples of Quasi-bialgebras and Quasi-Hopf Algebras

103 103 110 119

viii

Contents 3.4 3.5 3.6 3.7 3.8 3.9

The Rigid Monoidal Structure of H M fd and MHfd The Reconstruction Theorem for Quasi-Hopf Algebras Sovereign Quasi-Hopf Algebras Dual Quasi-Hopf Algebras Further Examples of (Dual) Quasi-Hopf Algebras Notes

125 128 131 135 141 146

4

Module (Co)Algebras and (Bi)Comodule Algebras 4.1 Module Algebras over Quasi-bialgebras 4.2 Module Coalgebras over Quasi-bialgebras 4.3 Comodule Algebras over Quasi-bialgebras 4.4 Bicomodule Algebras and Two-sided Coactions 4.5 Notes

147 147 154 162 168 176

5

Crossed Products 5.1 Smash Products 5.2 Quasi-smash Products and Generalized Smash Products 5.3 Endomorphism H-module Algebras 5.4 Two-sided Smash and Crossed Products 5.5 H ∗ -Hopf Bimodules 5.6 Diagonal Crossed Products 5.7 L–R-smash Products 5.8 A Duality Theorem for Quasi-Hopf Algebras 5.9 Notes

177 177 185 188 191 196 201 214 220 223

6

Quasi-Hopf Bimodule Categories 6.1 Quasi-Hopf Bimodules 6.2 The Dual of a Quasi-Hopf Bimodule 6.3 Structure Theorems for Quasi-Hopf Bimodules 6.4 The Categories H MHH and H M 6.5 A Structure Theorem for Comodule Algebras 6.6 Coalgebras in H MHH 6.7 Notes

225 225 230 235 239 246 249 251

7

Finite-Dimensional Quasi-Hopf Algebras 7.1 Frobenius Algebras 7.2 Integral Theory 7.3 Semisimple Quasi-Hopf Algebras 7.4 Symmetric Quasi-Hopf Algebras 7.5 Cointegral Theory 7.6 Integrals, Cointegrals and the Fourth Power of the Antipode 7.7 A Freeness Theorem for Quasi-Hopf Algebras 7.8 Notes

253 253 261 268 273 279 288 299 303

8

Yetter–Drinfeld Module Categories 8.1 The Left and Right Center Constructions

305 305

Contents 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 9

Yetter–Drinfeld Modules over Quasi-bialgebras fd The Rigid Braided Category H H YD Yetter–Drinfeld Modules as Modules over an Algebra The Quantum Double of a Quasi-Hopf Algebra The Quasi-Hopf Algebras Dω (H) and Dω (G) Algebras within Categories of Yetter–Drinfeld Modules Cross Products of Algebras in H M , H MH , H H YD Notes

ix 310 318 325 330 335 342 347 351

Two-sided Two-cosided Hopf Modules 9.1 Two-sided Two-cosided Hopf Modules 9.2 Two-sided Two-cosided Hopf Modules versus Yetter–Drinfeld Modules H H 9.3 The Categories H H MH and H YD 9.4 A Structure Theorem for Bicomodule Algebras H 9.5 The Structure of a Coalgebra in H H MH H 9.6 A Braided Monoidal Structure on H MHH H 9.7 Hopf Algebras within H H MH 9.8 Biproduct Quasi-Hopf Algebras 9.9 Notes

353 353 355 360 362 363 369 371 376 379

10

Quasitriangular Quasi-Hopf Algebras 10.1 Quasitriangular Quasi-bialgebras and Quasi-Hopf Algebras 10.2 Further Examples of Monoidal Algebras 10.3 The Square of the Antipode of a QT Quasi-Hopf Algebra 10.4 The QT Structure of the Quantum Double 10.5 The Quantum Double D(H) when H is Quasitriangular 10.6 Notes

381 381 386 388 394 400 406

11

Factorizable Quasi-Hopf Algebras 11.1 Reconstruction in Rigid Monoidal Categories 11.2 The Enveloping Braided Group of a QT Quasi-Hopf Algebra 11.3 Bosonisation for Quasi-Hopf Algebras 11.4 The Function Algebra Braided Group 11.5 Factorizable QT Quasi-Hopf Algebras 11.6 Factorizable Implies Unimodular 11.7 The Quantum Double of a Factorizable Quasi-Hopf Algebra 11.8 Notes

407 407 414 419 421 433 440 443 450

12

The Quantum Dimension and Involutory Quasi-Hopf Algebras 12.1 The Integrals of a Quantum Double 12.2 The Cointegrals of a Quantum Double 12.3 The Quantum Dimension 12.3.1 The Quantum Dimension of H 12.3.2 The Quantum Dimension of D(H) 12.4 The Trace Formula for Quasi-Hopf Algebras

451 451 457 462 462 466 469

x

Contents 12.5 12.6 12.7

13

Involutory Quasi-Hopf Algebras Representations of Involutory Quasi-Hopf Algebras Notes

472 474 479

Ribbon Quasi-Hopf Algebras 13.1 Ribbon Categories 13.2 Ribbon Categories Obtained from Rigid Monoidal Categories 13.3 Ribbon Quasi-Hopf Algebras 13.4 A Class of Ribbon Quasi-Hopf Algebras 13.5 Some Ribbon Elements for Dω (H) and Dω (G) 13.6 Notes

481 481 488 496 505 508 512

Bibliography Index

515 525

Preface

Some basic ideas in mathematics are very generic and almost omnipresent. Let us just mention “operators on some structure,” an idea going back to symmetry of geometric configurations, and also “duality.” These ideas are also at the roots of the modern theory of quasi-Hopf algebras, which is the topic of this book. Geometry is at the root of many developments in mathematics, and for our topic of interest we may go back to algebraic geometry and the theory of (affine) algebraic varieties, which may be seen as sets of solutions of polynomial equations in some affine space over some field. One then studies such varieties via the ring of functions on them with values in the base field; in fact one restricts attention to polynomial functions forming the coordinate ring of the variety. There one observes the fundamental duality between commutative (affine) algebra and the algebraic geometry of (affine) algebraic varieties, later better phrased in the more general scheme theory. The other generic idea of operators acting on geometric structures led directly to actions or transformation groups and operator algebras. The idea of group actions and their invariants is deeply embedded in the philosophy of mathematics; for example, in the “Erlangen Program” of F. Klein, geometry was redefined as the study of properties invariant for actions of transformation groups. On the more algebraic side, actions of groups of automorphisms of fields were used by E. Galois to solve some problems about solutions to polynomial equations over a field. In the resulting Galois theory another duality appeared, namely the duality between subgroups of the Galois group of some field extension and the lattice of subfields of the field. This Galois duality originally was considered for finite-dimensional separable field extensions but it was extended to inseparable extensions by using derivations and higher derivations, leading to Lie algebra actions and their invariants. Thus, a more general Galois theory mixing Lie actions (of derivations) and group actions (of automorphisms) resulted, immediately leading to a Galois theory for Hopf algebra actions. Further extensions of the Galois theory were in the direction of continuous groups, later called Lie groups. So here the generic ideas of action and duality met, and Hopf algebras appeared naturally. But also the geometric line of development showed a similar phenomenon with the study of abelian varieties and algebraic groups. Roughly stated, an algebraic group is an algebraic variety with a group structure on its points;

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interesting examples are matrix groups, that is, groups embedded in a matrix ring and having the structure of an algebraic variety, like GLn (k) and SLn (k), the general and special linear groups over the field k, respectively. The group structure on the variety translates into a structure of the coordinate ring given by a comultiplication, a counit and an antipode satisfying suitable conditions that turn it into a commutative Hopf algebra. Hopf algebras got their name because they appeared first in a celebrated paper by H. Hopf on algebraic topology. In fact the structure was discovered on the cohomology ring of an H-space; roughly stated, that is a topological space with a multiplication on it together with a special element such that left and right multiplication by this element defines a map which is homotopic to the identity map (so a kind of neutral element up to homotopy). Group actions on vector spaces may be studied by looking at modules over the group algebra k[G] of the acting group G over the base field k; similarly, Lie algebra actions of a Lie algebra g on a vector space may be studied by looking at the universal enveloping algebra of g over k, say Uk (g). Now both k[G] and Uk (g) are Hopf algebras but not commutative anymore; instead, they are cocommutative. So aspects of group actions and Lie algebra actions become unified in a theory of actions of arbitrary Hopf algebras on general algebras or vector spaces or modules, and this received extensive interest in ring theory. Let us point out one important “generality” for general Hopf algebras: they need not be commutative or cocommutative, as many of the early examples of Hopf algebras were. In his famous address to the International Congress of Mathematicians in 1986, Drinfeld introduced the term “quantum group,” roughly referring to a quasitriangular Hopf algebra, that is, a Hopf algebra endowed with a so-called R-matrix, satisfying certain axioms that represent a relaxation of the cocommutativity condition and implying the (equally famous) quantum Yang–Baxter equation. Drinfeld proved that any finite-dimensional Hopf algebra can be embedded in a quasitriangular one, called its quantum (or Drinfeld) double. There is a vast literature on quantum groups and many examples could be obtained from deforming well-known easier Hopf algebras. Combined with the restriction to special Hopf algebras it also makes sense to restrict to special categories of modules like so-called Yetter–Drinfeld modules, to name just one. Essential for the transition from Hopf algebras to quasi-Hopf algebras was the concept of monoidal category, roughly stated a category with a product (called the “tensor product”) generalizing the tensor product of vector spaces in a suitable way and satisfying natural conditions. For example, the category of sets is a monoidal category, the “tensor product” being the Cartesian product of sets. One of the axioms of a monoidal category is the so-called “associativity constraint,” which for the categories of vector spaces and of sets is “trivial;” for instance, for vector spaces this boils down to saying that, if U, V , W are vector spaces, then (U ⊗ V ) ⊗ W and U ⊗ (V ⊗W ) can be identified in the usual (or “trivial”) way. One of the fundamental features of a Hopf algebra, H, is that its category of (left) representations is a monoidal category, with tensor product inherited from the

Preface

xiii

category of vector spaces, and the tensor product of two left H-modules is again a left H-module via the comultiplication of H. The associativity constraint is, again, “trivial.” If one is not interested in an a priori given type of algebra but wants to make sure that there is a “product” on the category of its representations, then one finds the motivation for the introduction of quasi-Hopf algebras as Drinfeld did in his seminal paper [80]. Roughly, a quasi-Hopf algebra is an algebra for which its category of left modules is monoidal, but maybe with non-trivial associativity constraint. More precisely, what Drinfeld did was to weaken the coassociativity condition for a Hopf algebra so that the comultiplication is only coassociative up to conjugation by an invertible element of H ⊗ H ⊗ H (which is a sort of 3-cocycle). Moreover, examples of quasi-Hopf algebras can be obtained by “twisting” the comultiplication of a Hopf algebra via a so-called “gauge transformation” (only if the gauge transformation is a sort of 2-cocycle is the twisted object again a Hopf algebra). After specialization to quantum groups, sometimes just taken to be non-commutative non-cocommutative Hopf algebras but usually with extra conditions like quasitriangularity, the generalization in terms of non-coassociativity became popular too and it found several applications as well. Again, the fundamental property is that the relaxation of coassociativity still makes the representation category into a monoidal category, and moreover the rigidity (i.e. the existence of dual objects) of the category of finite-dimensional representations of a Hopf algebra, owing to the presence of an antipode, is preserved by replacing the notion of an antipode by a suitable analogue. Categorically speaking, passing from the category of Hopf algebras to the one of quasi-Hopf algebras does not (in principle) really add to the complexity; in fact the latter is in some sense more manageable because of the presence of a kind of gauge group. Monoidal categories were present, if hidden, in the classical ideas mentioned before and they have been very useful in obtaining a unified theory. One of the early facts that stimulated interest in monoidal categories stemmed from their applicability in rational conformal field theory (RCFT). The monoidal categories in RCFT could, by Tannaka–Krein reconstruction, be considered as module categories over some “Hopf-like” algebras. Back in 1984 Drinfeld and Jimbo introduced a quantum group by deforming a universal enveloping algebra U(g) for some Lie algebra g; in fact for every semisimple Lie algebra they constructed what was called afterwards the Drinfeld–Jimbo algebra. For the study of some categories of modules over the Drinfeld–Jimbo algebras, a relation with the so-called KZ-equations had to be used; these equations were introduced by Knizhnik and Zamolodchikov in 1984. The KZequations are linear differential equations satisfied by two-dimensional conformal field theories associated with affine Lie algebras. Such KZ-equations may be used to obtain a quantization of universal enveloping algebras, and Drinfeld used KZequations to construct a quasi-Hopf algebra for some Lie algebra g, say Qg , so that some categories of modules over Qg are equivalent to similar ones over the Drinfeld– Jimbo algebra of the Lie algebra g. Further interesting applications of KZ-equations follow, for example, from the fact that their monodromy along closed paths yields

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a representation of the braid group. We refer to the specialized literature for more detail concerning applications in physics. We do the same for some deep relations with number theory in the sense of A. Grothendieck’s “Esquisse.” In this book we aim to develop the theory of quasi-Hopf algebras from scratch, or almost, dealing mainly with algebraic methods. Knowledge of Hopf algebras will benefit the reader but we do introduce the necessary concepts. Using monoidal categories as the main tool makes for a rather abstract treatment of the material, but we hope the unifying effect of it will expose well the beautiful generalization from Hopf algebras to quasi-Hopf algebras; moreover, the categorical point of view also stays close to the applications in physics, as indicated by the foregoing remarks. We now outline the content of the book (more historical and bibliographical remarks can be found in the Notes section at the end of each chapter). In Chapters 1 and 2 we present the basic categorical concepts and tools needed for the rest of the book (monoidal, rigid and braided categories and algebras, coalgebras and Hopf algebras in such categories). We included detailed definitions and proofs; we do not assume that the reader has prior knowledge of these topics. In particular, we introduce the concepts of coalgebra, bialgebra and Hopf algebra in the usual sense (over a field), so we do not assume from the reader a knowledge of these concepts either. In Chapter 3 we introduce the main objects of our study, quasi-bialgebras and quasi-Hopf algebras (as well as the dual concepts), present their basic properties and some classes of examples. We have two warnings for the reader: (1) the concept of quasi-bialgebra is introduced in Definition 3.4, but afterwards we make a reduction, and the axioms of a quasi-bialgebra that will be used from there on are the ones presented in equations (3.1.7)–(3.1.10); (2) unlike Drinfeld, we do not include the bijectivity of the antipode in the definition of a quasi-Hopf algebra, and we shall see in later chapters that the bijectivity is automatic in the finite-dimensional and the quasitriangular case. In Chapter 4 we study “(co)actions” of quasi-bialgebras and quasi-Hopf algebras, namely we introduce the concepts of module (co)algebra and (bi)comodule algebra over a quasi-bialgebra, we give some examples and present some connections that exist between these structures. In Chapter 5 we introduce various types of crossed products that appear in the context of quasi-Hopf algebras (smash products, diagonal crossed products, etc.), we study the relations between them and as an application we present a duality theorem for finite-dimensional quasi-Hopf algebras. In Chapter 6 we introduce so-called quasi-Hopf bimodules over a quasi-Hopf algebra H, prove some structure theorems for them leading to the fact that their category is monoidally equivalent to the category of left H-modules and, as an application, we prove a structure theorem for quasi-Hopf comodule algebras. In Chapter 7 we study finite-dimensional quasi-Hopf algebras, more precisely integrals and cointegrals for them. We use the machinery provided by Frobenius algebras, and we present some basic results about Frobenius, symmetric and Frobe-

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xv

nius augmented algebras (so again we do not assume from the reader a knowledge of these topics). A consequence of the theory we develop is that the antipode of a finite-dimensional quasi-Hopf algebra is bijective. We end the chapter with a section containing a freeness result for quasi-Hopf algebras (for that section the reader is assumed to have some knowledge of module theory). In Chapter 8 we introduce the four categories of Yetter–Drinfeld modules over a quasi-Hopf algebra, prove that they are all braided isomorphic and, when restricted to finite-dimensional objects, rigid. Then we introduce the quantum double of a finitedimensional quasi-Hopf algebra (for the moment, only as a quasi-Hopf algebra), and two particular cases, objects denoted by Dω (H) and Dω (G) (for the latter, G is a finite group and Dω (G) is called the twisted quantum double of G). We end the chapter with some properties and examples of algebras in Yetter–Drinfeld categories. In Chapter 9 we define so-called two-sided two-cosided Hopf modules over a quasi-Hopf algebra, prove that their category is monoidally equivalent to a category of Yetter–Drinfeld modules and use this equivalence to prove some structure theorems for bicomodule algebras and bimodule coalgebras. We characterize Hopf algebras within the category of two-sided two-cosided Hopf modules and use this to define biproduct quasi-Hopf algebras. In Chapter 10 we study quasitriangular quasi-Hopf algebras, QT for short. We show that the antipode of a QT quasi-Hopf algebra is inner, hence bijective. We prove that the quantum double of a finite-dimensional quasi-Hopf algebra is a QT quasiHopf algebra and we characterize the quantum double of a QT finite-dimensional quasi-Hopf algebra as a certain biproduct quasi-Hopf algebra. In Chapter 11 we introduce the concept of factorizable quasi-Hopf algebra, prove that the quantum double of a finite-dimensional quasi-Hopf algebra is factorizable, and describe the quantum double of a factorizable quasi-Hopf algebra. We prove also that any factorizable quasi-Hopf algebra is unimodular (i.e. the spaces of left and right integrals coincide). In Chapter 12 we describe the integrals of a quantum double of a finite-dimensional quasi-Hopf algebra (reproving that it is unimodular). We define the quantum dimension of an object in a braided rigid category, apply this to the category of finitedimensional modules over a quasi-Hopf algebra and compute the quantum dimension of a finite-dimensional quasi-Hopf algebra H and of its quantum double D(H) regarded as left D(H)-modules. We present a trace formula for quasi-Hopf algebras, and then we introduce the concept of involutory quasi-Hopf algebra. In Chapter 13 we introduce the concepts of balanced and ribbon categories, leading to the concept of ribbon quasi-Hopf algebra, which is a QT quasi-Hopf algebra endowed with an element (called ribbon element) satisfying some axioms. In the final two sections, we present two classes of examples of ribbon quasi-Hopf algebras. We have tried to make this book as self-contained as possible, providing as many details (in definitions and proofs) as we could. Owing to lack of space, we had to leave aside some other topics on quasi-Hopf algebras that would have deserved to be presented here (we intentionally left aside Drinfeld’s theory of quantum enveloping

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algebras, because this is very well presented in C. Kassel’s book [127]). In order to help the reader to get an idea of what else can be said about quasi-Hopf algebras, we have included in the bibliography a number of papers on (or related to) quasi-Hopf algebras that we did not cite or use in the book. We have also included some papers or books about Hopf algebras or category theory or other topics that we considered relevant for us or for the subject of the book. This book is an outcome of the long-term scientific cooperation between the Noncommutative Algebra groups from Antwerp, Brussels and Bucharest. We wholeheartedly thank our colleagues from the University of Antwerp, the University of Brusssels, the University of Bucharest and the Institute of Mathematics of the Romanian Academy for the scientific discussions we had with them over the years. Finally, the authors would like to thank Paul Taylor and Bodo Pareigis for sharing their “diagrams” programs, which were intensively used in this book.

1 Monoidal and Braided Categories

In this chapter we introduce the basic categorical language that will be used throughout this book. We define the concepts of monoidal and braided monoidal category and prove that any monoidal category is monoidally equivalent to a strict one.

1.1 Monoidal Categories Recall that a category C consists of the following: • a collection Ob(C ), whose elements are called the objects of C ; if X is an object of C , we write either X ∈ Ob(C ) or simply X ∈ C ; • for every two objects X,Y ∈ Ob(C ), a set HomC (X,Y ), whose elements are denoted by f : X → Y and called the morphisms from X to Y in C ; • for every object X of C , a specified morphism IdX ∈ HomC (X, X), called the identity morphism of X; • for every three objects X,Y, Z of C , a function ◦ : HomC (X,Y ) × HomC (Y, Z) → HomC (X, Z), called the composition function, that maps a pair ( f , g) to ◦( f , g) := g ◦ f , where f : X → Y and g : Y → Z are morphisms in C . These data are subject to the following axioms: (A) Associativity axiom: for all morphisms f : X → Y , g : Y → Z and h : Z → T in C we have (h ◦ g) ◦ f = h ◦ (g ◦ f ). (I) Identity axiom: f ◦ IdX = f = IdY ◦ f , for every morphism f : X → Y in C . f

A morphism f : X → Y in C will also be denoted by X → Y . Note that, when there is no danger of confusion, the composition of two morphisms f : X → Y and g : Y → Z in C will often be written as g f instead of g ◦ f . A morphism f : X → Y in C is called an isomorphism if there exists a morphism g : Y → X in C , called the inverse of f , such that g ◦ f = IdX and f ◦ g = IdY . Note that the inverse is unique.

2

Monoidal and Braided Categories

If X ∈ Ob(C ), we denote EndC (X) := HomC (X, X). A subcategory D of a category C is a collection of some objects and some morphisms of C in such a way that D becomes a category with composition and identities from C . Furthermore, we say that D is a full subcategory when HomD (X,Y ) = HomC (X,Y ), for all X,Y ∈ Ob(D). Recall also that a functor F between two categories C and D consists of: • a map Ob(F) : Ob(C ) → Ob(D); we will denote Ob(F)(X) = F(X), for all X ∈ Ob(C ); • a function HomF (X,Y ) : HomC (X,Y ) → HomD (F(X), F(Y )) for any objects X,Y of C ; we will denote HomF (X,Y )( f ) = F( f ), for any morphism f : X → Y in C . These data are subject to the following axioms: (A1) Identities are preserved by F, that is, F(IdX ) = IdF(X) , for all X ∈ C . (A2) Composition is preserved by F, i.e. F(g ◦ f ) = F(g) ◦ F( f ), for any morphisms f : X → Y and g : Y → Z in C . If F : C → D and G : D → E are two functors then the pointwise composition defines a functor from C to E . It will be denoted by G ◦ F, or simply GF when there is no danger of confusion. If C is a category, there exists a functor IdC : C → C , called the identity functor on C , which is the identity on both objects and morphisms in C . A functor F : C → D is called an isomorphism if there exists a functor G : D → C such that FG = IdD and GF = IdC . Such a functor G, if it exists, is unique and is called the inverse of F. Two categories are isomorphic if there exists an isomorphism between them. If F : C → D is a functor, we call the full image of F (denoted Im(F)) the full subcategory of D whose objects are (F(X))X∈Ob(C ) . A natural transformation μ between two functors F, G : C → D consists of a family of morphisms in D, μ = (μX : F(X) → G(X))X∈Ob(C ) , having the property that G( f ) ◦ μX = μY ◦ F( f ), for any morphism f : X → Y in C . If, moreover, μX is an isomorphism in D, for all X ∈ Ob(C ), then μ is called a natural isomorphism between F and G. Finally, if C , D are categories then C × D is the category whose • objects are pairs (X,Y ), where X is an object of C and Y is an object of D; • morphisms between (X,Y ) and (X  ,Y  ) are pairs ( f , g) consisting of a morphism f : X → X  in C and a morphism g : Y → Y  in D. The identity morphisms and the composition functions in C × D are canonically defined in terms of those of C and D. The new category C × D is called the product of C and D.

1.1 Monoidal Categories

3

We can now introduce the concept of monoidal category, which is roughly a category C endowed with an associative “tensor product” ⊗ : C × C → C , with a unit object 1 and coherence. Rigorously, we have the following: Definition 1.1 A monoidal category consists of a category C endowed with a functor ⊗ : C × C → C (called the tensor product), a distinguished object 1 ∈ C (called the unit object of C ) and natural isomorphisms (X,Y, Z are arbitrary objects of C ) aX,Y,Z : (X ⊗Y ) ⊗ Z → X ⊗ (Y ⊗ Z) (the associativity constraint), lX : 1 ⊗ X → X (the left unit constraint), rX : X ⊗ 1 → X (the right unit constraint), satisfying the so-called Pentagon Axiom and Triangle Axiom, namely for any objects X,Y, Z, T ∈ C the following diagrams are commutative: ((X ⊗Y ) ⊗ Z) ⊗ T

aX⊗Y,Z,T

(X ⊗Y ) ⊗ (Z ⊗ T )

aX,Y,Z⊗T

X ⊗ (Y ⊗ (Z ⊗ T ))

aX,Y,Z ⊗IdT

(1.1.1)

IdX ⊗aY,Z,T

(X ⊗ (Y ⊗ Z)) ⊗ T

X ⊗ ((Y ⊗ Z) ⊗ T ),

aX,Y ⊗Z,T

(X ⊗ 1) ⊗Y

aX,1,Y

X ⊗ (1 ⊗Y )

rX ⊗IdY

(1.1.2)

IdX ⊗lY

X ⊗Y

.

The monoidal category (C , ⊗, 1, a, l, r) is called strict if all the natural isomorphisms a, l and r are defined by identity morphisms in C . f

g

f

g

Remark 1.2 Let X → Y → Z and X  → Y  → Z  be morphisms in C . The fact that ⊗ : C × C → C is a functor implies the following equality: (g ◦ f ) ⊗ (g ◦ f  ) = (g ⊗ g ) ◦ ( f ⊗ f  ) : X ⊗ X  → Z ⊗ Z  . Also, for all objects X,Y of C we have IdX⊗Y = IdX ⊗ IdY . If C is a monoidal category and X,Y, Z, T are objects of C , there are two different ways to go from ((X ⊗ Y ) ⊗ Z) ⊗ T to X ⊗ (Y ⊗ (Z ⊗ T )). The Pentagon Axiom says that these two ways coincide. Then it is automatic that all the other consistency problems of this type are solved as well; see Remark 1.35 below. Proposition 1.3 Let (C , ⊗, 1, a, l, r) be a monoidal category and consider the switch functor τ : C × C → C × C , defined by τ (X,Y ) = (Y, X), for any X,Y ∈ C , and f

g

τ ( f , g) = (g, f ), for any morphisms X → X  and Y → Y  in C . Then C := (C , ⊗ := ⊗ ◦ τ , a, 1, l := r, r := l) is a monoidal category, where aX,Y,Z := a−1 Z,Y,X , for all X,Y, Z ∈ C . In what follows C will be called the reverse monoidal category associated to C .

4 Proof

Monoidal and Braided Categories All the axioms for C to be a monoidal category follow from those of C and f

g

the fact that (g ◦ f )−1 = f −1 ◦ g−1 , for any isomorphisms X → Y → Z in C . For example, the Pentagon Axiom for C reduces to the commutativity of the diagram T ⊗ (Z ⊗ (Y ⊗ X))

a−1 T,Z,Y ⊗X

(T ⊗ Z) ⊗ (Y ⊗ X)

a−1 T ⊗Z,Y,X

((T ⊗ Z) ⊗Y ) ⊗ X

IdT ⊗a−1 Z,Y,X

a−1 T,Z,Y ⊗IdX

T ⊗ ((Z ⊗Y ) ⊗ X)

(T ⊗ (Z ⊗Y )) ⊗ X,

a−1 T,Z⊗Y,X

which holds because of (1.1.1). Similarly, for a as above, l = r and r = l, the Triangle Axiom is satisfied because of (1.1.2). Remark 1.4 Apart from C , to a monoidal category C we can associate a new one that will be denoted by C opp and called the opposite category associated to C . As a category, C opp has the same objects as C and HomC opp (X,Y ) = HomC (Y, X), for any objects X,Y of C . If f ∈ HomC opp (X,Y ) and g ∈ HomC opp (Y, Z) then the composition ◦opp between g and f in C opp is g ◦opp f = f ◦ g, the latest composition being in C . If C is monoidal then so is C opp , with the monoidal structure induced by that of C , namely C opp = (C opp , ⊗, 1, a−1 , l −1 , r−1 ). The Triangle Axiom in Definition 1.1 gives the compatibility between the left and right unit constraints. There also exist other compatibilities of this type: Proposition 1.5

Let (C , ⊗, 1, a, l, r) be a monoidal category. Then the diagrams (X ⊗Y ) ⊗ 1

aX,Y,1

X ⊗ (Y ⊗ 1) IdX ⊗rY

rX⊗Y

X ⊗Y and (1 ⊗ X) ⊗Y

a1,X,Y

1 ⊗ (X ⊗Y ) lX⊗Y

lX ⊗IdY

X ⊗Y are commutative, for any objects X,Y ∈ C . Moreover, we have that l1 = r1 . Proof

Since a is natural, the following diagrams are commutative: (X ⊗ (Y ⊗ 1)) ⊗ T

aX,Y ⊗1,T

(IdX ⊗rY )⊗IdT

(X ⊗Y ) ⊗ T

X ⊗ ((Y ⊗ 1) ⊗ T ) IdX ⊗(rY ⊗IdT )

aX,Y,T

X ⊗ (Y ⊗ T ),

(1.1.3)

1.1 Monoidal Categories (X ⊗Y ) ⊗ (1 ⊗ T )

aX,Y,1⊗T

5

X ⊗ (Y ⊗ (1 ⊗ T ))

IdX⊗Y ⊗lT

(1.1.4)

IdX ⊗(IdY ⊗lT ) aX,Y,T

(X ⊗Y ) ⊗ T

X ⊗ (Y ⊗ T ),

for all X,Y, T ∈ C . Then we have: aX,Y,T ((IdX ⊗ rY ) ⊗ IdT )(aX,Y,1 ⊗ IdT ) (1.1.3)

= (IdX ⊗ (rY ⊗ IdT ))aX,Y ⊗1,T (aX,Y,1 ⊗ IdT )

(1.1.2)

= (IdX ⊗ (IdY ⊗ lT ))(IdX ⊗ aY,1,T )aX,Y ⊗1,T (aX,Y,1 ⊗ IdT )

(1.1.1)

= (IdX ⊗ (IdY ⊗ lT ))aX,Y,1⊗T aX⊗Y,1,T

(1.1.4)

= aX,Y,T (IdX⊗Y ⊗ lT )aX⊗Y,1,T

(1.1.2)

= aX,Y,T (rX⊗Y ⊗ IdT ).

Using that aX,Y,T is an isomorphism we get, for all X,Y, T ∈ C , (IdX ⊗ rY )aX,Y,1 ⊗ IdT = rX⊗Y ⊗ IdT .

(1.1.5)

Now, by the naturality of r the diagrams ((X ⊗Y ) ⊗ 1) ⊗ 1 (IdX ⊗rY )aX,Y,1 ⊗Id1

r(X⊗Y )⊗1

(X ⊗Y ) ⊗ 1

(IdX ⊗rY )aX,Y,1

rX⊗Y ⊗Id1

(X ⊗Y ) ⊗ 1

rX⊗Y

X ⊗Y

rX⊗Y

are commutative, so by (1.1.5) (with T = 1) we obtain that (IdX ⊗ rY )aX,Y,1 r(X⊗Y )⊗1 = rX⊗Y r(X⊗Y )⊗1 , and therefore the first triangle in the proposition is commutative because r(X⊗Y )⊗1 is an isomorphism. If we express the commutativity of the first triangle for C , the reverse monoidal category associated to C as in Proposition 1.3, we obtain the commutativity of the second triangle in the proposition. So it remains to prove l1 = r1 . For this, note that the naturality of r implies that (X ⊗ 1) ⊗ 1

rX⊗1

X ⊗1

rX ⊗Id1

X ⊗1

rX rX

X

is commutative, for any X ∈ C . Since rX is an isomorphism we deduce that rX⊗1 = rX ⊗ Id1 .

(1.1.6)

6

Monoidal and Braided Categories

Note that, by applying equation (1.1.6) in C , we obtain in C the relation l1⊗X = Id1 ⊗ lX .

(1.1.7)

Now, by (1.1.2) we have r1⊗1 = r1 ⊗ Id1 = (Id1 ⊗ l1 )a1,1,1 , and by the commutativity of the first triangle in the proposition we get r1⊗1 = (Id1 ⊗ r1 )a1,1,1 . Since a1,1,1 is an isomorphism we obtain Id1 ⊗ l1 = Id1 ⊗ r1 . By the naturality of l the diagrams l1⊗1

1 ⊗ (1 ⊗ 1) Id1 ⊗l1

Id1 ⊗r1

1⊗1

1⊗1 r1

l1

1

l1

are commutative. Using that l1⊗1 is an isomorphism and Id1 ⊗ l1 = Id1 ⊗ r1 we get l1 = r1 , and this finishes the proof. Proposition 1.6 Let 1 be the unit object of a monoidal category C . Then EndC (1) is a commutative monoid, and if we identify 1 ⊗ 1 with 1 via l1 = r1 then the tensor product of two morphisms in EndC (1) coincides with their composition. Proof It can be easily checked that the composition endows EndC (1) with a monoid structure, the unit element being Id1 . Thus, we only need to show that f ⊗ g = r1−1 ◦ ( f ◦ g) ◦ r1 = r1−1 ◦ (g ◦ f ) ◦ r1 , for all f , g ∈ EndC (1). To this end, note that the naturality of l and r imply the commutativity of the following diagrams: 1⊗1

l1

Id1 ⊗g

1⊗1

and

1 g

l1

1

1⊗1

r1

1

f ⊗Id1

1⊗1

f r1

1.

Now, since l1 and r1 are isomorphisms we obtain Id1 ⊗ g = l1−1 ◦ g ◦ l1 and f ⊗ Id1 = r1−1 ◦ f ◦ r1 . Since l1 = r1 it follows that f ⊗ g = ( f ⊗ Id1 ) ◦ (Id1 ⊗ g) = r1−1 ◦ ( f ◦ g) ◦ r1 , f ⊗ g = (Id1 ⊗ g) ◦ ( f ⊗ Id1 ) = r1−1 ◦ (g ◦ f ) ◦ r1 . Thus, we proved the equalities f ⊗ g = r1−1 ◦ ( f ◦ g) ◦ r1 = r1−1 ◦ (g ◦ f ) ◦ r1 . Note that by interchanging f and g in the above relation we also obtain f ⊗ g = g ⊗ f , for all f , g ∈ EndC (1).

1.2 Examples of Monoidal Categories

7

1.2 Examples of Monoidal Categories 1.2.1 The Category of Sets We denote the category of sets by Set, and by {∗} a fixed singleton, that is, a fixed set with one element. Furthermore, by × we denote the direct product of sets, that is, for any sets X and Y , X × Y is the set of ordered pairs (x, y) with x ∈ X and f

f

y ∈ Y , and by f × f  the direct product of two functions X → Y and X  → Y  , that is, f × f  : X × X  → Y ×Y  is defined by f × f  (x, x ) = ( f (x), f  (x )), for all x ∈ X and x ∈ X  . If follows that × defines a functor from Set × Set to Set. For any sets X,Y and Z, we have canonical isomorphisms, defined for all x ∈ X, y ∈ Y and z ∈ Z, by aX,Y,Z : (X ×Y ) × Z → X × (Y × Z), aX,Y,Z ((x, y), z) = (x, (y, z)), lX : {∗} × X → X, lX (∗, x) = x, rX : X × {∗} → X, rX (x, ∗) = x. The proof of the next result is straightforward, so it is left to the reader. Proposition 1.7 With notation as above, (Set, ×, {∗}, a, l, r) is a monoidal category.

1.2.2 The Category of Vector Spaces One of the most important examples of a monoidal category for what follows is the category k M of vector spaces over a base field k. The tensor product in k M is the usual tensor product of vector spaces, the unit object 1 is the field k itself, and the associativity and unit constraints are the natural isomorphisms (for all X,Y, Z ∈ k M ) aX,Y,Z : (X ⊗Y ) ⊗ Z → X ⊗ (Y ⊗ Z), aX,Y,Z ((x ⊗ y) ⊗ z) = x ⊗ (y ⊗ z), lX : k ⊗ X → X, lX (λ ⊗ x) = λ x, rX : X ⊗ k → X, rX (x ⊗ λ ) = λ x, for all λ ∈ k, x ∈ X, y ∈ Y and z ∈ Z. The above statement remains valid if we consider k a commutative ring and take k M equal to the category of modules over k.

1.2.3 The Category of Bimodules We present now the noncommutative version of Subsection 1.2.2. Let k be a field (or, more generally, a commutative ring) and R a k-algebra. Denote by R MR the category of R-bimodules and R-bimodule maps. Then R MR is monoidal with the following structure: • The tensor product functor is ⊗R : R MR × R MR → R MR defined as follows. On objects, we have ⊗R (M, N) := M ⊗R N, the tensor product over R between M and N. It becomes an R-bimodule in the canonical way: r · (m ⊗R n) · r = rm ⊗R nr ,

8

Monoidal and Braided Categories

for all r, r ∈ R, m ∈ M and n ∈ N. If f : M → N, g : P → Q are morphisms in R MR then the map ( f ⊗R g)(m ⊗R p) = f (m) ⊗R g(p), for all m ∈ M and p ∈ P, is a morphism in R MR . • The unit is R, considered as an R-bimodule via its multiplication. • The associativity and unit constraints are defined as follows: aX,Y,Z : (X ⊗R Y ) ⊗R Z → X ⊗R (Y ⊗R Z), aX,Y,Z ((x ⊗R y) ⊗R z) = x ⊗R (y ⊗R z), lX : R ⊗R X → X, lX (r ⊗R x) = rx, rX : X ⊗R R → X, rX (x ⊗R r) = xr, for all r ∈ R, x ∈ X, y ∈ Y and z ∈ Z. We leave it to the reader to check that this defines a monoidal structure on R MR . Note that if R = k then R MR coincides with k M as a monoidal category.

1.2.4 The Category of G-graded Vector Spaces Throughout this subsection G is a group written multiplicatively and with neutral element e, k is a field and k∗ = k\{0}. Definition 1.8 A G-graded vector space over k is a k-vector space V which decom poses into a direct sum of the form V = g∈G Vg , where each Vg is a k-vector space. For a given g ∈ G the elements of Vg are called homogeneous elements of degree g. If v ∈ V is a homogeneous element then we denote the degree of v by | v |∈ G.  Let W = g∈G Wg be another G-graded vector space. Then a k-linear map f : V → W is called a G-graded morphism if it preserves the degree of homogeneous elements, that is, f (Vg ) ⊆ Wg , for all g ∈ G. VectG denotes the category of G-graded vector spaces and G-graded morphisms. 



If V = g∈G Vg and W = g∈G Wg are G-graded vector spaces then V ⊗W is also a G-graded vector space with the grading defined by (V ⊗W )g :=

 σ τ =g

Vσ ⊗Wτ ,

(1.2.1)

for all g ∈ G. Indeed, it is an elementary fact that in k M the tensor product commutes with arbitrary direct sums. Hence ⎞     ⎛      (V ⊗W )g = Vσ ⊗Wτ = Vg ⊗ ⎝ Wg ⎠ = V ⊗W, g∈G

g∈G

σ τ =g

g∈G

g ∈G

as required. Furthermore, if f : V → V  and g : W → W  are morphisms in VectG then f ⊗ g becomes a morphism in VectG . Thus, the tensor product ⊗ of the category of k-vector spaces induces a tensor product on VectG . Also, k can be viewed as a G-graded vector space via the trivial grading, that is, ke = k and kg = 0, for all G g = e. In this way the left and right unit constraints l and r of k M become graded morphisms, that is, morphisms in VectG .

1.2 Examples of Monoidal Categories

9

Our next aim is to describe the monoidal structures of VectG , somehow induced by the monoidal structure of k M . To this end we first need some group cohomology, with a particular emphasis on the third cohomology group of a group G with coefficients in k∗ , the group of units of a field k, viewed trivially as a Z[G]-module. Here Z is the ring of integers and Z[G] is the group algebra associated to G over the commutative ring Z. More generally, for G a (multiplicative) group with neutral element e and R a commutative ring we denote by R[G] the free R-module with basis {g | g ∈ G}, so any element of R[G] has the form ∑g∈G αg g with (αg )g∈G a family of elements of R having only a finite number of non-zero elements. Then R[G] with multiplication defined by (αh h)(βg g) = αh βg hg, extended by linearity, and unit e, is called the group algebra associated to G over R. It is easy to see that R[G] is a unital associative R-algebra, and that R[G] is a G-graded vector space with grading defined by R[G]g = Rg, for all g ∈ G. Coming back to the survey on group cohomology, let K n (G, k∗ ) be the set of maps from Gn to k∗ . Then one can easily see that K n (G, k∗ ) is a group under pointwise multiplication. There exist maps Δn : K n (G, k∗ ) → K n+1 (G, k∗ ), which for n ∈ {2, 3} are given by the formulas Δ2 (g)(x, y, z) = g(y, z)g(xy, z)−1 g(x, yz)g(x, y)−1 , Δ3 ( f )(x, y, z,t) = f (y, z,t) f (xy, z,t)−1 f (x, yz,t) f (x, y, zt)−1 f (x, y, z). It is known that Bn (G, k∗ ) := ImΔn−1 ⊆ Z n (G, k∗ ) := Ker(Δn ). The nth cohomology group is defined as H n (G, k∗ ) = Z n (G, k∗ )/Bn (G, k∗ ), and two elements of H n (G, k∗ ) are called cohomologous if they lie in the same equivalence class. The elements of Z 3 (G, k∗ ) are called 3-cocycles, and the elements of B3 (G, k∗ ) are called 3-coboundaries. We have the following. Definition 1.9 A 3-cocycle on G with coefficients in k∗ is a map φ : G×G×G → k∗ such that

φ (y, z,t)φ (x, yz,t)φ (x, y, z) = φ (x, y, zt)φ (xy, z,t),

(1.2.2)

for all x, y, z,t ∈ G. A 3-cocycle φ is called normalized if φ (x, e, y) = 1, for all x, y ∈ G. Remarks 1.10 (1) If φ is a normalized 3-cocycle, then φ (e, y, z) = φ (x, y, e) = 1, for all x, y, z ∈ G. Indeed, by taking z = e in (1.2.2), we find that φ (x, y, e) = 1. By taking y = e, we find that φ (e, z,t) = 1. (2) A coboundary Δ2 (g) is normalized if and only if g(e, x) = g(z, e), for all x, z ∈ G. As we shall see, H 3 (G, k∗ ) is completely determined by the normalized 3-cocycles. Lemma 1.11

Every 3-cocycle φ is cohomologous to a normalized 3-cocycle.

Proof By taking y = z = e in (1.2.2) we find φ (x, e,t) = φ (e, e,t)φ (x, e, e). In particular, by taking x = t = e, it follows that φ (e, e, e) = 1. Then we consider the map

10

Monoidal and Braided Categories

f : G × G → k∗ , f (x, y) = φ (e, e, y)−1 φ (x, e, e), and compute: Δ2 ( f )(x, e, y) = f (e, y) f (x, y)−1 f (x, y) f (x, e)−1 = φ (e, e, y)−1 φ (e, e, e)φ (e, e, e)φ (x, e, e)−1 = φ (x, e, y)−1 . It then follows that φ Δ2 ( f ) is normalized. Let B3n (G, k∗ ) and Zn3 (G, k∗ ) be the subgroups of B3 (G, k∗ ) and Z 3 (G, k∗ ) consisting of normalized elements. We have a well-defined group morphism Zn3 (G, k∗ )/B3n (G, k∗ ) φˆ → φ ∈ Z 3 (G, k∗ )/B3 (G, k∗ ) which is surjective by Lemma 1.11. One can see that it is also injective, and therefore H 3 (G, k∗ ) = Zn3 (G, k∗ )/B3n (G, k∗ ). Example 1.12 If k is a field of characteristic different from 2 and C2 is the cyclic group of order 2 then H 3 (C2 , k∗ ) = C2 . If char(k) = 2, then H 3 (C2 , k∗ ) = {e}. Proof Write C2 = {1, σ }. A straightforward computation shows that all normalized coboundaries are trivial. If φ is a normalized 3-cocycle, then the only value of φ (x, y, z) that is possibly different from 1 is φ (σ , σ , σ ). By substituting x = y = z = t = σ in (1.2.2), we find that φ (σ , σ , σ ) = ±1. If φ (σ , σ , σ ) = 1, then φ is trivial. The only possibly non-trivial normalized 3-cocycle is given by φ (σ , σ , σ ) = −1. Consequently, if char(k) = 2 then any normalized 3-cocycle is trivial, and so 3 H (C2 , k∗ ) = {e}. One can now provide the connection between H 3 (G, k∗ ) and some monoidal structures on VectG . Proposition 1.13 Let G be a group, k a field and VectG the category of G-graded kvector spaces. There is a bijective correspondence between the monoidal structures on VectG of the form (VectG , ⊗, a, k, l, r) and the set of normalized 3-cocycles on G, where ⊗ is defined by (1.2.1) and l, r are the constraints of the monoidal category k M as defined in Subsection 1.2.2. More precisely, any associativity constraint a on VectG is completely determined by a normalized 3-cocycle φ ∈ H 3 (G, k∗ ), in the sense that, for any U,V,W ∈ VectG and any homogeneous elements u ∈ U, v ∈ V and w ∈ W , aU,V,W is the k-linear map aU,V,W ((u ⊗ v) ⊗ w) = φ (| u |, | v |, | w |)u ⊗ (v ⊗ w). G We denote by VectG φ the category Vect with monoidal structure determined by φ .

Proof If φ is a normalized 3-cocycle on G then, clearly, the morphism aU,V,W defined above preserves the degree of homogeneous elements, so it is a morphism in VectG . The Pentagon Axiom (1.1.1) follows now from (1.2.2), while the Triangle Axiom in (1.1.2) follows because φ is normalized. The details are straightforward, so they are left to the reader.

1.2 Examples of Monoidal Categories

11

Conversely, suppose that VectG has a monoidal structure of the form mentioned in the statement. For any U ∈ VectG and any k-linear map f : U → k define θ f : U → k[G] by θ f (u) = ∑x∈G f (ux )x, for all u ∈ U, where u = ∑x∈G ux is the decomposition of u in homogeneous components. Obviously, θ f is a graded morphism. Likewise we define θg and θh , for any V,W ∈ VectG and all g : V → k and h : W → k. Now take ε : k[G] → k defined by ε (g) = 1, for all g ∈ G, extended by linearity. A simple computation shows that (ε ⊗ ε )(θ f ⊗ θg ) = f ⊗ g. We are now able to show that an associativity constraint a of VectG is completely determined by ak[G],k[G],k[G] . More precisely, for a, an associativity constraint on VectG , define φ (x, y, z) := (ε ⊗ (ε ⊗ ε ))ak[G],k[G],k[G] ((x ⊗ y) ⊗ z), for all x, y, z ∈ G. By the naturality of a, for any U,V,W and f , g, h as above, we have (θ f ⊗ (θg ⊗ θh ))aU,V,W ((u ⊗ v) ⊗ w) = ak[G],k[G],k[G] ((θ f ⊗ θg ) ⊗ θh )((u ⊗ v) ⊗ w), where u ∈ U, v ∈ V and w ∈ W are arbitrary elements. Assume that u, v, w are homogeneous of degrees x, y and z, respectively, and write aU,V,W ((u ⊗ v) ⊗ w) = ∑ ui ⊗ (vi ⊗ wi ), i

for some homogeneous elements ui ∈ U, vi ∈ V and wi ∈ W . We obtain that

∑ f (ui )g(vi )h(wi ) | ui | ⊗ (| vi | ⊗ | wi |) = f (u)g(v)h(w)ak[G],k[G],k[G] ((x ⊗ y) ⊗ z). i

By applying ε ⊗ (ε ⊗ ε ) on both sides of the above equality we obtain

∑ f (ui )g(vi )h(wi ) = φ (x, y, z) f (u)g(v)h(w). i

Since f , g, h are arbitrary we get aU,V,W ((u⊗v)⊗w) = φ (| u |, | v |, | w |)u⊗(v⊗w), as stated. Note that the bijectivity of the associativity constraint implies that φ (x, y, z) = 0, for all x, y, z ∈ G. It is clear now that a satisfies the Pentagon and Triangle Axioms if and only if φ is a normalized 3-cocycle on G, and so we are done. Remark 1.14 If φ is the trivial 3-cocycle on G then the monoidal structure on VectG is entirely induced by the monoidal structure of k M described in Subsection 1.2.2. In this case VectG is strict monoidal and the grading is relevant only in the definition of the tensor product of VectG . A non-strict monoidal structure on VectG when G is cyclic of order 2 can be obtained by considering Example 1.12. Note that in this case we get the so-called category of super vector spaces. Example 1.15 (Super vector spaces) Let k be a field, Z2 = {0, 1} the cyclic group of order 2, this time written additively, and consider the category VectZ2 of Z2 -graded k-vector spaces. It can be identified with the category whose objects are pairs V = (V0 ,V1 ) of k-vector spaces. A morphism from (V0 ,V1 ) to (V0 ,V1 ) in VectZ2 is a pair ( f0 , f1 ) of k-linear maps with fi : Vi → Vi , i ∈ {0, 1}.

12

Monoidal and Braided Categories Then VectZ2 is monoidal with tensor product defined by

 (V0 ,V1 ) ⊗ (W0 ,W1 ) = (V0 ⊗W0 ) ⊕ (V1 ⊗W1 ), (V0 ⊗W1 ) ⊕ (V1 ⊗W0 ) ,

 ( f0 , f1 ) ⊗ (g0 , g1 ) = ( f0 ⊗ g0 ) ⊕ ( f1 ⊗ g1 ), ( f0 ⊗ g1 ) ⊕ ( f1 ⊗ g0 ) .

The associativity constraint is (aV,W,Z )V,W,Z∈VectZ2 , defined on homogeneous elements v ∈ V , w ∈ W and z ∈ Z by −v ⊗ (w ⊗ z) for | v |, | w |, | z | all odd, aV,W,Z ((v ⊗ w) ⊗ z) = v ⊗ (w ⊗ z) otherwise. The unit object is (k, 0) and the left and right unit constrains of VectZ2 are defined by the left and right unit constraints of k M . As we have already mentioned, VectZ2 with the monoidal structure described above is called the category of super vector spaces. Proof In Proposition 1.13 take G = Z2 and φ the (possibly) non-trivial 3-cocycle constructed in Example 1.12, that is, φ is 1 everywhere except on (1, 1, 1) where its value is −1. More generally, examples of monoidal structures on VectZn are given by the following family of normalized 3-cocycles on Zn . Example 1.16 Let Zn = {0, 1, . . . , n − 1} be the cyclic group of order n ≥ 2 written additively and q be an nth root of unity in k. Then 1 if y + z < n φq (x, y, z) = , ∀ x, y, z ∈ {0, 1, . . . , n − 1}, qx if y + z ≥ n defines a normalized 3-cocycle on Zn . Moreover, φq is a coboundary if and only if q = 1. Consequently, φq and φq are cohomologous if and only if q = q , whenever q is another nth root of unity in k. Proof If y = 0 then clearly y + z < n, and so φq (x, 0, z) = 1. Clearly we have φq (0, y, z) = φq (x, y, 0) = 1, for all x, y, z ∈ {0, 1, . . . , n − 1}. Thus we only have to show that, for all x, y, z,t ∈ {1, 2, . . . , n − 1},

φq (y, z,t)φq (x, y + z,t)φq (x, y, z) = φq (x, y, z + t)φq (x + y, z,t).

(1.2.3)

To see this, consider the following situations: 1. If y + z + t < n then z + t < n and y + z < n, and so (1.2.3) reduces to 1 = 1. 2. If y + z + t ≥ n then there are the following possibilities: (i) If z + t < n then (1.2.3) reduces to φq (x, y + z,t)φq (x, y, z) = qx . It is clearly satisfied when y + z < n. In the case when y + z ≥ n we have y + z = n + u, for some u ∈ {0, . . . , n−2}, and we will prove that φq (x, u,t) = 1. Indeed, u+t = y+z+t −n < y < n and this implies the desired equality. (ii) If z +t ≥ n we write z +t = n + v, for some v ∈ {0, . . . , n − 2}. Then (1.2.3) reduces to φq (x, y + z,t)φq (x, y, z) = qx φq (x, y, v). To prove this, consider the following cases:

1.2 Examples of Monoidal Categories

13

(ii.1) If y + z < n then we have to prove that φq (x, y, v) = 1. This follows from y + v = y + z + t − n < t < n and from the definition of φq . (ii.2) If y+z ≥ n, there exists w ∈ {0, . . . , n−2} such that y+z = w+n, and we will prove that φq (x, w,t) = φq (x, y, v). To this end observe that w +t < n ⇔ y + z +t < 2n and, similarly, y + v < n ⇔ y + z + t < 2n, which implies that w + t < n ⇔ y + v < n. Thus, we have proved that φq is a normalized 3-cocycle on Zn . Obviously, it is a coboundary if q = 1. Conversely, if φq is a coboundary then there exists g ∈ K 2 (Zn , k∗ ) such that g(0, x) = g(z, 0) and 1 if y + z < n −1 −1 g(y, z)g(x + y, z) g(x, y + z)g(x, y) = qx if y + z ≥ n, for all x, y, z ∈ {0, 1, . . . , n − 1}. By taking in the above relation x = 1, y = n − 1 and z = 1 we get that q = g(n − 1, 1)g(1, n − 1)−1 . If we take x = 1, y = n − k and z = 1, where k ∈ {2, . . . , n − 1}, we obtain g(n − k + 1, 1)g(1, n − k + 1)−1 = g(n − k, 1)g(1, n − k)−1 . We conclude that q = g(n − 1, 1)g(1, n − 1)−1 = g(n − 2, 1)g(1, n − 2)−1 = · · · = g(1, 1)g(1, 1)−1 = 1, as needed. Finally, since φq φq = φqq , for any q and q nth roots of unity in k, it follows that φq and φq are cohomologous if and only if φqq−1 is a coboundary, if and only if q = q . This finishes the proof.

1.2.5 The Category of Endo-functors Let C , D be arbitrary categories and [C , D] the category whose objects are functors F : C → D. A morphism in [C , D] between two functors F, G : C → D is a natural transformation μ : F → G. The composition ν ◦ μ of two natural transformations μ ν F → G → H is the “vertical” composition of ν and μ , that is, (ν ◦ μ )X := νX ◦ μX , for any object X of C . One can easily see that this composition is indeed a natural transformation, and that it is associative. Moreover, for any natural transformation μ : F → G we have 1G ◦ μ = μ = μ ◦ 1F , where, in general, for a functor T : C → D we denote by 1T : T → T the identity natural transformation of T , (1T )X = IdT (X) , for any object X of C . Thus [C , D] is a category. Consider now C = D. The composition of functors defines a tensor product ⊗ on [C , C ] (which is called the category of endo-functors on C ), that is, F ⊗ G := F ◦ G, and for μ : F → G and μ  : F  → G natural transformations, μ ⊗ μ  : F ⊗ F  = F ◦ F  → G ⊗ G = G ◦ G is the natural transformation defined, for any X ∈ C , by the diagram FF  (X) μF  (X)

GF  (X)

F(μX )

( μ ⊗ μ  )X G(μX )

FG (X) μG (X)

GG (X) .

14

Monoidal and Braided Categories

Note that the above diagram is commutative because of the naturality of μ . The natural transformation μ ⊗ μ  is called the “horizontal” composition or the Godement product of the natural transformations μ and μ  . It can be defined for any given functors and natural transformations F ↓ μ - C G

B

F ↓ μ - D. G

To show that μ ⊗ μ  is indeed a natural transformation consider the diagram FF  (X)

F(μX )

FF  ( f )

FF  (Y )

FG (X)

μG (X)

FG ( f ) F(μY )

GG (X)

GG ( f )

FG (Y )

μG (Y )

GG (Y ) ,

f

where X → Y is an arbitrary morphism in C . The left-handed square is commutative because μ  is natural and F is a functor, the right-handed square also commutes because μ is natural and G ( f ) : G (X) → G (Y ) is a morphism in C , and horizontally the compositions are (μ ⊗ μ  )X and (μ ⊗ μ  )Y , respectively. These facts show that μ ⊗ μ  is natural, as claimed. Furthermore, the horizontal composition can be rewritten, using vertical composition, as

μ ⊗ μ  = (1G ⊗ μ  ) ◦ (μ ⊗ 1F  ) = (μ ⊗ 1G ) ◦ (1F ⊗ μ  ). To see this, note that for any object T of [C , C ] we have (1T ⊗ μ )X = T (μX ) and (μ ⊗ 1T )X = μT (X) , for all X ∈ C . The identity for the composition ⊗ is 1C : IdC → IdC , the identity natural transformation of the identity functor IdC to itself. This means that μ ⊗ 1C = μ = 1C ⊗ μ , for any morphism μ : F → G in [C , C ]. Proposition 1.17

([C , C ], ⊗, IdC ) is a strict monoidal category.

Proof Clearly, (F ⊗ G) ⊗ H = F ⊗ (G ⊗ H) and F ⊗ IdC = F = IdC ⊗ F, for any functors F, G, H ∈ [C , C ]. Also, it can be easily checked that the composition ⊗ is associative. Thus, we only need to prove that for any given functors and natural transformations G F C

G

↓ μ ↓ ν -

G

C

F F 

↓ μ↓ ν-

C,

we have (ν ◦ μ ) ⊗ (ν  ◦ μ  ) = (ν ⊗ ν  ) ◦ (μ ⊗ μ  ). Indeed, we compute:

 (ν ◦ μ ) ⊗ (ν  ◦ μ  ) X = F  ((ν  ◦ μ  )X )(ν ◦ μ )G(X)

1.2 Examples of Monoidal Categories

15

= F  (νX )F  (μX )νG(X) μG(X) = F  (νX )νG (X) F  (μX )μG(X) = (ν ⊗ ν  )X ( μ ⊗ μ  )X

 = (ν ⊗ ν  ) ◦ ( μ ⊗ μ  ) X , for any object X of C , and this finishes the proof.

1.2.6 A Strict Category Associated to a Monoidal Category Let (C , ⊗, 1, a, l, r) be a monoidal category. The aim of this subsection is to associate to C a strict monoidal category, denoted by e(C ). The objects of e(C ) are pairs (F, ρ F ), where F : C → C is a functor and ρ F is a family of natural isomorphisms μ F : F(X) ⊗ Y → F(X ⊗ Y ), indexed by (X,Y ) ∈ C × C . A morphism (F, ρ F ) → ρX,Y (G, ρ G ) in e(C ) consists of a natural transformation μ : F → G such that, for any objects X,Y of C , the diagram below is commutative: F ρX,Y

F(X) ⊗Y

F(X ⊗Y ) μX⊗Y

μX ⊗IdY

G(X) ⊗Y

(1.2.4)

G(X ⊗Y ).

G ρX,Y

μ

μ

If (F, ρ F ) → (G, ρ G ) → (H, ρ H ) are morphisms in e(C ) then μ  ◦ μ : (F, ρ F ) → (H, ρ H ) is clearly a morphism in e(C ). Moreover, the identity morphism associated to the object (F, ρ F ) is 1F : F → F, the identity natural transformation associated to F. One can easily see that 1F is a morphism in e(C ). The category e(C ) is strict monoidal. The tensor product is given by (F, ρ F ) ⊗ (G, ρ G ) = (FG, ρ FG ), where, for all X,Y ∈ C , FG ρX,Y : FG(X) ⊗Y

F ρG(X),Y

F(G(X) ⊗Y )

G ) F(ρX,Y

FG(X ⊗Y ),

and the unit object is (IdC , ρ IdC = (IdX⊗Y )X,Y ∈C ). To see this, note that (FG)H

ρX,Y

H FG H G F = FG(ρX,Y )ρH(X),Y = FG(ρX,Y )F(ρH(X),Y )ρGH(X),Y F(GH)

H G F GH F = F(G(ρX,Y )ρH(X),Y )ρGH(X),Y = F(ρX,Y )ρGH(X),Y = ρX,Y

,

for any (F, ρ F ), (G, ρ G ), (H, ρ H ) ∈ e(C ) and X,Y ∈ C . μ



μ



If (F, ρ F ) → (G, ρ G ), (F  , ρ F ) → (G , ρ G ) are morphisms in e(C ) then μ ⊗ μ  , the horizontal composition defined in Subsection 1.2.5, is a morphism in e(C ) from

16

Monoidal and Braided Categories 



(FF  , ρ FF ) to (GG , ρ GG ). To see this, we use the commutativity of the diagrams F(F  (X) ⊗Y )



F(μX ⊗IdY )

F(G (X) ⊗Y )

μF  (X)⊗Y

G ) F(ρX,Y

μG (X)⊗Y

G(F  (X) ⊗Y )

μG (X⊗Y )

G(G (X) ⊗Y )

G(μX ⊗IdY )

G(μX )⊗IdY

G(F  (X)) ⊗Y

FG (X ⊗Y )



G ) G(ρX,Y

GG (X ⊗Y ),

G(G (X)) ⊗Y

ρFG (X),Y

ρGG (X),Y

G(F  (X) ⊗Y )

G(G (X) ⊗Y ),

G(μX ⊗IdY )

G are natural transformations. This fact allows which follow by using that μ and ρ−,Y to compute: 



FF F = (μ ⊗ μ  )X⊗Y F(ρX,Y )ρFF (X),Y (μ ⊗ μ  )X⊗Y ρX,Y 

 F = μG (X⊗Y ) F(μX⊗Y ρX,Y )ρFF (X),Y 

G = μG (X⊗Y ) F(ρX,Y )F(μX ⊗ IdY )ρFF (X),Y 

G = G(ρX,Y )μG (X)⊗Y F(μX ⊗ IdY )ρFF (X),Y 

G = G(ρX,Y (μX ⊗ IdY ))μF  (X)⊗Y ρFF (X),Y 

G = G(ρX,Y )G(μX ⊗ IdY )ρFG (X),Y (μF  (X) ⊗ IdY ) 

G )ρGG (X),Y (G(μX ) ⊗ IdY )(μF  (X) ⊗ IdY ) = G(ρX,Y 

G )ρGG (X),Y ((μ ⊗ μ  )X ⊗ IdY ) = G(ρX,Y 

GG ((μ ⊗ μ  )X ⊗ IdY ), = ρX,Y

as needed.

1.3 Monoidal Functors The aim of this section is to introduce and study functors between monoidal categories that behave well with respect to their monoidal structures. This will allow us to extend, for instance, the notion of equivalent categories to the monoidal setting. Definition 1.18 A functor F : C → D is called an equivalence of categories if there exists a functor G : D → C such that FG is naturally isomorphic to IdD and GF is naturally isomorphic to IdC . We say that C and D are equivalent categories if there exists a functor F : C → D which is an equivalence of categories.

1.3 Monoidal Functors

17

We have a criterion for a functor F : C → D to be an equivalence of categories. Proposition 1.19

F : C → D is an equivalence of categories if and only if

• F is essentially surjective, that is, for any object U of D there exists an object X of C such that U ∼ = F(X); • F is fully faithful, that is, for any X,Y ∈ C the following map is bijective: HomC (X,Y ) f → F( f ) ∈ HomD (F(X), F(Y )). Proof Suppose that F : C → D defines an equivalence of categories and let G : D → C be a functor such that FG is naturally isomorphic to IdD , say via μ : IdD → FG, and GF is naturally isomorphic to IdC , say via ν : GF → IdC . Then, for any object U of D, μU : U → FG(U) defines an isomorphism in D, and so F is essentially surjective. If f , f  ∈ HomC (X,Y ) such that F( f ) = F( f  ) then the naturality of ν implies f νX = νY GF( f ) = νY GF( f  ) = f  νX , and therefore f = f  . Similarly, by using the naturality of μ we get that the map HomD (U,V ) g → G(g) ∈ HomC (G(U), G(V )) is injective, for any objects U,V of D. Now let g ∈ HomD (F(X), F(Y )). By the naturality of ν , for f = νY G(g)νX−1 : X → Y it follows that

νY GF( f ) = f νX = νY G(g). Thus GF( f ) = G(g), and so F( f ) = g. This shows that F is fully faithful, as needed. Conversely, assume that F is essentially surjective and fully faithful. Then, for any object U of D we fix an object G(U) of C and an isomorphism μU : U → FG(U) in g D. Moreover, if U → V is a morphism in D then μV gμU−1 ∈ HomD (FG(U), FG(V )), and since F is fully faithful there is a unique morphism G(g) : G(U) → G(V ) in C such that FG(g) = μV gμU−1 . Hence, the correspondences U → G(U) and g → G(g) define a functor from D to C that turns the family of isomorphisms μ = (μU : U → FG(U))U∈D into a natural isomorphism from IdD to FG. It remains to prove that GF is naturally isomorphic to IdC . For any X ∈ C consider −1 μF(X) : FGF(X) → F(X). The functor F is fully faithful, so there exists a unique

−1 morphism νX : GF(X) → X in C such that F(νX ) = μF(X) . Actually, since μF(X) is an isomorphism in D it follows that νX is an isomorphism in C . In addition, the f

naturality of μ implies that, for any X → Y in C , −1 −1 F( f νX ) = F( f )F(νX ) = F( f )μF(X) = μF(Y ) FGF( f )

= F(νY )FGF( f ) = F(νY GF( f )). From here we conclude that f νX = νY GF( f ). Thus ν = (νX : GF(X) → X)X∈C is a natural isomorphism from GF to IdC , and this finishes the proof.

18

Monoidal and Braided Categories

Remark 1.20 Let F : C → D be an equivalence of categories. If G : D → C , μ : IdD → FG and ν : GF → IdC are as in the second part of the proof of Proposition 1.19 then, for all X ∈ C and Y ∈ D, we have −1 −1 F(νX ) = μF(X) and G(μY ) = νG(Y ).

(1.3.1)

Indeed, the first equality is just the definition of ν . For the second one, we use the −1 ), and since F is fully faithful definition of G to see that FG(μY ) = μFG(Y ) = F(νG(Y ) −1 it follows that G(μY ) = νG(Y , as desired. )

Proposition 1.21 Let C be a monoidal category and e(C ) the strict monoidal category associated to C ; see Subsection 1.2.6. Then the functor F : C → e(C ), F(X) = (X ⊗ −, aX,−,− ) and F( f ) = f ⊗ − is fully faithful. Consequently, C is equivalent to the full image of F. Proof

Let X,Y be objects of C and X

g f

Y morphisms in C such that F( f ) =

F(g). Then f ⊗IdZ = g⊗IdZ , for any Z ∈ C . Together with the naturality of r this implies f rX = rY ( f ⊗ Id1 ) = rY (g ⊗ Id1 ) = grX . Since rX is an isomorphism we get f = g, and so the map f → F( f ) is injective. It is also surjective, that is, any natural transformation θ : F(X) → F(Y ) satisfying (1.2.4) has the form θ = (θZ = fθ ⊗ IdZ )Z∈C , for some morphism fθ : X → Y in C . Actually, fθ : X → Y is the morphism in C defined by the composition r−1

θ1

r

Y X fθ : X → X ⊗1 →Y ⊗1 → Y.

To see this observe that the three (small) rectangular diagrams below are commutative, for all X,Y, Z ∈ C : X ⊗Z

rX−1 ⊗IdZ

fθ ⊗IdZ

Y ⊗Z

rY−1 ⊗IdZ

(X ⊗ 1) ⊗ Z

aX,1,Z

θ1 ⊗IdZ

(Y ⊗ 1) ⊗ Z

X ⊗ (1 ⊗ Z)

IdX ⊗lZ

θ1⊗Z aY,1,Z

Y ⊗ (1 ⊗ Z)

X ⊗Z θZ

IdY ⊗lZ

Y ⊗ Z.

Indeed, the first one is commutative because of the definition of fθ , the second one is commutative because of (1.2.4), and the last one is commutative because of the naturality of θ . Therefore, the exterior rectangular diagram is commutative. According to (1.1.2) we have aX,1,Z = (IdX ⊗ lZ−1 )(rX ⊗ IdZ ), and consequently we obtain (IdX ⊗ lZ )aX,1,Z (rX−1 ⊗ IdZ ) = IdX⊗Z . Similarly, (IdY ⊗ lZ )aY,1,Z (rY−1 ⊗ IdZ ) = IdY ⊗Z . Hence θZ = fθ ⊗ IdZ = F( fθ )Z , for all Z ∈ C , and this finishes the proof. An (op)monoidal functor between two monoidal categories is a functor that respects the two monoidal structures. More precisely:

1.3 Monoidal Functors

19

Definition 1.22 Let (C , ⊗, 1, a, l, r) and (D, , I, a , l  , r ) be monoidal categories and F : C → D a functor. (i) F is called monoidal if there exists a family of morphisms in D

ϕ2 = (ϕ2,X,Y : F(X)F(Y ) → F(X ⊗Y ))X,Y ∈C , natural in X and Y , and ϕ0 : I → F(1) a morphism in D such that, for all X,Y, Z ∈ C , the corresponding diagrams in (1.3.2) are commutative. If ϕ0 and ϕ2 are defined by identity morphisms in D then we call the monoidal functor (F, ϕ0 , ϕ2 ) strict monoidal. (ii) F is called opmonoidal if there exists a family of morphisms in D

ψ2 = (ψ2,X,Y : F(X ⊗Y ) → F(X)F(Y ))X,Y ∈C , natural in X and Y , and ψ0 : F(1) → I a morphism in D such that, for all X,Y, Z ∈ C , the corresponding diagrams in (1.3.2) are commutative. If ψ0 and ψ2 are defined by identity morphisms in D then the opmonoidal functor (F, ψ0 , ψ2 ) is actually strict monoidal. (iii) F is called a strong monoidal functor if it is monoidal and, moreover, ϕ0 and ϕ2 are defined by isomorphisms in D. Equivalently, F is strong monoidal if it is opmonoidal and, moreover, ψ0 and ψ2 are defined by isomorphisms in D. (F(X)F(Y ))F(Z) ϕ2,X,Y IdF(Z)

aF(X),F(Y ),F(Z)

ψ2,X,Y IdF(Z)

F(X)(F(Y )F(Z)) IdF(X) ψ2,Y,Z

F(X ⊗Y )F(Z) ϕ2,X⊗Y,Z

F(X)F(Y ⊗ Z)

ψ2,X⊗Y,Z

ψ2,X,Y ⊗Z

F((X ⊗Y ) ⊗ Z)

F(aX,Y,Z )

ϕ0 IdF(X)

IF(X)  lF(X)

F(X)

F(lX )

ϕ2,X,Y ⊗Z

F(X ⊗ (Y ⊗ Z)), IdF(X) ϕ0

F(1)F(X) ψ0 IdF(X)

IdF(X) ϕ2,Y,Z

ψ2,1,X

ϕ2,1,X

F(1 ⊗ X) ,

F(X)I  rF(X)

F(X)

F(X)F(1) IdF(X) ψ0

F(rX )

ψ2,X,1

ϕ2,X,1

F(X ⊗ 1). (1.3.2)

Example 1.23 If C is a monoidal category then the identity functor IdC is a strict monoidal functor. Remark 1.24 Many times in what follows, unless otherwise specified, if F : C → D is an (op)monoidal functor, we will denote by the same symbols ⊗, 1, a, l, r the tensor product, unit, etc. both in C and D.

20

Monoidal and Braided Categories

Remark 1.25

If C

(F,ϕ0 ,ϕ2 )

(G,ψ0 ,ψ2 )

D

E are monoidal functors then so is GF with G(ϕ0 )

ψ

0 ξ0 : 1 −→ G(I) −→ GF(1) and

ξ2,X,Y : GF(X) ⊗ GF(Y )

ψ2,F(X),F(Y )

−→

G(ϕ2,X,Y )

G(F(X)F(Y )) −→ GF(X ⊗Y ).

If F, G are opmonoidal or strong monoidal functors, the same is true for GF. Definition 1.26 Let C , D be monoidal categories. They are called monoidally isomorphic if there exist monoidal functors (F, ϕ0 , ϕ2 ) : C → D and (G, ψ0 , ψ2 ) : D → C such that FG = IdD and GF = IdC as monoidal functors (this implies that F and G are automatically strong monoidal). Proposition 1.27 Let C , D be monoidal categories. Then they are monoidally isomorphic if and only if there exists a strong monoidal functor F : C → D which is also an isomorphism of categories. Proof We only have to prove the converse. Let G : D → C be the inverse of the −1 ), monoidal functor (F, ϕ0 , ϕ2 ); define ψ0 := G(ϕ0−1 ) and ψ2,U,V := G(ϕ2,G(U),G(V ) for all U,V ∈ D. We leave it to the reader to check that (G, ψ0 , ψ2 ) is indeed a monoidal functor, and that FG = IdD and GF = IdC as monoidal functors. An example of a strong monoidal functor is the following. Proposition 1.28 Proof

The functor F defined in Proposition 1.21 is strong monoidal.

The commutativity of the diagram X ⊗Y

IdX⊗Y

X ⊗Y −1 lX⊗Y

lX−1 ⊗IdY

(1 ⊗ X) ⊗Y

a1,X,Y

1 ⊗ (X ⊗Y )

is equivalent to the fact that the second diagram in the statement of Proposition 1.5 is commutative. So ϕ0 := l −1 : (IdC , ρ IdC = (IdX⊗Y )X,Y ∈C ) → F(1) = (1 ⊗ −, a1,−,− ) is an isomorphism in e(C ). For any X,Y ∈ C we have F(X) ⊗ F(Y ) = (X ⊗ (Y ⊗ −), (IdX ⊗ aY,−,− )aX,Y ⊗−,− ) . Since for any Z, T ∈ C the commutativity of the diagram (X ⊗ (Y ⊗ Z)) ⊗ T

(IdX ⊗aY,Z,T )aX,Y ⊗Z,T

a−1 X,Y,Z ⊗IdT

((X ⊗Y ) ⊗ Z) ⊗ T

X ⊗ (Y ⊗ (Z ⊗ T )) a−1 X,Y,Z⊗T

aX⊗Y,Z,T

(X ⊗Y ) ⊗ (Z ⊗ T )

1.3 Monoidal Functors

21

is equivalent to the commutativity of (1.1.1), it follows that

ϕ2,X,Y := a−1 X,Y,− : F(X) ⊗ F(Y ) → F(X ⊗Y ) = ((X ⊗Y ) ⊗ −, aX⊗Y,−,− ) is an isomorphism in e(C ). We now prove that (F, ϕ0 , (ϕ2,X,Y )X,Y ∈C ) is monoidal. The commutativity of the first diagram in Definition 1.22 comes out as

ϕ2,X,Y ⊗Z (idX⊗− ⊗ ϕ2,Y,Z ) = (aX,Y,Z ⊗ −)ϕ2,X⊗Y,Z (ϕ2,X,Y ⊗ idZ⊗− ), an equality between two natural transformations from X ⊗ (Y ⊗ (Z ⊗ −)) to (X ⊗ (Y ⊗ Z)) ⊗ −. This equality holds since −1 (ϕ2,X,Y ⊗Z )T (idX⊗− ⊗ ϕ2,Y,Z )T = a−1 X,Y ⊗Z,T (IdX ⊗ aY,Z,T ) (1.1.1)

−1 = (aX,Y,Z ⊗ IdT )a−1 X⊗Y,Z,T aX,Y,Z⊗T

= (aX,Y,Z ⊗ −)T (ϕ2,X⊗Y,Z )T (ϕ2,X,Y ⊗ idZ⊗− )T , for all T ∈ C . Similarly, by using again the commutativity of the second triangle in Proposition 1.5 and the fact that e(C ) is strict monoidal we compute −1 F(lX )Y (ϕ2,1,X )Y (ϕ0 ⊗ idX⊗− )Y = (lX ⊗ IdY )a−1 1,X,Y lX⊗Y = IdX⊗Y = (lX⊗− )Y ,

for all Y ∈ C . By (1.1.2) we have −1 F(rX )Y (ϕ2,X,1 )Y (idX⊗− ⊗ ϕ0 )Y = (rX ⊗ IdY )a−1 X,1,Y (IdX ⊗ lY ) = IdX⊗Y = (rX⊗− )Y ,

for all Y ∈ C , and therefore the other two (square) diagrams in Definition 1.22 are commutative as well. Thus the proof is finished. The notion of natural transformation extends to the monoidal setting as follows. Definition 1.29 Let C , D be monoidal categories and (F, ϕ0F , ϕ2F ), (G, ϕ0G , ϕ2G ) : C → D monoidal functors. A monoidal natural transformation ω from (F, ϕ0F , ϕ2F ) to (G, ϕ0G , ϕ2G ) is a natural transformation ω : F → G such that, for any objects X,Y of C , the following diagrams are commutative: F(X) ⊗ F(Y )

F ϕ2,X,Y

1

G ϕ2,X,Y

G(X ⊗Y )

ϕ0G

ϕ0F

ωX⊗Y

ωX ⊗ωY

G(X) ⊗ G(Y )

F(X ⊗Y )

and

F(1)

ω1

G(1).

The transformation ω is called a monoidal natural isomorphism if ω is both a monoidal natural transformation and a natural isomorphism. Reversing the appropriate arrows in the above diagrams leads to the definition of an opmonoidal natural transformation between two opmonoidal functors. We are now able to define the concept of monoidal equivalence.

22

Monoidal and Braided Categories

Definition 1.30 Let C , D be monoidal categories and F : C → D a monoidal (resp. opmonoidal) functor. We call F a monoidal (resp. opmonoidal) equivalence if there exists a monoidal (resp. opmonoidal) functor G : D → C such that FG is monoidally (resp. opmonoidally) naturally isomorphic to IdD and GF is monoidally (resp. opmonoidally) naturally isomorphic to IdC . If F and G as above are both strong monoidal then F is called a strong monoidal equivalence between C and D. If a functor F : C → D defines an (op)monoidal (resp. strong monoidal) equivalence between C and D we say that the categories C and D are (op)monoidally (resp. strong monoidally) equivalent. In the strong monoidal case the above definition can be reformulated. Proposition 1.31 A functor F : C → D defines a strong monoidal equivalence if and only if F is strong monoidal and an equivalence of categories. Proof The direct implication is immediate. For the converse, let G : D → C be a functor as in Remark 1.20. If μ : IdD → FG and ν : GF → IdC are the natural isomorphisms satisfying (1.3.1), we show that G admits a unique strong monoidal structure with respect to which μ and ν become monoidal transformations. F )X,Y ∈C , ϕ0F ) the strong monoidal In what follows we denote by (ϕ2F := (ϕ2,X,Y structure of F, and by ϕ 2F , ϕ 0F the inverse morphisms of ϕ2F , ϕ0F , respectively. Assume that G admits a strong monoidal structure (ϕ2G , ϕ0G ) with respect to which ν becomes a monoidal natural transformation. Remark 1.25 implies that GF is strong GF = G(ϕ F G GF F G monoidal via ϕ2GF = (ϕ2,X,Y 2,X,Y )ϕ2,F(X),F(Y ) )X,Y ∈C and ϕ0 = G(ϕ0 )ϕ0 , and from here we get that −1 GF F G νX⊗Y (νX ⊗ νY ) = ϕ2,X,Y = G(ϕ2,X,Y )ϕ2,F(X),F(Y )

or, equivalently, G −1

F ϕ2,F(X),F(Y ) = G(ϕ2,X,Y )νX⊗Y (νX ⊗ νY ).

(1.3.3)

Let U,V ∈ D. By the naturality of ϕ2G the diagram GFG(U) ⊗ GFG(V )

G ϕ2,FG(U),FG(V )

G(μU )⊗G(μV )

G(FG(U)FG(V )) G(μU μV )

G(U) ⊗ G(V )

G ϕ2,U,V

G(UV )

is commutative, and therefore G ϕ2,U,V

= (1.3.1),(1.3.3)

=

G G(μU−1 μV−1 )ϕ2,FG(U),FG(V ) (G( μU ) ⊗ G( μV )) F −1 G((μU−1 μV−1 )ϕ 2,G(U),G(V ) )νG(U)⊗G(V ) ,

(1.3.4)

for all U,V ∈ D. Together with ν1 ϕ0GF = Id1 or, equivalently, with ϕ0G = G(ϕ 0F )ν1−1 , this guarantees the uniqueness of the stated strong monoidal strucure of G.

1.3 Monoidal Functors

23

Conversely, define ϕ2G as in (1.3.4) and ϕ0G := G(ϕ 0F )ν1−1 . We prove that with this structure G is strong monoidal and μ , ν become monoidal natural transformations. To show the commutativity of the diagrams in (1.3.2) we need the equality G F ) = μUV (μU−1 μV−1 )ϕ 2,G(U),G(V F(ϕ2,U,V ),

(1.3.5)

F which holds for any U,V ∈ D. Indeed, the naturality of μ applied to ϕ2,G(U),G(V ) and respectively to μU μV gives the following commutative diagrams:

FG(FG(U)FG(V ))

F FG(ϕ2,G(U),G(V ))

FGF(G(U) ⊗ G(V ))

μFG(U)FG(V )

μF(G(U)⊗G(V ))

FG(U)FG(V ) FG(U)FG(V )

F ϕ2,G(U),G(V )

μFG(U)FG(V )

F(G(U) ⊗ G(V )) ,

FG(FG(U)FG(V )) FG(μU μV )

μU μV

UV

μUV

FG(UV ).

Therefore, G F −1 ) = FG(μU−1 μV−1 )FG(ϕ 2,G(U),G(V F(ϕ2,U,V ) )F(νG(U)⊗G(V ) ) (1.3.1)

F = FG(μU−1 μV−1 )FG(ϕ 2,G(U),G(V ) ) μF(G(U)⊗G(V )) F = FG(μU−1 μV−1 )μFG(U)FG(V ) ϕ 2,G(U),G(V ) F = μUV (μU−1 μV−1 )ϕ 2,G(U),G(V ),

for all U,V ∈ D, as stated. We can compute now that  G G )F(ϕ2,UV,W )F(ϕ2,U,V ⊗ IdG(W ) ) FG(aU,V,W  −1 F G = FG(aU,V,W )μ(UV )W (μUV μW−1 )ϕ 2,G(UV ),G(W ) F(ϕ2,U,V ⊗ IdG(W ) )  −1 G F )μ(UV )W (μUV μW−1 )(F(ϕ2,U,V )IdFG(W ) )ϕ 2,G(U)⊗G(V = FG(aU,V,W ),G(W ) (1.3.5)

 = FG(aU,V,W )μ(UV )W ((μU−1 μV−1 )μW−1 ) F F

2,G(U)⊗G(V (ϕ 2,G(U),G(V ) IdFG(W ) )ϕ ),G(W )  = μU(V W ) aU,V,W ((μU−1 μV−1 )μW−1 ) F F

2,G(U)⊗G(V (ϕ 2,G(U),G(V ) IdFG(W ) )ϕ ),G(W )

= μU(V W ) (μU−1 (μV−1 μW−1 ))aFG(U),FG(V ),FG(W ) F F

2,G(U)⊗G(V (ϕ 2,G(U),G(V ) IdFG(W ) )ϕ ),G(W ) F = μU(V W ) (μU−1 (μV−1 μW−1 ))(IdFG(U) ϕ 2,G(V ),G(W ) ) F ϕ 2,G(U),G(V )⊗G(W ) F(aG(U),G(V ),G(W ) ),

where we applied: in the second equality the naturality of ϕ2F to the morphisms

24

Monoidal and Braided Categories

G  ϕ2,U,V and IdG(W ) , in the fourth equality the naturality of μ to the morphism aU,V,W , −1 −1 −1  in the fifth equality the naturality of a to the morphisms μU , μV , μW , and in the last equality the fact that ϕ2F closes commutatively the diagram (1.3.2). Similar arguments allow us to calculate that G G F(ϕ2,U,V W )F(IdG(U) ⊗ ϕ2,V,W )F(aG(U),G(V ),G(W ) ) G

F = μU(V W ) (μU−1 μV−1 W )ϕ2,G(U),G(V W ) F(IdG(U) ⊗ ϕ2,V,W )F(aG(U),G(V ),G(W ) ) G = μU(V W ) (μU−1 μV−1 W )(IdFG(U) F(ϕ2,V,W )) F ϕ 2,G(U),G(V )⊗G(W ) F(aG(U),G(V ),G(W ) )

F = μU(V W ) (μU−1 (μV−1 μW−1 ))(IdFG(U) ϕ 2,G(V ),G(W ) ) F ϕ 2,G(U),G(V )⊗G(W ) F(aG(U),G(V ),G(W ) ).

Thus, since F is fully faithful it follows that our ϕ2G closes commutatively the corresponding hexagonal diagram in (1.3.2). Together with ϕ0G as above, it also closes commutatively the two square diagrams in (1.3.2), since F is fully faithful: G )F(ϕ0G ⊗ IdG(U) ) FG(lU )F(ϕ2,I,U F F(ϕ0G ⊗ IdG(U) ) = FG(lU )μIU (μI−1 μU−1 )ϕ 2,G(I),G(U) F = FG(lU )μIU (μI−1 F(ϕ0G )μU−1 )ϕ 2,1,G(U) F = FG(lU )μIU (ϕ 0F μU−1 )ϕ 2,1,G(U) F = μU lU (IdI μU−1 )(ϕ 0F IdFG(U) )ϕ 2,1,G(U)  F = lFG(U) (ϕ 0F IdFG(U) )ϕ 2,1,G(U) = F(lG(U) ), G )F(Id G and similarly FG(rU )F(ϕ2,U,I G(U) ⊗ ϕ0 ) = F(rG(U) ), for all U ∈ D. The remaining details are left to the reader. We check that μ , ν are monoidal natural transformations. For all X,Y ∈ C , GF F G ) = FG(ϕ2,X,Y )F(ϕ2,F(X),F(Y F(ϕ2,X,Y )) (1.3.5)

F −1 −1 F = FG(ϕ2,X,Y )μF(X)F(Y ) (μF(X) μF(Y ) )ϕ2,GF(X),GF(Y )

(1.3.1)

F F = μF(X⊗Y ) ϕ2,X,Y (F(νX )F(νY ))ϕ 2,GF(X),GF(Y ) (1.3.1)

−1 = μF(X⊗Y ) F(νX ⊗ νY ) = F(νX⊗Y (νX ⊗ νY )),

and since F is fully faithful this proves that ν is a monoidal natural transformation (clearly, ϕ0GF = ν1−1 since ϕ0G = G(ϕ˜ 0F )ν1−1 ). From the definitions we have FG G F −1 −1 ϕ2,U,V = F(ϕ2,U,V )ϕ2,G(U),G(V ) = μUV ( μU  μV ), ∀ U, V ∈ D.

So μ is a monoidal natural transformation (it is easy to see that ϕ0FG = μI ). Proposition 1.32 The functor F : C → e(C ) defined in Proposition 1.28 provides a strong monoidal equivalence between C and the full image of F. Proof

This follows from Propositions 1.21, 1.28 and 1.31.

1.4 Mac Lane’s Strictification Theorem for Monoidal Categories

25

1.4 Mac Lane’s Strictification Theorem for Monoidal Categories We will prove that any monoidal category C is strong monoidally equivalent to a strict category, C str . The construction associating to a monoidal category a strict one which is strong monoidally equivalent to it is called strictification. To this end, we first need some concepts and preliminary results. Let (C , ⊗, 1, a, l, r) be a monoidal category. For any positive integer m, a word of length m with objects of C is a sequence S of the form S = (X1 , . . . , Xm ), where X1 , . . . , Xm ∈ C . By convention, the word of length zero with objects of C is the empty sequence φ . To any non-empty word S = (X1 , . . . , Xm ) with objects of C we associate an object F(S) of C defined by F(S) = ((· · · ((X1 ⊗ X2 ) ⊗ X3 ) ⊗ · · · ) ⊗ Xm−1 ) ⊗ Xm . If S = φ then F(φ ) = 1, the unit object of C . Definition 1.33 Let C be a monoidal category. By C str we denote the category whose objects are words of finite length with objects of C . If S and S are objects of C str then we define HomC str (S, S ) := HomC (F(S), F(S )). The identities and composition of morphisms are taken from C . By the above definition it is immediate that F : C str → C , S → F(S), defines a functor (F acts as identity on morphisms). We define the monoidal structure of C str . If S = (X1 , . . . , Xm ) and S = (Y1 , . . . ,Yn ) are non-empty words of length m and n, respectively, with objects of C , we define the tensor product SS between S and S as follows: SS = (X1 , . . . , Xm ,Y1 , . . . ,Yn ). By convention, φ S = Sφ = S, for any word S of finite length with objects of C . Clearly, (SS )S = S(S S ) := SS S , for any objects S, S , S of C str . First we define ϕ2,φ ,φ : F(φ ) ⊗ F(φ ) = 1 ⊗ 1 → 1 = F(φ ) = F(φ φ ), ϕ2,φ ,φ = l1 = r1 . For a non-empty word S with objects of C define

ϕ2,φ ,S : F(φ ) ⊗ F(S) = 1 ⊗ F(S) → F(S) = F(φ S), ϕ2,φ ,S = lF(S) , ϕ2,S,φ : F(S) ⊗ F(φ ) = F(S) ⊗ 1 → F(S) = F(Sφ ), ϕ2,S,φ = rF(S) , ϕ2,S,(Y ) : F(S) ⊗ F((Y )) = F(S) ⊗Y → F(S) ⊗Y = F(S(Y )), ϕ2,S,(Y ) = IdF(S)⊗Y , and, inductively, if S = (Y1 , . . . ,Ym ) is a word of length m ≥ 2 with objects of C and S∗ = (Y1 , . . . ,Ym−1 ) then 

ϕ2,S,S : F(S) ⊗ F(S ) =

a−1 F(S),F(S∗ ),Ym  F(S) ⊗ (F(S∗ ) ⊗Ym ) −→

ϕ2,S,S ⊗IdYm ∗ −→

(F(S) ⊗ F(S∗ )) ⊗Ym

F(SS∗ ) ⊗Ym = F(SS∗ (Ym )) = F(SS ).

By induction on the length of S it follows that ϕ2 := (ϕ2,S,S : F(S) ⊗ F(S ) →

26

Monoidal and Braided Categories f

F(SS ))S,S ∈C str is a natural isomorphism. Then for any morphisms S → T and f

f

f

S → T  in C str , that is, for any morphisms F(S) → F(T ) and F(S ) → F(T  ) in ff

C , define the tensor product morphism SS −→ T T  in C str as being the unique morphism that makes the diagram ϕ2,S,S

F(S) ⊗ F(S )

F(SS )

f⊗f

ff

ϕ2,T,T 

F(T ) ⊗ F(T  )

F(T T  )

commutative. It is clear at this point that (C str , , φ ) is a strict monoidal category. Theorem 1.34 The categories C and C str are strong monoidally equivalent. Proof We show that (F, Id1 , ϕ2 ) : C str → C defines a strong monoidal equivalence. Since X = F((X)), for any object of C , we get that F is essentially surjective. Also, since HomC str (S, S ) = HomC (F(S), F(S )), for any objects S, S of C str , we obtain that F is fully faithful. Thus, according to Proposition 1.19, F : C str → C is an equivalence of categories. We will prove that (F, Id1 , ϕ2 ) is a monoidal functor. For this, we only need to check that the first diagram that appears in Definition 1.22, specialized for our situation, is commutative (the commutativity of the other two diagrams is immediate). This reduces, for any S, S , S ∈ C str , to

ϕ2,S,S S (IdF(S) ⊗ ϕ2,S ,S )aF(S),F(S ),F(S ) = ϕ2,SS ,S (ϕ2,S,S ⊗ IdF(S ) ). (1.4.1) We prove the relation in (1.4.1) by induction on the length of S . By the naturality of r, the diagram rF(S)⊗F(S )

(F(S) ⊗ F(S )) ⊗ 1

F(S) ⊗ F(S ) ϕ2,S,S

ϕ2,S,S ⊗Id1

F(SS ) ⊗ 1

rF(SS )

F(SS )

is commutative. This fact allows us to compute

ϕ2,S,S (IdF(S) ⊗ ϕ2,S ,φ )aF(S),F(S ),F(φ ) = ϕ2,S,S (IdF(S) ⊗ rF(S ) )aF(S),F(S ),1 (by Proposition 1.5) = ϕ2,S,S rF(S)⊗F(S ) = rF(SS ) (ϕ2,S,S ⊗ Id1 ) = ϕ2,SS ,φ (ϕ2,S,S ⊗ IdF(φ ) ), and this shows that (1.4.1) holds for S = φ . Assume that (1.4.1) is valid for any S of length p − 1 and let S = (Z1 , . . . , Z p ) be

1.4 Mac Lane’s Strictification Theorem for Monoidal Categories

27

a word of length p ≥ 1 with objects of C . If S∗ = (Z1 , . . . , Z p−1 ) we have that aF(S),F(S )⊗F(S ),Z ∗

(F(S) ⊗ (F(S ) ⊗ F(S∗ ))) ⊗ Z p (IdF(S) ⊗ϕ2,S ,S )⊗IdZ p

∗



S∗ )) ⊗ Z p

aF(S),F(S S ),Z ∗

∗

(F(S) ⊗ F(S )) ⊗ (F(S∗ ) ⊗ Z p ) ∗

∗

(F(SS

p

ϕ2,S,S ⊗IdF(S )⊗Z p

(ϕ2,S,S ⊗IdF(S ) )⊗IdZ p

) ⊗ F(S∗ )) ⊗ Z p

F(S) ⊗ (F(S S∗ ) ⊗ Z p ),

p

aF(S)⊗F(S ),F(S ),Z

((F(S) ⊗ F(S )) ⊗ F(S∗ )) ⊗ Z p 

F(S) ⊗ ((F(S ) ⊗ F(S∗ )) ⊗ Z p )

IdF(S) ⊗(ϕ2,S ,S ⊗IdZ p )

∗

(F(S) ⊗ F(S

p

aF(SS ),F(S ),Z ∗

p

F(SS ) ⊗ (F(S∗ ) ⊗ Z p )

are commutative, because of the naturality of a. Therefore

ϕ2,S,S S (IdF(S) ⊗ ϕ2,S ,S )aF(S),F(S ),F(S ) = (ϕ2,S,S S∗ ⊗ IdZ p )a−1 F(S),F(S S ),Z ∗

p

  (IdF(S) ⊗ (ϕ2,S ,S∗ ⊗ IdZ p ))(IdF(S) ⊗ a−1 )a F(S ),F(S∗ ),Z p F(S),F(S ),F(S )  −1

= ϕ2,S,S S∗ (IdF(S) ⊗ ϕ2,S ,S∗ ) ⊗ IdZ p aF(S),F(S )⊗F(S ),Z ∗

p

  (IdF(S) ⊗ a−1 )a F(S ),F(S∗ ),Z p F(S),F(S ),F(S )

  ϕ2,S,S S∗ (IdF(S) ⊗ ϕ2,S ,S∗ )aF(S),F(S ),F(S∗ ) ⊗ IdZ p a−1 F(S)⊗F(S ),F(S∗ ),Z p   = ϕ2,SS ,S∗ (ϕ2,S,S ⊗ IdF(S∗ ) ) ⊗ IdZ p a−1 F(S)⊗F(S ),F(S ),Z

(1.1.1)

=

∗

=

 (ϕ2,SS ,S∗ ⊗ IdZ p )a−1 (ϕ F(SS ),F(S∗ ),Z p 2,S,S

p

⊗ IdF(S∗ )⊗Z p )

= ϕ2,SS ,S (ϕ2,S,S ⊗ IdF(S ) ), where in the first and the last equality we used the inductive definition of ϕ2 and in the fourth equality the inductive hypothesis. By using also the inductive definition of ϕ2 , it follows immediately that F is actually a strong monoidal functor. Since F is an equivalence of categories as well, we are in a position to apply Proposition 1.31 and we obtain that F is a strong monoidal equivalence. The equivalence inverse G : C → C str of F is defined by G(X) = (X), the word of length 1 defined by X ∈ C , and G( f ) = f , for any morphism f in C . Also by the proof of Proposition 1.31 it follows that the strong monoidal structure of G is defined by ψ0 = Id1 : φ → (1) and

ψ2,X,Y : G(X)G(Y ) = (X,Y ) → G(X ⊗Y ) = (X ⊗Y ), ψ2,X,Y = IdX⊗Y , for all X,Y ∈ C . To any strong monoidal functor (T,t0 ,t2 ) : C → D we associate a strict monoidal functor T str : C str → D str as follows.

28

Monoidal and Braided Categories

Let (F, Id1 , ϕ2 ) : C str → C and (G, Id1 , ψ2 ) : D str → D be the functors from Theorem 1.34 that provide the monoidal equivalences. If S = (X1 , . . . , Xm ) is a nonempty word with objects of C define T str (S) = T (S) := (T (X1 ), . . . , T (Xm )) ∈ D str . In addition, T str (φ ) = φ . In order to define T str on morphisms observe first that T (F(S)) ∼ = G(T (S)), for any S ∈ C str . Indeed, if S = (X1 , . . . , Xm ) ∈ C str is a nonempty word and S∗ = (X1 , . . . , Xm−1 ) then T (F(S)) = T (F(S∗ (Xm ))) = T (F(S∗ ) ⊗ Xm ) ∼ = T (F(S∗ )) ⊗ T (Xm ), because T is a strong monoidal functor. Since T str (φ ) = φ , by induction on the length of S it follows now that T (F(S)) ∼ = G(T (S)), for any S ∈ C str . f

Let now S → S be a morphism in C str , that is, f ∈ HomC (F(S), F(S )). We then define T str ( f ) : T (S) → T (S ) in D str (i.e. T str ( f ) : G(T (S)) → G(T (S )) in D) by requiring the commutativity of the diagram G(T (S))

T str ( f )

∼ =

∼ =

T (F(S))

G(T (S ))

T(f)

T (F(S )).

In this way we have produced a strict monoidal functor T str : C str → D str , as claimed. Remark 1.35 The strictification theorem C → C str allows us also to obtain Mac Lane’s coherence theorem, which states that in any monoidal category C the commutativity of a diagram built from identities, associativity, and left and right unit constraints of C , by tensoring and composing, is equivalent to the Pentagon and Triangle Axioms. Moreover, the coherence theorem gives a unique natural transformation between tensor words of the same length, constructed out of the constraints a, l, r of C . By a tensor word of length m in C we mean an expression like ((· · · ((X1 ⊗ X2 ) ⊗ X3 ) ⊗ · · · ) ⊗ Xm−1 ) ⊗ Xm ,

(1.4.2)

built with m objects X1 , . . . , Xm of C ; note that in (1.4.2) the order of the parentheses can vary and that the tensor word of length 0 is 1, the unit object of the category. Thus, in diagrammatic computations, this fact allows us to omit the parentheses in expressions such as (1.4.2). Also, any diagram in C produces a diagram in C str , whose commutativity implies the commutativity of the original diagram in C . Hence, in practice, it is enough to prove that a certain diagram commutes when it is interpreted in a strict monoidal category.

1.5 (Pre-)Braided Monoidal Categories If C is a monoidal category and X,Y are objects of C then X ⊗ Y and Y ⊗ X are not necessarily isomorphic objects of C . The definition of a braiding on a monoidal

1.5 (Pre-)Braided Monoidal Categories

29

category C requires the existence of isomorphisms between X ⊗Y and Y ⊗ X, for all objects X,Y of C that are compatible with the monoidal structure of C . Recall that if C is a category then the switch functor τ : C × C → C × C is defined by τ (X,Y ) = (Y, X) and τ ( f , g) = (g, f ). Definition 1.36 A pre-braiding on a monoidal category C is a natural transformation c : ⊗ → ⊗ ◦ τ satisfying the so-called Hexagon Axiom, namely for any objects X,Y, Z ∈ C the following diagrams are commutative: (Y ⊗ X) ⊗ Z

aY,X,Z

Y ⊗ (X ⊗ Z)

cX,Y ⊗IdZ

(1.5.1) IdY ⊗cX,Z

(X ⊗Y ) ⊗ Z

Y ⊗ (Z ⊗ X)

aX,Y,Z

aY,Z,X

X ⊗ (Y ⊗ Z)

X ⊗ (Z ⊗Y )

cX,Y ⊗Z

a−1 X,Z,Y

(Y ⊗ Z) ⊗ X,

(X ⊗ Z) ⊗Y

IdX ⊗cY,Z

(1.5.2) cX,Z ⊗IdY

X ⊗ (Y ⊗ Z) a−1 X,Y,Z

(Z ⊗ X) ⊗Y

(X ⊗Y ) ⊗ Z

cX⊗Y,Z

Z ⊗ (X ⊗Y ).

a−1 Z,X,Y

A pre-braiding c is called a braiding if it is a natural isomorphism. A (pre-)braided category is a pair (C , c) consisting of a monoidal category C and a (pre-)braiding c on C . Assuming C strict, we have that c is a pre-braiding on C if c is a natural transformation c : ⊗ → ⊗ ◦ τ , satisfying the conditions X Y

Z

(a) cX,Y ⊗Z =

X Y

Z

and (b) cX⊗Y,Z = Y

Z X

,

(1.5.3)

Z X Y X Y

where, from now on, for any two objects X and Y of C , we will denote cX,Y by

. Y X

Y X

If c is a braiding on C , by X Y

we denote c−1 X,Y , the inverse of cX,Y .

30

Monoidal and Braided Categories In what follows, for C a (strict) monoidal category, we will denote by X

X

X

, X

h,

f

Y

X

νh 

 and μh

Y

Z

Y

Z

the following morphisms in C : IdX : X → X, f : X → Y , μ : X ⊗ Y → Z and ν : X → Y ⊗ Z, respectively. In diagrammatic notation, the fact that c is a natural transformation comes out as M N

h fh=

M N

h gh , ∀ f : M → U and g : N → V in C .

f

g

V U

Y

V U

For (C , c) a braided category, we have

Proposition 1.37 X

Z

X

Y

Z

X

 =  , (b) νh νh

(a)

(1.5.4)

U

U Y

Y

X

Y

Z

 , νh

=  νh U

Y

Y

Z

(1.5.5)

Y U

for any morphism ν : X ⊗ Z → U in C , and X

(a)

Y

μh = 

V

W X

X

Y

Y X

h 

Y

X

h 

μ

μ

= μh 

, (b)

X V

V W X Y X

W Y

,

X V W X

(1.5.6)

μh 

= μh 

(c)

X V

W

X V W

for any morphism μ : Y → V ⊗W in C . Proof This follows from (1.5.4) and (1.5.3). For example, to prove the first equality in (1.5.5) we have to consider in (1.5.4) f = IdY , g = ν , and then use the relation (a) in (1.5.3). We next provide some examples of (pre-)braided categories. Example 1.38 For a braided category C , let C in be equal to C as a monoidal −1 . Then (C in , c) is category, equipped with the mirror-reversed braiding cX,Y = cY,X

1.5 (Pre-)Braided Monoidal Categories

31

also a braided category. To see this, note that (1.5.1) and (1.5.2) are obtained from (1.5.2) and (1.5.1), respectively, by replacing c with c. Definition 1.39 We call a braided category (C , c) symmetric if C = C in , as braided −1 = cX,Y , for any X,Y ∈ C . categories. That is, cY,X Example 1.40 If (C , c) is (pre-)braided then the reverse monoidal category C associated to C is (pre-)braided with cX,Y = cY,X : X⊗Y = Y ⊗ X → X ⊗Y = Y ⊗X. Indeed, it can be easily checked that (1.5.1) for (C , c) follows from (1.5.2) for (C , c), and that (1.5.2) for (C , c) follows from (1.5.1) for (C , c). Example 1.41 If (C , c) is braided then so is C opp , the opposite category associated opp to C , with cX,Y := c−1 X,Y . Example 1.42 The monoidal category Set of sets is symmetric with c defined by cX,Y : X ×Y → Y × X, cX,Y (x, y) = (y, x). Example 1.43 For k a field the category k M of vector spaces over k is symmetric with cX,Y (x ⊗ y) = y ⊗ x, for any X,Y ∈ k M and x ∈ X, y ∈ Y . We describe the braided structures on a category of G-graded vector spaces over a field k, G being a multiplicative group with neutral element e. Proposition 1.44 Let G be a group, φ a normalized 3-cocycle on G and VectG φ the category of G-graded vector spaces endowed with the monoidal structure induced by φ as in Proposition 1.13. Then VectG φ is braided if and only if G is abelian and there exists R : G × G → k∗ such that, for all x, y, z ∈ G, we have R(xy, z)φ (x, z, y) = φ (x, y, z)R(x, z)φ (z, x, y)R(y, z),

(1.5.7)

φ (x, y, z)R(x, yz)φ (y, z, x) = R(x, y)φ (y, x, z)R(x, z).

(1.5.8)

Proof Let k[G] be the group algebra associated to G, and let ε : k[G] → k be defined by ε (g) = 1, for all g ∈ G, extended by linearity. If c is a braiding for VectG φ we claim that the restriction of R := (ε ⊗ ε )ck[G],k[G] : k[G] ⊗ k[G] → k at G × G (also denoted by R) satisfies the two conditions in (1.5.7) and (1.5.8). To see this we first show that c is completely determined by R, in the sense that cV,W (v ⊗ w) = R(|v|, |w|)w ⊗ v,

(1.5.9)

for any V,W ∈ VectG and homogeneous elements v ∈ V and w ∈ W . Indeed, as in the proof of Proposition 1.13, for any f : V → k define θ f : V → k[G] by θ f (v) = ∑x∈G f (vx )x, for all v ∈ V , where v = ∑x∈G vx is the decomposition of v in homogeneous components. We have already remarked that θ f is a morphism in VectG . Also, for g : W → k let θg : W → k[G]. We recall that (ε ⊗ ε )(θg ⊗ θ f ) = g ⊗ f .

32

Monoidal and Braided Categories

Now, by the naturality of c we have (θg ⊗ θ f )cV,W = ck[G],k[G] (θ f ⊗ θg ), and composing both sides of this equality to the left with ε ⊗ ε we get that f (v)g(w)R(|v|, |w|) = (g ⊗ f )cV,W (v ⊗ w), for any homogeneous elements v ∈ V , w ∈ W . If we write cV,W (v ⊗ w) = ∑i wi ⊗ vi we obtain ∑i g(wi ) f (vi ) = f (v)g(w)R(|v|, |w|), for all f : V → k, g : W → k. Thus cV,W (v ⊗ w) = ∑ wi ⊗ vi = R(|v|, |w|)w ⊗ v, i

and this proves (1.5.9). If |v| = x and |w| = y then we must have w ⊗ v ∈ (W ⊗V )xy = z∈G Wz ⊗ Vz−1 xy . But w ⊗ v ∈ Wy ⊗ Vx and this forces xy = yx. Hence G must be abelian. It is easy to see now that the two conditions on R in (1.5.7) and (1.5.8) are equivalent to the commutativity of the diagrams in (1.5.2) and (1.5.1), respectively, and that cV,W is an isomorphism if and only if R takes values in k∗ . Conversely, if G is abelian and there is an R satisfying the two conditions above then it can be easily checked that c defined by (1.5.9) is a braiding for VectG φ. 

Definition 1.45 Let G be an abelian group and k a field. A pair (φ , R) consisting of a normalized 3-cocycle φ on G with values in k∗ and a map R : G × G → k∗ satisfying (1.5.7) and (1.5.8) is called an abelian 3-cocycle on G with values in k∗ . G We denote by VectG (φ ,R) the category Vect endowed with the braided structure given by the abelian 3-cocycle (φ , R). Remark 1.46 For an abelian group G, examples of pairs (φ , R) satisfying the conditions of Proposition 1.44 are given by the so-called normalized coboundary abelian 3-cocycles. More precisely, let g ∈ K 2 (G, k∗ ) such that g(x, e) = g(e, z), for all x, z ∈ G, and define Rg : G×G → k∗ by Rg (x, y) = g(x, y)−1 g(y, x), for all x, y ∈ G. Then the assumption on G to be abelian ensures that the pair (Δ2 (g), Rg ) satisfies the equalities (1.5.7) and (1.5.8). In what follows, a pair (Δ2 (g), Rg ) as above will be called a coboundary abelian 33 (G, k∗ ) is the abelian group of abelian cocycle on G with coefficients in k∗ . Then Hab 3-cocycles modulo their coboundaries. Example 1.47 The strict monoidal category VectZ2 is braided via the following structure. If V,W ∈ VectZ2 we define cV,W : V ⊗W → W ⊗V by cV,W (v ⊗ w) = (−1)|v||w| w ⊗ v, for homogeneous elements v ∈ V , w ∈ W . We denote this braided category by VectZ−12 . Proof Define R : Z2 × Z2 → k∗ by R(x, y) = (−1)xy , for all x, y ∈ {0, 1}. If φ is the trivial 3-cocycle on Z2 , then it can be easily checked that (φ , R) is an abelian 3-cocycle on Z2 . Moreover, one has VectZ(φ2,R) = VectZ−12 , as braided categories. More generally, for VectZn we have the following braided structures.

1.5 (Pre-)Braided Monoidal Categories

33

Example 1.48 Let G = Zn be the cyclic group of order n ≥ 2 written additively 2 and ν ∈ k such that ν n = ν 2n = 1. If φν n is the normalized 3-cocycle on G defined in Example 1.16 and Rν : G × G → k∗ is given by Rν (x, y) = ν xy , for all x, y ∈ {0, . . . , n − 1}, then (φν n , Rν ) endows the category of Zn -graded vector spaces with a braided structure. Proof

Let us first prove that the relations in (1.5.7) hold, that is,

φν n (x, z, y)Rν (x + y, z) = Rν (x, z)Rν (y, z)φν n (z, x, y)φν n (x, y, z), for all x, y, z ∈ {0, . . . , n − 1}. For this we consider the following cases: 1. If x + y < n the above relation reduces to φν n (x, z, y) = φν n (x, y, z), and this follows directly from the definition of φν n . 2. If x + y ≥ n we write x + y = u + n, for some 0 ≤ u ≤ n − 2. We must show that

φν n (x, z, y)ν uz = ν (x+y)z ν nz φν n (x, y, z). By the definition of φν n this is equivalent to ν uz = ν (x+y)z+nz , and the latter follows since uz − (x + y)z − nz = −2nz and ν 2n = 1. The relation (1.5.8) can be proved in a similar way. This time we have to show

φν n (y, z, x)φν n (x, y, z)Rν (x, y + z) = Rν (x, z)Rν (x, y)φν n (y, x, z), for all x, y, z ∈ {0, . . . , n − 1}. To this end consider the following possibilities: 1 . If y + z < n the above equality comes out as

ν x(y+z) φν n (y, z, x) = ν xz ν xy φν n (y, x, z), which is clearly satisfied. 2 . If y+z ≥ n we write y+z = v+n, for some 0 ≤ v ≤ n−2. The required equality reduces to ν nx ν xv = ν x(y+z) , and this holds since nx + xv = x(y + z). Thus, the category VectZn is braided with the monoidal structure from Example 1.16 (where q = ν n ), and braiding given by cV,W (v ⊗ w) = ν |v||w| w ⊗ v, for any Zn graded vector spaces V and W and homogeneous elements v ∈ V , w ∈ W . Coming back to the full generality, we next express the compatibilities between braidings and unit constraints. Proposition 1.49

If (C , c) is a braided category then the diagrams cX,1

X ⊗1 rX

c1,X

1 ⊗ X and 1 ⊗ X lX

X

X ⊗1 rX

lX

X

are commutative, for any X ∈ C . Proof It suffices to prove the commutativity of the first triangle. The second triangle can be viewed as the first one, but now in (C , c) instead of (C , c).

34

Monoidal and Braided Categories If Y is another object of C we have cX,Y (lX ⊗ IdY )(cX,1 ⊗ IdY ) = cX,Y lX⊗Y a1,X,Y (cX,1 ⊗ IdY ) = lY ⊗X (Id1 ⊗ cX,Y )a1,X,Y (cX,1 ⊗ IdY ) = lY ⊗X a1,Y,X cX,1⊗Y aX,1,Y = (lY ⊗ IdX )cX,1⊗Y aX,1,Y = cX,Y (IdX ⊗ lY )aX,1,Y = cX,Y (rX ⊗ IdY ),

where in the first equality we used Proposition 1.5, in the second one the naturality of l, in the third one (1.5.1), in the fourth one again Proposition 1.5, in the fifth one the naturality of c, and in the last one (1.1.2). Thus, since cX,Y is an isomorphism, we get that lX cX,1 ⊗ IdY = rX ⊗ IdY , for all Y ∈ C . By the naturality of r the two diagrams below are commutative rX⊗1

(X ⊗ 1) ⊗ 1 rX ⊗Id1

lX cX,1 ⊗Id1

X ⊗1

X ⊗1 rX

rX

lX cX,1

X.

Therefore rX rX⊗1 = rX (rX ⊗ Id1 ) = rX (lX cX,1 ⊗ Id1 ) = lX cX,1 rX⊗1 . But rX⊗1 is an isomorphism in C , hence rX = lX cX,1 . Corollary 1.50

If (C , c) is a braided category then cX,1 = c−1 1,X , for all X ∈ C .

The next result is known as the categorical version of the Yang–Baxter equation. In any (pre-)braided category (C , c) the diagram

Proposition 1.51 X ⊗ (Y ⊗ Z)

IdX ⊗cY,Z

X ⊗ (Z ⊗Y )

a−1 X,Z,Y

(X ⊗ Z) ⊗Y

cX,Z ⊗IdY

aX,Y,Z

aZ,X,Y

(X ⊗Y ) ⊗ Z

Z ⊗ (X ⊗Y )

cX,Y ⊗IdZ

IdZ ⊗cX,Y

(Y ⊗ X) ⊗ Z

Z ⊗ (Y ⊗ X)

aY,X,Z

Y ⊗ (X ⊗ Z)

(Z ⊗ X) ⊗Y

IdY ⊗cX,Z

aZ,Y,X

Y ⊗ (Z ⊗ X)

−1 aY,Z,X

(Y ⊗ Z) ⊗ X

cY,Z ⊗IdX

(Z ⊗Y ) ⊗ X

is commutative, for any X,Y, Z objects of C . Proof As the picture suggests, we split the dodecagon diagram into two hexagons and a square. Then the upper dashed arrow is cX⊗Y,Z and the lower dashed arrow is

1.5 (Pre-)Braided Monoidal Categories

35

cY ⊗X,Z , because of (1.5.2). Now, the square diagram is commutative because of the naturality of c, and the proof is finished. In diagrammatic notation, the categorical version of the Yang–Baxter equation can be expressed as X Y

Z

X Y

Z

= Z Y X

,

(1.5.10)

Z Y X

and holds for any objects X,Y and Z of C . We end this section by introducing the concept of a braided monoidal functor. Definition 1.52 Let (C , c) and (D, d) be (pre-)braided categories and (F, ϕ0 , ϕ2 ) : C → D a strong monoidal functor. We call F (pre-)braided monoidal if, for all objects X,Y ∈ C , the following diagram is commutative: F(X) ⊗ F(Y )

ϕ2,X,Y

dF(X),F(Y )

F(Y ) ⊗ F(X)

ϕ2,Y,X

F(X ⊗Y )

(1.5.11)

F(cX,Y )

F(Y ⊗ X).

Example 1.53 If C is a (pre-)braided category then the identity functor IdC is a (pre-)braided monoidal functor. (F,ϕ0 ,ϕ2 )

(G,ψ0 ,ψ2 )

Remark 1.54 If C D E are (pre-)braided monoidal functors then so is GF, considered as a strong monoidal functor as in Remark 1.25. Definition 1.55 Let C , D be (pre-)braided categories. (i) C and D are called isomorphic as (pre-)braided categories if there exist two (pre-)braided monoidal functors (F, ϕ0 , ϕ2 ) : C → D and (G, ψ0 , ψ2 ) : D → C such that FG = IdD and GF = IdC , as (pre-)braided monoidal functors. (ii) C and D are called equivalent as (pre-)braided categories if there exist two (pre-)braided monoidal functors (F, ϕ0 , ϕ2 ) : C → D and (G, ψ0 , ψ2 ) : D → C such that (F, G) gives a monoidal equivalence between C and D. We also say that F (and G as well) is a braided equivalence functor. Similar to the monoidal case, one can prove that two braided categories are isomorphic (resp. equivalent) as braided categories if and only if there exists a braided monoidal functor between them which is also an isomorphism (resp. equivalence) of categories. The case of the isomorphism is clear and we will prove below only the equivalence case. Proposition 1.56 Let (F, ϕ0 , ϕ2 ) : C → D be a strong monoidal functor that provides an equivalence of categories. If C is braided then there exists a unique braiding on D such that F becomes a braided monoidal functor.

36

Monoidal and Braided Categories

Proof Let G : D → C be the equivalence inverse of F and μ : IdD → FG and ν : GF → IdC the required natural monoidal isomorphisms. Assume, moreover, that −1 μF(X) = F(νX ) : FGF(X) → F(X),

(1.5.12)

for all X ∈ C , which is always possible in view of Remark 1.20 and the proof of Proposition 1.31. Any braiding c for C defines a braiding d on D as follows. For any objects U,V of D take dU,V to be the following composition: UV

μU μV

FG(U)FG(V )

ϕ2,G(U),G(V )

F(G(U) ⊗ G(V )) F(cG(U),G(V ) )

dU,V

V U

(1.5.13)

FG(V )FG(U)

μV−1 μU−1

F(G(V ) ⊗ G(U)),

−1 ϕ2,G(V ),G(U)

where  is the tensor product of D. Then (D, d = (dU,V )U,V ∈D ) is a braided category and F : (C , c) → (D, d) becomes a braided monoidal functor. Indeed, by the naturality of ϕ2 and c, the three rectangle diagrams below are commutative: F(X)F(Y )

F(νX−1 )F(νY−1 )

FGF(X)FGF(Y )

ϕ2,GF(X),GF(Y )

F(GF(X) ⊗ GF(Y ))

F(νX )F(νY )

F(X)F(Y )

F(GF(Y ) ⊗ GF(X))

F(GF(X) ⊗ GF(Y )) F(νX ⊗νY )

FGF(Y )FGF(X)

F(νY ⊗νX ) F(cX,Y )

F(X ⊗Y ) ,

ϕ2,X,Y −1 ϕ2,GF(Y ),GF(X)

F(cGF(X),GF(Y ) )

F(X ⊗Y )

F(νX ⊗νY )

F(Y ⊗ X)

F(νY )F(νX ) −1 ϕ2,Y,X

F(Y )F(X) ϕ2,Y,X

F(Y ⊗ X) . Together with (1.5.12) this guarantees that (1.5.11), specialized for our F : (C , c) → (D, d), is commutative. Since c is a braiding for C , it follows that d is a braiding on D, and consequently that F : (C , c) → (D, d) is a braided monoidal functor. We point out that the required commutativity of the two hexagonal diagrams corresponding to d follows from those of c, by using that μ is a monoidal natural isomorphism, the naturality of ϕ2 and its compatibility relations with the monoidal structures on C and D, respectively. We leave the verification of the details to the reader. We prove now the uniqueness part. Indeed, if d  is a braiding on D making F braided monoidal then  −1 dF(X),F(Y ) = ϕ2,Y,X F(cX,Y )ϕ2,X,Y , ∀ X, Y ∈ C .

(1.5.14)

1.5 (Pre-)Braided Monoidal Categories

37

Since d  is a natural transformation and μU , μV are morphisms in D, we get  dU,V

= (1.5.14)

=

 (μV−1 μU−1 )dFG(U),FG(V ) ( μU  μV ) (1.5.13)

−1 (μV−1 μU−1 )ϕ2,G(V ),G(U) F(cG(U),G(V ) )ϕ2,G(U),G(V ) ( μU  μV ) = dU,V ,

for all U,V ∈ D, as desired. Corollary 1.57 Let (C , c) and (D, d) be braided categories. A functor F : C → D is a braided equivalence if and only if F is braided and an equivalence of categories. Proof We have only to prove that if F is braided and an equivalence then F has a monoidal equivalence inverse which is also a braided functor. Indeed, by the uniqueness of d in Proposition 1.56 we have that d is completely determined by (1.5.13). It is enough to show that, with respect to this braiding, the equivalence inverse G of F considered in the proof of Proposition 1.56 is a braided functor, that is, G G G(dU,V )ϕ2,U,V = ϕ2,V,U cG(U),G(V ) , for all U,V ∈ D, where ϕ2G is defined by the relations in (1.3.4). Note that (F, G) provides a monoidal equivalence; see Proposition 1.31. As F is injective on morphisms, the last relation follows from G ) FG(dU,V )F(ϕ2,U,V

(1.3.5)

=

−1 FG(dU,V )μUV (μU−1 μV−1 )ϕ2,G(U),G(V )

=

−1 μV U dU,V (μU−1 μV−1 )ϕ2,G(U),G(V )

(1.5.13)

−1 μV U (μV−1 μU−1 )ϕ2,G(V ),G(U) F(cG(U),G(V ) )

(1.3.5)

G F(ϕ2,V,U )F(cG(U),G(V ) ),

= =

for all U,V ∈ D, where in the second equality we used the fact that μ is a natural transformation and dU,V is a morphism in D. By Proposition 1.56 and Corollary 1.57 we obtain the following. Corollary 1.58 If C is a braided category then so is C str . Thus any braided category is braided equivalent to a strict one. Example 1.59 If (C , c) is a braided category and (C , c) is the reverse braided category associated to it then (IdC , ϕ0 = Id1 , ϕ2 = (cY,X )Y,X∈C ) : (C , c) → (C , c) is a braided monoidal functor. Indeed, for (IdC , ϕ0 = Id1 , ϕ2 = (cY,X )Y,X∈C ) the first diagram in Definition 1.22 is commutative because of (1.5.1), (1.5.2) and Proposition 1.51, while the next two square diagrams in Definition 1.22 are commutative because of Proposition 1.49. Thus (IdC , ϕ0 = Id1 , ϕ2 = (cY,X )Y,X∈C ) is a (strong) monoidal functor. It is also braided since the commutativity of the diagram in Definition 1.52 reduces to the condition cX,Y cY,X = cX,Y cY,X , for any X,Y ∈ C .

38

Monoidal and Braided Categories

1.6 Rigid Monoidal Categories Throughout this section (C , ⊗, 1, a, l, r) is a monoidal category. Our aim is to define the concepts of left and right dual of an object of C . Definition 1.60 Let X,Y be objects of C . A pairing between Y and X is a morphism ε : Y ⊗ X → 1 in C . A copairing in C between Y and X is a morphism η : 1 → X ⊗Y in C . If ε : Y ⊗ X → 1 is a pairing between Y and X then ε˜ : HomC (Z, X ⊗ T ) → HomC (Y ⊗ Z, T ) is the map sending f ∈ HomC (Z, X ⊗ T ) to −1 aY,X,T

Id ⊗ f

ε ⊗Id

l

Y T ε˜ ( f ) : Y ⊗ Z −→ Y ⊗ (X ⊗ T ) −→ (Y ⊗ X) ⊗ T −→T 1 ⊗ T → T.

Definition 1.61 The pairing ε : Y ⊗ X → 1 is called exact if the associated map ε˜ is bijective, for all Z, T objects of C . We next see that an exact pairing is a pairing for which there exists a copairing that is compatible with it in the following sense. Proposition 1.62 A pairing ε : Y ⊗ X → 1 is exact if and only if there exists a copairing η : 1 → X ⊗Y such that the following diagrams are commutative: IdY ⊗η

Y ⊗1

Y ⊗ (X ⊗Y )

(1.6.1)

−1 aY,X,Y

rY lY

Y

1 ⊗Y

ε ⊗IdY

η ⊗IdX

1⊗X

(Y ⊗ X) ⊗Y, (X ⊗Y ) ⊗ X

(1.6.2)

aX,Y,X

lX rX

X

X ⊗1

IdX ⊗ε

X ⊗ (Y ⊗ X).

Moreover, if ε is exact, the copairing η for which the diagrams (1.6.1) and (1.6.2) are commutative is unique. Proof If ε is exact then for Z = 1 and T = Y let η : 1 → X ⊗ Y be the morphism in C satisfying ε˜ (η ) = rY ; this is exactly the condition that the diagram in (1.6.1) is commutative (in particular, this proves the uniqueness part in the statement). Let now Z, T be arbitrary objects of C . The morphism η induces a map η˜ : HomC (Y ⊗ Z, T ) → HomC (Z, X ⊗ T ), defined for any morphism g : Y ⊗ Z → T by

η˜ (g) : Z

lZ−1

- 1⊗Z

η ⊗IdZ

- (X ⊗Y ) ⊗ Z

aX,Y,Z

- X ⊗ (Y ⊗ Z)

IdX ⊗g

- X ⊗ T. (1.6.3)

We show that ε˜ and η˜ are inverses. It is enough to prove that ε˜ η˜ = IdHomC (Y ⊗Z,T ) . g Indeed, for any Y ⊗ Z - T we have −1 ε˜ η˜ (g) = lT (ε ⊗ IdT )aY,X,T (IdY ⊗ (IdX ⊗ g)aX,Y,Z (η ⊗ IdZ )lZ−1 )

1.6 Rigid Monoidal Categories

39

−1 −1 = lT (ε ⊗ IdT )(IdY ⊗X ⊗ g)aY,X,Y ⊗Z (IdY ⊗ aX,Y,Z (η ⊗ IdZ )lZ ) −1 −1 = lT (Id1 ⊗ g)(ε ⊗ IdY ⊗Z )aY,X,Y ⊗Z (IdY ⊗ aX,Y,Z (η ⊗ IdZ )lZ ) −1 −1 = glY ⊗Z (ε ⊗ IdY ⊗Z )aY,X,Y ⊗Z (IdY ⊗ aX,Y,Z (η ⊗ IdZ )lZ ) −1 −1 ⊗ IdZ )aY,X⊗Y,Z (IdY ⊗ (η ⊗ IdZ )lZ−1 ) = glY ⊗Z (ε ⊗ IdY ⊗Z )aY ⊗X,Y,Z (aY,X,Y −1 −1 ⊗ IdZ )aY,X⊗Y,Z (IdY ⊗ (η ⊗ IdZ )lZ−1 ) = glY ⊗Z a1,Y,Z ((ε ⊗ IdY ) ⊗ IdZ )(aY,X,Y −1 −1 (IdY ⊗ η ) ⊗ IdZ )aY,1,Z (IdY ⊗ lZ−1 ) = g(lY (ε ⊗ IdY )aY,X,Y −1 (IdY ⊗ η )rY−1 ⊗ IdZ ) = g(lY (ε ⊗ IdY )aY,X,Y

= g(IdY ⊗ IdZ ) = g, where we used: the naturality of a in the second equality, the naturality of l in the fourth equality, (1.1.1) in the fifth equality, again the naturality of a in the sixth equality, Proposition 1.5 and the naturality of a in the seventh equality, (1.1.2) in the penultimate equality, and (1.6.1) in the last one. Since ε˜ is bijective we also have η˜ ε˜ = IdHomC (Z,X⊗T ) . In particular, for Z = X,

ε ( f ) = f , which means T = 1 and f = rX−1 : X → X ⊗ 1 we have η −1 rX−1 = (IdX ⊗ l1 (ε ⊗ Id1 )aY,X,1 (IdY ⊗ rX−1 ))aX,Y,X (η ⊗ IdX )lX−1 .

From Proposition 1.5 and since r is a natural transformation we have −1 (IdY ⊗ rX−1 ) = l1 (ε ⊗ Id1 )rY−1 l1 (ε ⊗ Id1 )aY,X,1 ⊗X = ε ,

and therefore rX−1 = (IdX ⊗ ε )aX,Y,X (η ⊗ IdX )lX−1 , which means that the diagram in (1.6.2) is commutative. Conversely, assume that there exists a copairing η : 1 → X ⊗ Y such that (1.6.1) and (1.6.2) hold, and define η˜ as in (1.6.3). From the above computation we know that ε˜ η˜ = IdHomC (Z,X⊗T ) . Thus we only have to show that η˜ ε˜ = IdHomC (Z,X⊗T ) . For any f ∈ HomC (Z, X ⊗ T ) we compute: −1 η˜ ε˜ ( f ) = (IdX ⊗ lT (ε ⊗ IdT )aY,X,T (IdY ⊗ f ))aX,Y,Z (η ⊗ IdZ )lZ−1 −1 = (IdX ⊗ lT (ε ⊗ IdT )aY,X,T )aX,Y,X⊗T (IdX⊗Y ⊗ f )(η ⊗ IdZ )lZ−1 −1 = (IdX ⊗ lT (ε ⊗ IdT )aY,X,T )aX,Y,X⊗T (η ⊗ IdX⊗T )(Id1 ⊗ f )lZ−1 −1 = (IdX ⊗ lT (ε ⊗ IdT ))aX,Y ⊗X,T (aX,Y,X ⊗ IdT )a−1 X⊗Y,X,T (η ⊗ IdX⊗T )lX⊗T f −1 = (IdX ⊗ lT )aX,1,T ((IdX ⊗ ε )aX,Y,X ⊗ IdT )a−1 X⊗Y,X,T (η ⊗ IdX⊗T )lX⊗T f −1 = (rX (IdX ⊗ ε )aX,Y,X (η ⊗ IdX ) ⊗ IdT )a−1 1,X,T lX⊗T f

= (rX (IdX ⊗ ε )aX,Y,X (η ⊗ IdX )lX−1 ⊗ IdT ) f = (IdX ⊗ IdT ) f = f , as required. We used the naturality of a in the second and fifth equality, the naturality of l and (1.1.1) in the fourth equality, the naturality of a and (1.1.2) in the sixth equality, Proposition 1.5 in the last but one equality, and (1.6.2) in the last one. Definition 1.63

Let ε : Y ⊗ X → 1 be an exact pairing and η : 1 → X ⊗ Y the

40

Monoidal and Braided Categories

copairing satisfying (1.6.1) and (1.6.2). In this case we say that (η , ε ) is an adjunction between Y and X, and denote this fact by (η , ε ) : Y  X. Moreover, we call Y a left dual to X and X a right dual to Y . A monoidal category is called left (resp. right) rigid if any object X of C has a left (resp. right) dual. We call C rigid if it is left and right rigid. If C is a left rigid monoidal category then the left dual of X ∈ C will be denoted by X ∗ , and the corresponding adjunction will be denoted by (coevX , evX ) : X ∗  X. Thus evX : X ∗ ⊗ X → 1 and coevX : 1 → X ⊗ X ∗ are morphisms in C (called evaluation, respectively coevaluation) such that rX ◦ (IdX ⊗ evX ) ◦ aX,X ∗ ,X ◦ (coevX ⊗ IdX ) ◦ lX−1 = IdX , lX ∗ ◦ (evX ⊗ IdX ∗ ) ◦ a−1 X ∗ ,X,X ∗

◦ (IdX ∗ ⊗ coevX ) ◦ rX−1∗

= IdX ∗ .

X∗ X

1

1

X X∗

(1.6.4) (1.6.5)

 In this situation we denote evX = and coevX = . Hence, when C is strict monoidal the following relations hold: X



X∗

X

=



 X ∗ = .



and X

X

X∗

(1.6.6)

X∗

Similarly, C is right rigid if for any X ∈ C there exist an object ∗ X ∈ C and morphisms evX : X ⊗ ∗ X → 1 and coevX : 1 → ∗ X ⊗ X such that  −1 lX ◦ (evX ⊗ IdX ) ◦ a−1 X,∗ X,X ◦ (IdX ⊗ coevX ) ◦ rX = IdX ,

r∗ X ◦ (Id∗ X ⊗ evX ) ◦ a∗ X,X,∗ X

◦ (coevX X

In what follows we will denote

evX

⊗ Id∗ X ) ◦ l∗−1 X

∗X

1

1 ∗X

X

X

X

(1.6.8)

 • . Then if C is

:= and coevX := •

strict the relations (1.6.7) and (1.6.8) above can be written as X  • =

•

= Id∗ X .

(1.6.7)

 •

=

•

and ∗X

∗X

X

∗X

,

(1.6.9)

∗X

respectively. Thus a right dual for X in C is nothing else than a left dual for X in C , the reverse monoidal category associated to C . Example 1.64 For any field k the category k M fd of finite-dimensional k-vector spaces is rigid monoidal. Proof

For any k-vector space V denote by V ∗ = Homk (V, k) the linear dual space

1.6 Rigid Monoidal Categories

41

of V . If, moreover, V is finite dimensional then for a basis {vi }i of V consider its dual basis {vi }i in V ∗ ; thus vi (v j ) = δi, j ,

∑ v∗ (vi )vi = v∗

and

i

for all

v∗

∈ V ∗,

∑ vi (v)vi = v, i

v ∈ V , where δi, j is the Kronecker delta symbol: 1 if i = j δi, j = 0 otherwise.

Then it can be easily seen that V ∗ together with evV : V ∗ ⊗ V → k, evV (v∗ ⊗ v) = v∗ (v), for all v∗ ∈ V ∗ and v ∈ V , and with coevV : k → V ⊗V ∗ defined by coevV (1) = ∑i vi ⊗ vi , is a left dual object of V . Similarly, a right dual for V is the same vector space V ∗ coming now with the linear maps evV : V ⊗ V ∗ v ⊗ v∗ → v∗ (v) ∈ k and coevV : k → V ∗ ⊗ V given by coevV (1) = ∑i vi ⊗ vi . The verification of all these details is left to the reader. Example 1.65 Let G be a group and φ ∈ H 3 (G, k∗ ) a normalized 3-cocycle on G with coefficients in k∗ , k a field. Then vectG φ , the category of finite-dimensional G-graded vector spaces, endowed with the monoidal structure of VectG φ from Proposition 1.13, is a rigid monoidal category. Proof If V is a finite-dimensional G-graded vector space with a basis {i v}i and dual basis {i v}i in V ∗ , the left dual of V is V ∗ = Homk (V, k), the vector space of k-linear maps from V to k, with homogeneous component of degree g ∈ G defined by      −1 Vg∗ = v → v∗ (vg ) | v∗ ∈ V ∗ = v∗ ∈ V ∗ | v∗|Vσ = 0, ∀ σ = g−1 , −1

where by vg we have denoted the component of degree g−1 of v ∈ V . The evaluation and coevaluation maps are respectively given by evV : V ∗ ⊗V → k, evV (v∗ ⊗ v) = v∗ (v), ∀ v∗ ∈ V ∗ , v ∈ V, coevV : k → V ⊗V ∗ , coevV (1) =

∑

φ (g, g−1 , g)−1 (i v)g ⊗ i v.

i;g∈G ∗V

The right dual of V coincides, as a G-graded vector space, with V ∗ described above, but the evaluation and coevaluation maps are now respectively defined by evV : V ⊗ ∗V → k, evV (v ⊗ ∗ v) = ∗ v(v), ∀ ∗ v ∈ ∗V, v ∈ V, coevV : k → ∗V ⊗V, coevV (1) =

∑

φ (g−1 , g, g−1 )−1 i v ⊗ (i v)g .

i;g∈G

For the moment we leave the verification of all these details to the reader. Later on we will see that this result follows from Corollary 3.53. We show that the left/right dual object, if it exists, is unique up to isomorphism. Proposition 1.66 In a left/right rigid monoidal category C a left/right dual of an object is unique up to an isomorphism in C .

42

Monoidal and Braided Categories

Proof Let Y and X ∗ be left duals for X, with notation as before for ε , η , evX , coevX . One can check directly that g : X ∗ → Y defined by the composition r−1∗

a−1 X ∗ ,X,Y

Id ∗ ⊗η

ev ⊗Id

l

X X Y Y X X ∗ ⊗ 1 −→ X ∗ ⊗ (X ⊗Y ) −→ (X ∗ ⊗ X) ⊗Y −→ 1 ⊗Y → Y X ∗ −→

is an isomorphism in C with inverse defined by the composition r−1

Y Y ⊗1 Y →

IdY ⊗coevX

−→

−1 aY,X,X ∗

ε ⊗Id

∗

l ∗

X Y ⊗ (X ⊗ X ∗ ) −→ (Y ⊗ X) ⊗ X ∗ −→X 1 ⊗ X ∗ → X ∗.

1 Y X  ◦ For instance, if we assume C strict and denote ε = and η = , then ◦ 1 X Y X∗

Y    ◦ ◦ = IdX ∗ and gg−1 = = IdY . g−1 g =





◦ ◦ X∗ Y The right-handed version is similar; the details are left to the reader. Strong monoidal functors preserve adjunctions, and consequently dual objects. Proposition 1.67 For (F, ϕ0 , ϕ2 ) : C → D a strong monoidal functor and (η , ε ) : Y  X an adjunction in C , (η F , ε F ) : F(Y )  F(X) is an adjunction in D, where ϕ2,Y,X

F(ε )

ϕ −1

0 ε F : F(Y ) ⊗ F(X) −→ F(Y ⊗ X) −→ F(1) −→ 1,

ϕ

F(η )

−1 ϕ2,X,Y

0 η F : 1 −→ F(1) −→ F(X ⊗Y ) −→ F(X) ⊗ F(Y ).

Consequently, if C is left rigid and (G, ψ0 , ψ2 ) : C → D is a strong monoidal functor, any monoidal natural transformation ω : F → G is a monoidal natural isomorphism. Proof

By the commutativity of the diagrams in Definition 1.22 we have F −1 lF(Y ) (ε F ⊗ IdF(Y ) )a−1 F(Y ),F(X),F(Y ) (IdF(Y ) ⊗ η )rF(Y )

= F(lY )ϕ2,1,Y (F(ε )ϕ2,Y,X ⊗ IdF(Y ) ) −1 −1 −1 a−1 F(Y ),F(X),F(Y ) (IdF(Y ) ⊗ ϕ2,X,Y F(η ))ϕ2,Y,1 F(rY ) −1 −1 = F(lY )ϕ2,1,Y (F(ε ) ⊗ F(IdY ))ϕ2,Y ⊗X,Y F(aY,X,Y ) −1 ϕ2,Y,X⊗Y (F(IdY ) ⊗ F(η ))ϕ2,Y,1 F(rY−1 ) −1 (IdY ⊗ η )rY−1 ) = F(IdY ) = IdF(Y ) , = F(lY (ε ⊗ IdY )aY,X,Y

where in the third equality we used the naturality of ϕ2 twice. The corresponding equality for (1.6.2) can be proved in a similar manner, so it is left to the reader. Assume now that C is left rigid and take ω : F → G a monoidal natural transformation. We claim that, for any object X of C , ωX is an isomorphism in D with −1 lG(X)

ωX−1 : G(X) −→ 1 ⊗ G(X)

η F ⊗IdG(X)

−→

(F(X) ⊗ F(Y )) ⊗ G(X)

(IdF(X) ⊗ωY )⊗IdG(X)

−→

1.7 The Left and Right Dual Functors (F(X) ⊗ G(Y )) ⊗ G(X) IdF(X) ⊗ε G

−→

aF(X),G(Y ),G(X)

−→

43

F(X) ⊗ (G(Y ) ⊗ G(X))

rF(X)

F(X) ⊗ 1 −→ F(X),

where (η , ε ) : Y  X is an adjunction in C . Indeed, we have −1 ωX ωX−1 = ωX rF(X) (IdF(X) ⊗ ε G )aF(X),G(Y ),G(X) ((IdF(X) ⊗ ωY )η F ⊗ IdG(X) )lG(X) −1 = rG(X) (ωX ⊗ ε G )aF(X),G(Y ),G(X) ((IdF(X) ⊗ ωY )η F ⊗ IdG(X) )lG(X)

= rG(X) (IdG(X) ⊗ ε G )(ωX ⊗ IdG(Y )⊗G(X) )aF(X),G(Y ),G(X) −1 ((IdF(X) ⊗ ωY )η F ⊗ IdG(X) )lG(X) −1 = rG(X) (IdG(X) ⊗ ε G )aG(X),G(Y ),G(X) ((ωX ⊗ ωY )η F ⊗ IdG(X) )lG(X) −1 −1 = rG(X) (IdG(X) ⊗ ε G )aG(X),G(Y ),G(X) (ψ2,X,Y ωX⊗Y F(η )ϕ0 ⊗ IdG(X) )lG(X) −1 −1 = rG(X) (IdG(X) ⊗ ε G )aG(X),G(Y ),G(X) (ψ2,X,Y G(η )ω1 ϕ0 ⊗ IdG(X) )lG(X) (1.6.2)

−1 = rG(X) (IdG(X) ⊗ ε G )aG(X),G(Y ),G(X) (η G ⊗ IdG(X) )lG(X) = IdG(X) ,

where we also used: the naturality of r in the second equality, the naturality of a in the fourth equality, the commutativity of the square diagram in Definition 1.29 in the fifth equality, the naturality of ω in the sixth equality, the commutativity of the triangle diagram in Definition 1.29 and the definition of η G in the last but one equality. Similarly, ωX−1 ωX = IdF(X) . So the proof is finished.

1.7 The Left and Right Dual Functors Let C be a monoidal category which is left/right rigid. Using the existence of left/right dual objects in C we will construct strong monoidal functors (−)∗ , ∗ (−) : C → C opp . Recall that C opp is our notation for the reverse of the opposite category associated to C , see Proposition 1.3 and Remark 1.4. Let us start by defining explicitly the two functors (−)∗ , ∗ (−) : C → C opp . Proposition 1.68 Let C be a left (resp. right) rigid monoidal category and for any object X of C consider a left (resp. right) dual object X ∗ (resp. ∗ X) for X. Then (−)∗ : C ∈ X → X ∗ ∈ C opp (resp. ∗ (−) : C ∈ X → ∗ X ∈ C opp ) defines a functor. Proof

We prove that F := (−)∗ is a functor. For f : X → Y morphism in C , let  r−1 ∗

Y f ∗ := Y ∗ → Y∗ ⊗1

aY−1 ∗ ,Y,X ∗

∗

IdY ∗ ⊗coevX

−→

−→ (Y ⊗Y ) ⊗ X

Y ∗ ⊗ (X ⊗ X ∗ )

∗ evY ⊗IdX ∗

IdY ∗ ⊗( f ⊗IdX ∗ )

−→

 ∗ lX ∗

−→ 1 ⊗ X → X

∗

.

Y ∗ ⊗ (Y ⊗ X ∗ )

44

Monoidal and Braided Categories

If we take F( f ) = f ∗ , F is a functor. To see this, assume C strict monoidal. Then Y∗

Y∗

Y∗

    gh fh (1.6.6) fh f∗ = , and so g∗ ◦ f ∗ = = = ( f ◦ g)∗ , h g fh







X∗

Z∗

Z∗

g f - X - Y in C . Likewise, (IdX )∗ = IdX ∗ . Thus, F is for any morphisms Z indeed a functor from C to C opp , as claimed. The right-handed version in the statement follows from the left-handed one by replacing C with the reverse category C ; the details are left to the reader.

We call f ∗ the left transpose of f in C . In the right-handed version the (right) ∗Y

 • fh : ∗Y → ∗ X. transpose of a morphism f : X → Y in C is ∗ f :=

• ∗X

We focus now on the strong monoidal structures of ∗ (−) and (−)∗ . Proposition 1.69 Let C be a monoidal category and X,Y objects of C that admit left duals. Then Y ∗ ⊗ X ∗ is a left dual for X ⊗Y . Proof For simplicity assume that C is strict monoidal. If η : 1 → X ⊗Y ⊗Y ∗ ⊗ X ∗ and ε : Y ∗ ⊗ X ∗ ⊗ X ⊗Y → 1 are defined by coev

η : 1 →X X ⊗ X ∗

IdX ⊗coevY ⊗IdX ∗

ε : Y ∗ ⊗ X ∗ ⊗ X ⊗Y

−→

IdY ∗ ⊗evX ⊗IdY

−→

X ⊗Y ⊗Y ∗ ⊗ X ∗ , ev

Y ∗ ⊗Y →Y 1,

then from the definition of a left dual it follows that (η , ε ) : Y ∗ ⊗ X ∗  X ⊗ Y is an adjunction in C . Thus Y ∗ ⊗ X ∗ is a left dual for X ⊗Y . Corollary 1.70 Let C be a left rigid monoidal category. Then for any pair of objects X,Y of C we have (X ⊗Y )∗ ∼ = Y ∗ ⊗ X ∗ as objects of C . Proof Since C is left rigid there exists an adjunction of the form (coevX⊗Y , evX⊗Y ) : (X ⊗ Y )∗  X ⊗ Y in C . If (η , ε ) : Y ∗ ⊗ X ∗  X ⊗ Y is the adjunction considered in Proposition 1.69 then, by Proposition 1.66, we get that (X ⊗Y )∗ ∼ = Y ∗ ⊗X ∗ as objects of C . Moreover, the isomorphism is produced by λX,Y : (X ⊗Y )∗ → Y ∗ ⊗ X ∗ , which in the case when C is strict monoidal reads

λX,Y = (evX⊗Y ⊗ IdY ∗ ⊗X ∗ ) ◦ (Id(X⊗Y )∗ ⊗ (IdX ⊗ coevY ⊗ IdX ∗ ) ◦ coevX ).

(1.7.1)

−1 Its inverse is defined by λX,Y : Y ∗ ⊗ X ∗ → (X ⊗Y )∗ , −1 λX,Y = (evY ◦ (IdY ∗ ⊗ evX ⊗ IdY ) ⊗ Id(X⊗Y )∗ ) ◦ (IdY ∗ ⊗X ∗ ⊗ coevX⊗Y ). −1 A direct computation shows also directly that λX,Y and λX,Y are inverses.

(1.7.2)

1.7 The Left and Right Dual Functors

45

If X1 , . . . , Xn are objects of a left rigid (strict) monoidal category, in diagrammatic notation we denote the evaluation and coevaluation morphisms of X1 ⊗ · · · ⊗ Xn by (X ⊗ · · · ⊗ Xn )∗

X1

···

Xn

···

evX1 ⊗···⊗Xn =

,

X1 ⊗···⊗Xn



1 1

 coevX1 ⊗···⊗Xn =

,

X1 ⊗···⊗Xn ···

X1

···

(X1 ⊗ · · · ⊗ Xn )∗

Xn

respectively. We adopt a similar notation for the case when C is right rigid monoidal. With this notation the morphism λ in (1.7.1) and its inverse take the form (X ⊗Y )∗

λX,Y =

Y ∗ X∗

  −1 , λX,Y =

X⊗Y

 Y ∗ X∗

 X⊗Y

.



(X ⊗Y )∗

 When C is right rigid there exists λX,Y : ∗ (X ⊗Y ) → ∗Y ⊗ ∗ X, an isomorphism in C : ∗ (X

 λX,Y =

•  • X⊗Y

⊗Y )

 •

∗X

∗X

X⊗Y

−1 with inverse λX,Y =

.

• •



• ∗Y

∗Y

∗ (X

⊗Y )

With the help of these isomorphisms in C the functors constructed in Proposition 1.68 become strong monoidal. To see this, observe first that C opp = C

opp

= (C opp , ⊗ ◦ τ , 1, (aZ,Y,X )X,Y,Z∈C , r−1 , l −1 )

for any monoidal category (C , ⊗, 1, a, l, r). Then we have the following. Proposition 1.71 Let C be a left, respectively right, rigid monoidal category. Then the functor (−)∗ , respectively ∗ (−), defined in Proposition 1.68 is strong monoidal if it is viewed as a functor from C to C opp . Proof We prove only the statement for (−)∗ ; the proof for ∗ (−) is similar. By (1.1.2) and Proposition 1.5 it follows that 1 is a self-dual object of C , that is, 1 is a left dual for itself. The adjunction (η , ε ) : 1  1 is given by ε = l1 = r1

46

Monoidal and Braided Categories

and η = l1−1 = r1−1 , respectively. So there exists ϕ0 : 1∗ → 1 an isomorphism in C . Concretely, according to Proposition 1.66 we have −1 −1 ∗ ϕ0 = l1 (ev1 ⊗ Id1 )a−1 1∗ ,1,1 (Id1 ⊗ l1 )r1∗ −1 −1 −1 −1 = l1 (ev1 ⊗ Id1 )(r1−1 ∗ ⊗ Id1 )r1∗ = l1 r1 ev1 r1∗ = ev1 r1∗ ,

where we used (1.1.2) in the second equality, the naturality of r and (1.1.6) in the third equality and Proposition 1.5 in the fourth equality. We claim that ((−)∗ , ϕ0 , ϕ2 = (λX,Y )X,Y ∈C ) is a strong monoidal functor from C to C opp . We assume, as usual, C strict monoidal and notice that the commutativity of the two square diagrams in Definition 1.22, specialized to our situation, reads (IdX ∗ ⊗ ϕ0 )λ1,X lX∗ = rX−1∗ and (ϕ0 ⊗ IdX ∗ )λX,1 rX∗ = lX−1∗ . We check only the second equality; the first one is similar. For this, note that

λX,1 rX∗ = λX,1 l(X⊗1)∗ (evX ⊗ Id(X⊗1)∗ )a−1 X ∗ ,X,(X⊗1)∗

 IdX ∗ ⊗ (rX ⊗ Id(X⊗1)∗ )coevX⊗1 rX−1∗ = l1∗ ⊗X ∗ (Id1 ⊗ λX,1 )(evX ⊗ Id(X⊗1)∗ )a−1 X ∗ ,X,(X⊗1)∗

 −1 IdX ∗ ⊗ (rX ⊗ Id(X⊗1)∗ )coevX⊗1 rX ∗ , because of the naturality of l, and therefore (ϕ0 ⊗ IdX ∗ )λX,1 rX∗ = l1⊗X ∗ (Id1 ⊗ (ϕ0 ⊗ IdX ∗ )λX,1 )(evX ⊗ Id(X⊗1)∗ )a−1 X ∗ ,X,(X⊗1)∗

 −1 IdX ∗ ⊗ (rX ⊗ Id(X⊗1)∗ )coevX⊗1 rX ∗ = l1⊗X ∗ (evX ⊗ Id1⊗X ∗ )(IdX ∗ ⊗X ⊗ (ϕ0 ⊗ IdX ∗ )λX,1 )

 −1 ∗ a−1 X ∗ ,X,(X⊗1)∗ IdX ⊗ (rX ⊗ Id(X⊗1)∗ )coevX⊗1 rX ∗ = l1⊗X ∗ (evX ⊗ Id1⊗X ∗ )a−1 X ∗ ,X,1⊗X ∗

 IdX ∗ ⊗ (IdX ⊗ (ϕ0 ⊗ IdX ∗ )λX,1 )(rX ⊗ Id(X⊗1)∗ )coevX⊗1 rX−1∗ = l1⊗X ∗ (evX ⊗ Id1⊗X ∗ )a−1 X ∗ ,X,1⊗X ∗ (IdX ∗ ⊗ (rX ⊗ Id1⊗X ∗ )(IdX⊗1 ⊗ (ϕ0 ⊗ IdX ∗ )λX,1 )coevX⊗1 ) rX−1∗ . We used the naturality of l (resp. a) in the first (resp. third) equality. 1

−1 , it follows that coevX⊗Y = Now, from the formula of λX,Y

  −1 λX,Y

X Y

, for

(X ⊗Y )∗

any objects X and Y of C . When C is arbitrary the above formula takes the form −1 −1 coevX⊗Y = (IdX⊗Y ⊗ λX,Y )aX⊗Y,Y ∗ ,X ∗ (a−1 X,Y,Y ∗ (IdX ⊗ coevY )rX ⊗ IdX ∗ )coevX .

1.7 The Left and Right Dual Functors

47

Thus the last computation continues as follows: (ϕ0 ⊗ IdX ∗ )λX,1 rX∗

 = l1⊗X ∗ (evX ⊗ Id1⊗X ∗ )a−1 X ∗ ,X,1⊗X ∗ IdX ∗ ⊗ (rX ⊗ Id1⊗X ∗ )(IdX⊗1 ⊗ (ϕ0 ⊗ IdX ∗ )) 

 −1 −1 ∗ coevX r ∗ aX⊗1,1∗ ,X ∗ a−1 (Id ⊗ coev )r ⊗ Id ∗ X X 1 X X X,1,1  = l1⊗X ∗ (evX ⊗ Id1⊗X ∗ )a−1 X ∗ ,X,1⊗X ∗ IdX ∗ ⊗ (rX ⊗ Id1⊗X ∗ )aX⊗1,1,X ∗    −1 −1 ∗ coevX r ∗ (IdX⊗1 ⊗ ϕ0 )a−1 X X,1,1∗ (IdX ⊗ coev1 )rX ⊗ IdX  = l1⊗X ∗ (evX ⊗ Id1⊗X ∗ )a−1 ∗ ∗ X ,X,1⊗X IdX ∗ ⊗ (rX ⊗ Id1⊗X ∗ )aX⊗1,1,X ∗    −1 −1 ∗ a−1 coev (Id ⊗ (Id ⊗ ϕ )coev )r ⊗ Id X X X rX ∗ 1 0 1 X X,1,1

−1 = l1⊗X ∗ (evX ⊗ Id1⊗X ∗ )a−1 X ∗ ,X,1⊗X ∗ (IdX ∗ ⊗ aX,1,X ∗ (rX ⊗ Id1 )aX,1,1   (IdX ⊗ (Id1 ⊗ ϕ0 )coev1 )rX−1 ⊗ IdX ∗ coevX rX−1∗



= l1⊗X ∗ (evX ⊗ Id1⊗X ∗ )a−1 X ∗ ,X,1⊗X ∗ IdX ∗ ⊗ aX,1,X ∗ (IdX ⊗ l1 (Id1 ⊗ ev1 )   −1 −1 −1 a1,1∗ ,1 r1⊗1 ∗ coev1 )rX ⊗ IdX ∗ coevX rX ∗

 −1 −1 = l1⊗X ∗ (evX ⊗ Id1⊗X ∗ )a−1 X ∗ ,X,1⊗X ∗ IdX ∗ ⊗ aX,1,X ∗ (rX ⊗ IdX ∗ )coevX rX ∗

 −1 −1 = l1⊗X ∗ (evX ⊗ Id1⊗X ∗ )a−1 X ∗ ,X,1⊗X ∗ IdX ∗ ⊗ (IdX ⊗ lX ∗ )coevX rX ∗ −1 = l1⊗X ∗ (evX ⊗ Id1⊗X ∗ )(IdX ∗ ⊗X ⊗ lX−1∗ )a−1 X ∗ ,X,X ∗ (IdX ∗ ⊗ coevX )rX ∗ −1 = l1⊗X ∗ (Id1 ⊗ lX−1∗ )(evX ⊗ IdX ∗ )a−1 X ∗ ,X,X ∗ (IdX ∗ ⊗ coevX )rX ∗

= l1⊗X ∗ (Id1 ⊗ lX−1∗ )lX−1∗ = lX−1∗ , where we applied the naturality of a in the second, third, fourth and eighth equality, (1.1.2) and Proposition 1.5 in the fifth equality, the naturality of r and (1.6.4) in the sixth equality, (1.1.2) in the seventh equality, that ⊗ is a functor in the ninth equality, (1.6.5) in the penultimate equality, and the naturality of l in the last one. Finally, the commutativity of the first diagram in (1.3.2) is equivalent to (Z ⊗Y ⊗ X)∗

Y ⊗X

=

Y ⊗X

 Z⊗Y ⊗X



(Z ⊗Y ⊗ X)∗

   



Z⊗Y

   Z⊗Y , 

Z⊗Y ⊗X



X ∗ Y ∗ Z∗

X∗

Y ∗ Z∗

for all objects X, Y , Z of C . The above equality is true because of (1.6.6). Definition 1.72 A monoidal category is called sovereign if it is left and right rigid such that the corresponding left and right dual functors (−)∗ , ∗ (−) : C → C opp are equal as strong monoidal functors.

48

Monoidal and Braided Categories

The next result presents some canonical isomorphisms between an object and its “double” duals. Proposition 1.73 Let C be a monoidal category, X an object of C and assume that X ∗ is a left, and ∗ X is a right, dual object of X in C . If X ∗ has a right dual object and ∗ X has a left dual object in C , then X ∼ = ∗ (X ∗ ). = (∗ X)∗ ∼ Proof

Consider θX : X → (∗ X)∗ defined by the following composition: r−1

X θX : X −→ X ⊗1

a−1 X,∗ X,(∗ X)∗

−→

IdX ⊗coev∗ X

−→

X ⊗ (∗ X ⊗ (∗ X)∗ )

(X ⊗ ∗ X) ⊗ (∗ X)∗

evX ⊗Id(∗ X)∗

−→

l(∗ X)∗

1 ⊗ (∗ X)∗ −→ (∗ X)∗ .

(1.7.3)

We claim that θX is an isomorphism in C with inverse given by r(−1 ∗ X)∗

θX−1 : (∗ X)∗ −→ (∗ X)∗ ⊗ 1 a−1 (∗ X)∗ ,∗ X,X

−→

Id(∗ X)∗ ⊗coevX

−→

((∗ X)∗ ⊗ ∗ X) ⊗ X X

(∗ X)∗ ⊗ (∗ X ⊗ X)

ev∗ X ⊗IdX

−→



l

X 1⊗X → X.

(1.7.4)

(∗ X)∗

and θX−1 = In graphical notation this means that θX = •

 • , where,



(∗ X)∗

X

as before, in order to avoid any possible confusion, for the evaluation and coevaluation morphisms of the left dual object of ∗ X in C we kept the standard notation and in the notation for evX and coevX we added a black dot. From (1.6.6) it is clear at this point that θX and θX−1 are indeed inverse to each other, as we claimed. In the same way one can construct an isomorphism θX : X → ∗ (X ∗ ).

1.8 Braided Rigid Monoidal Categories This section deals with braidings and duals. As we shall see, some conditions in the definition of a braided, or rigid, monoidal category are automatic. In this direction a first result is the following. Proposition 1.74 Let C be a braided category. If C is left or right rigid monoidal then it is rigid monoidal. Proof As we have seen before, C is left rigid if and only if its reverse monoidal category C is right rigid. Since C is braided as well, it suffices to prove only the left-handed version in the statement. Let C be a left rigid braided category. By Examples 1.59 the functor (IdC , ϕ0 = Id1 , ϕ2 = (cY,X )Y,X∈C ) : (C , c) → (C , c) is a braided monoidal functor. Thus, it carries left duals to left duals; see Proposition 1.67. In our situation this means that an

1.8 Braided Rigid Monoidal Categories

49

adjunction (coevX , evX ) : X ∗  X is mapped to an adjunction (coevX , evX ) : X ∗  X in C , that is, to an adjunction (coevX , evX ) : X  X ∗ in C . Thus, X ∗ together with X X∗

evX =

1



and



coevX =

X∗ X

1

is a right dual for X in C , so the proof is finished. Remark 1.75 Since the definition of rigidity is independent of the choice of the braiding it follows that X ∗ , the left dual of an object X of a left rigid braided category, is a right dual as well if it is considered together with coevX = cX,X ∗ ◦ coevX and evX = evX ◦ c−1 X ∗ ,X . Moreover, the isomorphism constructed in Proposition 1.66 between X ∗ and X ∗ viewed as a right dual of X in the two different ways presented above is just IdX ∗ , and this is because X∗

X∗



=

X∗

 X ∗

=

X∗



X∗

=





(1.6.6)



X∗

=

. X∗

X∗

X∗

We used in the first two equalities the naturality of c and Proposition 1.49. Corollary 1.76 If C is a braided rigid monoidal category then X ∗ ∼ = ∗ X for any ∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∼ ∼ ∼ ∼ object X of C . Consequently, X := (X ) = (X ) = X = ( X) = (∗ X) := ∗∗ X. Proof This result follows from Propositions 1.66, 1.74 and 1.73. Note that ΘX : ∗ ∗ X ∗ → ∗ X and Θ−1 X : X → X defined by X∗

 • ΘX =

X∗

=

∗X

∗X

 •

 and

∗X

Θ−1 X =

∗X



=



• • X∗

X∗

are inverses of each other. The black dots are used to distinguish the evaluation and coevaluation morphisms of the right dual of X from the ones of the left dual of X. By the above remark, another pair of inverse isomorphisms can be obtained by replacing in the above definitions of ΘX and Θ−1 X the braiding of C with its inverse, that is, by thinking of everything in terms of C in instead of C . We denote these by ΘX and Θ−1 X , respectively. The next result shows that the condition of invertibility for a braiding in a left rigid monoidal category is redundant. Theorem 1.77 A pre-braided left rigid monoidal category is braided, and so rigid monoidal as well.

50

Monoidal and Braided Categories

Proof Let C be a pre-braided left rigid monoidal category. If c is the pre-braiding of C we claim that c is invertible with inverse given, for V , W objects of C , by W V

 −1 = cV,W

,



(1.8.1)

V W

where the evaluation and coevaluation morphisms are those of V . To prove this assertion observe first that V∗ V W

W



V∗ V W

=

W

W

=

and

 ,

W V V∗

W

W V

V∗

relations that follow from (1.5.3), the naturality of c and Proposition 1.49. We can compute now V W

 −1 cV,W

◦ cV,W =

V W

 =



V W W V

 −1 cV,W ◦ cV,W

=

V

W V

, V W

W V



V W

=



W

=



(1.6.6)

(1.6.6)



W V

=

W V

. W V

So our claim is proved. The second assertion follows from Proposition 1.74. The dual functors (−)∗ and ∗ (−) behave well with respect to braidings. Proposition 1.78 If C is a braided category then the strong monoidal functors from Proposition 1.71 are, moreover, braided monoidal if they are viewed as functors from in C to C opp . Proof

−1 : X⊗Y = Y ⊗ X ← X ⊗Y = Y ⊗X, The braiding of C opp is cX,Y = cY,X = cY,X opp

opp

opp in

coincides with that of C . Now, the fact that (−)∗ : C → and so the braiding of C in C opp is braided means that Y ∗ ⊗ X ∗ = X ∗ ⊗Y ∗

cX ∗ ,Y ∗

λX,Y

(X ⊗Y )∗

X ∗ ⊗Y ∗ = Y ∗ ⊗X ∗ λY,X

c∗X,Y

(Y ⊗ X)∗

1.8 Braided Rigid Monoidal Categories

51

is commutative. In diagrammatic language this comes out as (Y ⊗ X)∗

(Y ⊗ X)∗

  X⊗Y  .

  =

Y ⊗X

Y ⊗X



X⊗Y

 

Y ∗ X∗

Y ∗ X∗

Now, by (1.6.6) the right-hand side of the above equality is equal to (Y ⊗ X)∗

∗

∗

(Y ⊗ X) (Y ⊗ X)      = = , Y ⊗X

Y ⊗X



Y ⊗X





Y ∗ X∗

Y ∗ X∗

Y ∗ X∗

X

proving the desired equality. Note that we used

X

  = and

Y X Y∗

Y X Y∗

X

 =

Y X Y∗

X



in the first and second equalities, respectively, and that these equalities Y X Y∗

follow from the naturality of c and Proposition 1.49. Thus the proof is finished. We will need later the mate of the braiding of a left rigid braided category. The notion “mates under adjunction” can be regarded as a “halfway” dualizing process. Definition 1.79 Let C be a monoidal category and X, Y , V , W objects of C such that X and W admit left dual objects. If f : V ⊗ W → X ⊗ Y is a morphism in C then the mate of f is the morphism f  : X ∗ ⊗V → Y ⊗W ∗ defined by the following composition: rX−1∗ ⊗V

f  : X ∗ ⊗V −→ (X ∗ ⊗V ) ⊗ 1 aX ∗ ,V,W ⊗W ∗

−→

a−1 X ∗ ,X,Y ⊗W ∗

−→

−→

X ∗ ⊗ (V ⊗ (W ⊗W ∗ ))

IdX ∗ ⊗( f ⊗IdW ∗ )

−→

IdX ∗ ⊗V ⊗coevW

−1 IdX ∗ ⊗aV,W,W ∗

X ∗ ⊗ ((X ⊗Y ) ⊗W ∗ )

(X ∗ ⊗ X) ⊗ (Y ⊗W ∗ )

(X ∗ ⊗V ) ⊗ (W ⊗W ∗ ) −→

X ∗ ⊗ ((V ⊗W ) ⊗W ∗ )

IdX ∗ ⊗aX,Y,W ∗

−→

evX ⊗IdY ⊗W ∗

−→

X ∗ ⊗ (X ⊗ (Y ⊗W ∗ )) lY ⊗W ∗

1 ⊗ (Y ⊗W ∗ ) −→ Y ⊗W ∗ .

When V = Y = 1, through the natural identifications, the mate of f is the transpose

52

Monoidal and Braided Categories

morphism f ∗ . In general, assuming C is strict monoidal, the mate of f is X∗ V

f =

V W

 , where

f

f=

.

f

Y W∗

X Y

Moreover, the mate of a morphism can be characterized as follows. Proposition 1.80 Let C be a monoidal category and V , W , X, Y objects of C such that there exist (coevW , evW ) : W ∗  W and (coevX , evX ) : X ∗  X adjunctions in C . If f : V ⊗ W → X ⊗ Y is a morphism in C then its mate f  is the unique morphism f  : X ∗ ⊗V → Y ⊗W ∗ in C making one (hence also the other one) of the following two diagrams commutative: rV−1

V

V ⊗1

−1 aV,W,W ∗ (IdV ⊗coevW )

(V ⊗W ) ⊗W ∗

lV−1

aX,Y,W ∗ ( f ⊗IdW ∗ )

1 ⊗V

aX,X ∗ ,V (coevX ⊗IdV )

X ∗ ⊗ (V ⊗W )

X ⊗ (X ∗ ⊗V )

a−1 (IdX ∗ ⊗ f ) X ∗ ,X,Y

IdX

⊗f

X ⊗ (Y ⊗W ∗ ), lY (evX ⊗IdY )

(X ∗ ⊗ X) ⊗Y

Y

a−1 X ∗ ,V,W

rY f  ⊗IdW

(X ∗ ⊗V ) ⊗W

(Y ⊗W ∗ ) ⊗W

(IdY ⊗evW )aY,W ∗ ,W

Y ⊗ 1.

Proof In diagrammatic notation we have to prove that f  is the unique morphism in C satisfying one of the following two equivalent identities: V

 f

=

X∗ V W

V

 f

,

X∗ V W

=

f

f

X Y W∗

X Y W∗

Y

.



Y

This is immediate if we use the relations in (1.6.6) in the definition of f  . Proposition 1.81 Let C be a monoidal category and V , W , X, Y objects of C such that X and W admit left dual objects. Then the correspondence HomC (V ⊗W, X ⊗Y ) f → f  ∈ HomC (X ∗ ⊗V,Y ⊗W ∗ ) is one to one. Consequently, we have HomC (V ⊗ W, X ⊗ Y ) ∼ = HomC (X ∗ ⊗ V,Y ⊗ ∗ ∗ ∗ W )∼ = HomC ((X ⊗V ) ⊗W,Y ) ∼ = HomC (V, (X ⊗Y ) ⊗W ).

1.8 Braided Rigid Monoidal Categories Proof

53

We show that the inverse of f → f  is g → g , where for any g : X ∗ ⊗ V → V W

 Y ⊗W ∗ define g : V ⊗W → X ⊗Y by g =

g

. To see this we compute



X Y V W

V

 ( f  ) =

f

W



=



X Y

V W



(1.6.6)

=

f

X

f

.



Y

X Y

On the other hand we have X∗ V  

(g ) =

X∗



V



=

g



 = g,

g



Y W∗

W∗

Y

as required. If we specialize this isomorphism for X = 1, which always has a left dual, we get that HomC (V ⊗ W,Y ) ∼ = HomC (V,Y ⊗ W ∗ ), for any other objects Y , V , W of C with W admitting a left dual object. It is clear now that we also have HomC (V ⊗ W, X ⊗ Y ) ∼ = HomC (V, (X ⊗ Y ) ⊗ W ∗ ) and HomC (X ∗ ⊗ V,Y ⊗ W ∗ ) ∼ = ∗ HomC ((X ⊗V ) ⊗W,Y ), and this completes the proof. We finally compute the mate of the braiding of a left rigid braided category. Proposition 1.82 If C is a left rigid braided category, then for all X,Y objects of ∗ ∗ C the mate of the braiding cX,Y : X ⊗Y → Y ⊗ X is c−1 X,Y ∗ : Y ⊗ X → X ⊗Y . Proof X

It is enough to show that cX,Y ∗ ◦ cX,Y = IdY ∗ ⊗X . For this we notice first that

1

 =

1

X



, an equality that follows from (1.5.3), the naturality of c and

Y Y∗ X Y Y∗ X

Proposition 1.49. This fact allows to compute Y∗ X

cX,Y ∗ ◦ cX,Y

 Y ∗

=

=

Y∗ X

as required.

X



Y∗ X

(1.6.6)

Y∗ X

=

= IdY ∗ ⊗X , Y∗

X

54

Monoidal and Braided Categories

1.9 Notes Monoidal categories were introduced by B´enabou in [30]; they are also known under the name of tensor categories. The notion of symmetric monoidal category goes back to Eilenberg and Kelly [84]. The concept of braided category was introduced by Joyal and Street in [119, 121]. The concept of rigidity abstracts the notion of duality, a concept intensively used for categories of vector spaces. The canonical isomorphisms constructed in the braided monoidal setting are natural generalizations of the wellknown isomorphisms that exist in a category of vector spaces. The concrete monoidal structures on some categories of graded vector spaces were taken from [119, 121], which were our main source of inspiration for many results of this chapter. We also took examples of normalized 3-cocycles from these papers and [5]. Note that H 3 (Zn , k∗ ) is completely determined by the nth roots of unity in k; see [192, Theorem 10.35] or [219, Theorem 6.2.2]. Thus H 3 (Zn , k∗ ) is completely determined by the cohomology classes defined by the cocycles φq from Example 1.16. Also, by [219, Example 6.1.4], H 3 (Z, k∗ ) = {e}, where Z is the infinite cyclic group written additively; this means that any normalized 3-cocycle on Z with coefficients in k∗ is a coboundary. This completes the description of the third cohomology group for any cyclic group. Apart from the papers already mentioned above we used in the presentation of this chapter the books of C. Kassel [127], S. Majid [148] and S. Mac Lane [141]. Proofs for the coherence theorem of Mac Lane can be found in [141, VII.2], [95, Theorem 2.9.2] and [217, Theorem 3.17]. The fact that a diagram in a monoidal category C is commutative if it is commutative when it is considered in a strict monoidal category can be proved for instance by using so-called “cliques” in C and morphisms between them; as in the case of the coherence theorem, a detailed proof involves categorical techniques that are not related to the topic of this book, and this is why we skip it.

2 Algebras and Coalgebras in Monoidal Categories

We define the notions of algebra and coalgebra, and of bialgebra and Hopf algebra, respectively, within a monoidal category and a (pre-)braided monoidal category, respectively. We also present several constructions associated to them.

2.1 Algebras in Monoidal Categories Recall that an algebra over a field k is a vector space A over k that also has a unital ring structure such that κ (ab) = (κ a)b = a(κ b), for all κ ∈ k and a, b ∈ A. Equivalently, an algebra over k, or a k-algebra, is a k-vector space for which there exist two k-linear maps mA : A ⊗ A → A and ηA : k → A making the diagrams below commutative: A⊗A⊗A

mA ⊗IdA

A⊗A

IdA ⊗mA

A⊗A ηA ⊗IdA

k⊗A

A ⊗ A, mA

lA

A

mA mA

A, IdA ⊗ηA

A⊗k rA

A ⊗ A. mA

A

Here lA : k ⊗ A → A, rA : A ⊗ k → A are the natural isomorphisms and all the unadorned tensor products are over k. Note that mA and ηA are obtained from the unital ring structure of A as follows: mA (a ⊗ b) = ab and η (κ ) = κ 1A , for all κ ∈ k, where 1A is the unit of A. Thus a k-algebra identifies with a triple (A, mA , ηA ) as above. Recall also that for two k-algebras A, B an algebra morphism from A to B is a k-linear map f : A → B such that f (1A ) = 1B and f (ab) = f (a) f (b), for all a, b ∈ A. If we consider A, B as triples (A, mA , ηA ) and (B, mB , ηB ), respectively, then the definition of a k-algebra morphism f from A to B can be rephrased as follows: f is a k-linear morphism such that f mA = mB ( f ⊗ f ) and f ηA = ηB . The diagrammatic reformulation of the definition of a k-algebra and a k-algebra

56

Algebras and Coalgebras in Monoidal Categories

morphism allows to define these concepts in any monoidal category, leading thus to the concepts of monoidal algebra and morphism of monoidal algebras. Definition 2.1 Let C = (C , ⊗, 1, a, l, r) be a monoidal category. An algebra in C is a triple (A, mA , η A ), where A ∈ C and mA : A ⊗ A → A (called the multiplication of A) and η A : 1 → A (called the unit of A) are morphisms in C such that: • mA is associative up to the associativity constraint a of C , that is, the diagram (A ⊗ A) ⊗ A

mA ⊗IdA

mA

A⊗A

aA,A,A

A

mA

A ⊗ (A ⊗ A)

IdA ⊗mA

A⊗A

is commutative; • η A is a unit for mA , that is, the following triangle diagrams are commutative: η A ⊗IdA

1⊗A

A⊗A

and

mA

lA

IdA ⊗η A

A⊗1

A⊗A mA

rA

A

A.

If (A, mA , η A ) and (B, mB , η B ) are algebras in C then an algebra morphism between A and B is a morphism f : A → B in C such that the diagrams A⊗A

f⊗f

mA

B⊗B mB

A

f

B

and

f

A ηA

B ηB

1

are commutative. If, moreover, f is an isomorphism in C then we say that A and B are isomorphic as algebras in C . Clearly, a k-algebra is nothing but an algebra in the category of k-vector spaces. In general, the meaning of an algebra in an arbitrary monoidal category C might be far from the classical one, as it is strongly connected with the monoidal structure of C . Also, since the multiplication of A is associative up to the associativity constraint a of C , it might happen that A is not associative in the usual sense. For a good understanding of these ideas we next supply a list of examples. Examples 2.2 (1) The unit object of a monoidal category C is an algebra in C with multiplication m1 = l1 = r1 : 1 ⊗ 1 → 1 and unit η 1 = Id1 . (2) An algebra A in Set is a monoid, and a morphism of algebras in Set is nothing but a morphism of monoids. Indeed, A is an algebra in Set if and only if there exist functions mA : A×A → A and η A : {∗} → A such that the first three diagrams in Definition 2.1 are commutative. If

2.1 Algebras in Monoidal Categories

57

we denote mA (a, b) = ab and η A (∗) = e, the commutativity of these diagrams reads (ab)c = a(bc) and ae = a = ea, for all a, b, c ∈ A. Thus A is a monoid. It is clear that if A, A are algebras in Set then f : A → A is an algebra morphism if and only if f (ab) = f (a) f (b), for all a, b ∈ A, and f (e) = e , where e is the neutral element of A . Thus f has to be a morphism of monoids. (3) Let [C , C ] be the category of endo-functors associated to a category C , endowed with the monoidal structure from Proposition 1.17. An algebra in [C , C ], called a monad in C , is a triple (T, m, η ) where T : C → C is a functor and m : T ◦ T → T and η : IdC → T are natural transformations such that, for any object X of C , mX mT (X) = mX T (mX ) and mX η T (X) = mX T (η X ) = IdT (X) . Indeed, if 1T is the identity natural transformation from T to T , the two sets of conditions stated above are equivalent to m ◦ (m ⊗ 1T ) = m ◦ (1T ⊗ m) and m ◦ (η ⊗ 1T ) = m ◦ (1T ⊗ η ) = 1T , as natural transformations. These conditions are equivalent to the commutativity of the diagrams in Definition 2.1, specialized for our situation. (4) Let G be a group and φ ∈ H 3 (G, k∗ ). If VectG φ is the category of G-graded vector spaces endowed with the monoidal structure from Proposition 1.13 then an algebra in VectG φ is a G-graded vector space A together with a multiplication • and a usual unit 1A ∈ Ae satisfying Ax Ay ⊆ Axy , for all x, y ∈ G, and (a • b) • c = φ (|a|, |b|, |c|)a • (b • c),

(2.1.1)

for all homogeneous elements a, b, c ∈ A. Indeed, let A be an object of VectG φ , that is, a G-graded k-vector space. Then A has G an algebra structure in Vectφ if and only if there exist morphisms mA : A ⊗ A → A and η A : k → A in VectG φ that make the diagrams in Definition 2.1 commutative. Now, mA preserves the degree of homogeneous elements if and only if Ax Ay ⊆ Axy and so does η A if and only if 1A := η (1) ∈ Ae . Moreover, those diagrams are commutative if and only if 1A is a usual unit for mA and mA is associative in the sense of (2.1.1). In what follows an algebra in VectG φ will be called a G-graded quasialgebra with reassociator φ . If φ is trivial then we simply call such an algebra a G-graded algebra. (5) Let k be a commutative ring and R a k-algebra. Then giving an algebra A in M R R is equivalent to giving a k-algebra A and a k-algebra morphism i : R → A. Such a pair (A, i) is called an R-ring. To see this, we consider A an algebra in R MR . Thus A is an R-bimodule and there exist R-bilinear morphisms mA : A ⊗R A → A and η A : R → A such that mA is associative and η A is a unit for it. Otherwise stated, we have a k-linear map mA := mA qRA,A : A ⊗ A a ⊗ b → ab ∈ A that is associative and unital in k M and such that r(ab) = (ra)b, (ar)b = a(rb) and (ab)r = a(br), for all a, b ∈ A and r ∈ R. Here qRA,A : A ⊗ A → A ⊗R A is the canonical projection. Furthermore, the fact that η R is R-bilinear is equivalent to r1A = 1A r, for all r ∈ R, where 1A := η A (1R ) is the unit for mA . So A becomes a k-algebra and η A : R → A turns into a k-algebra morphism since η A (1R ) = 1A and, for all r, r ∈ R,

η A (rr ) = (rr )1A = r(r 1A ) = r(1A r ) = r(1A (1A r )) = (r1A )(r 1A ) = η A (r)η A (r ).

58

Algebras and Coalgebras in Monoidal Categories

Conversely, if A admits a k-algebra structure such that there exists a k-algebra morphism i : R → A, then A is an R-bimodule via rar = i(r)ai(r ), for all a ∈ A and r, r ∈ R. Moreover, in this way the multiplication of A becomes R-balanced and so we have a well-defined morphism mA : A ⊗R A → A given by mA (a ⊗R b) = mA (a ⊗ b) = ab, for all a, b ∈ A, where mA is the multiplication of A in k M . It then follows that (A, mA , i) is an algebra in R MR , as required. The two correspondences defined above are inverse to each other, so we are done. A monoidal functor takes algebras to algebras. Let (F, ϕ0 , ϕ2 ) : C → D be a monoidal functor and A an algebra f in C . Then F(A) has an algebra structure in D. Moreover, if A - B is an algebra Proposition 2.3

morphism in C then F(A) Proof

- F(B) is an algebra morphism in D.

F( f )

We show that F(A) with mF(A) and η F(A) given by mF(A) : F(A) ⊗ F(A)

η F(A) : 1

ϕ0

- F(1)

ϕ2,A,A

- F(A ⊗ A)

- F(A),

F(mA )

F(η A )

- F(A),

is an algebra in D. For this, since ϕ2 is a natural transformation, the diagrams F(A) ⊗ F(A ⊗ A) IdF(A) ⊗F(mA )

F(A) ⊗ F(A)

ϕ2,A,A⊗A

F(A ⊗ (A ⊗ A))

F(IdA ⊗mA )

ϕ2,A,A

F(1) ⊗ F(A) F(η A )⊗IdF(A)

F(A ⊗ A),

F(A) ⊗ F(A)

F((A ⊗ A) ⊗ A)

F(A) ⊗ F(1)

ϕ2,1,A

F(1 ⊗ A)

F(η A ⊗IdA )

ϕ2,A,A

F(A ⊗ A)

and F(A ⊗ A) ⊗ F(A) F(mA )⊗IdF(A)

F(A) ⊗ F(A)

ϕ2,A⊗A,A

F(mA ⊗IdA )

ϕ2,A,A

IdF(A) ⊗F(η A )

F(A ⊗ A),

F(A) ⊗ F(A)

ϕ2,A,1

F(A ⊗ 1)

F(IdA ⊗η A )

ϕ2,A,A

F(A ⊗ A)

are all commutative. By using this, we compute: mF(A) (IdF(A) ⊗ mF(A) )aF(A),F(A),F(A) = F(mA )ϕ2,A,A (IdF(A) ⊗ F(mA )ϕ2,A,A )aF(A),F(A),F(A) = F(mA (IdA ⊗ mA ))ϕ2,A,A⊗A (IdF(A) ⊗ ϕ2,A,A )aF(A),F(A),F(A) = F(mA (IdA ⊗ mA )aA,A,A )ϕ2,A⊗A,A (ϕ2,A,A ⊗ IdF(A) ) = F(mA )F(mA ⊗ IdA )ϕ2,A⊗A,A (ϕ2,A,A ⊗ IdF(A) ) = F(mA )ϕ2,A,A (F(mA )ϕ2,A,A ⊗ IdF(A) ) = mF(A) (mF(A) ⊗ IdF(A) ), where in the third equality we used the commutativity of the first diagram in Definition 1.22 and in the fourth equality the associativity of mA .

2.1 Algebras in Monoidal Categories

59

The fact that η F(A) is a unit for mF(A) follows from the computations: mF(A) (η F(A) ⊗ IdF(A) ) = F(mA )ϕ2,A,A (F(η A ) ⊗ IdF(A) )(ϕ0 ⊗ IdF(A) ) = F(mA (η A ⊗ IdA ))ϕ2,1,A (ϕ0 ⊗ IdF(A) ) = F(lA )ϕ2,1,A (ϕ0 ⊗ IdF(A) ) = lF(A) (in the last but one equality we used the fact that η A is a unit for mA , and in the last equality we used the commutativity of the second diagram in Definition 1.22), and mF(A) (IdF(A) ⊗ η F(A) ) = F(mA )ϕ2,A,A (IdF(A) ⊗ F(η A ))(IdF(A) ⊗ ϕ0 ) = F(mA (IdA ⊗ η A ))ϕ2,A,1 (IdF(A) ⊗ ϕ0 ) = F(rA )ϕ2,A,1 (IdF(A) ⊗ ϕ0 ) = rF(A) , where, this time, in the last equality we used the commutativity of the third diagram in Definition 1.22. f Finally, let A - B be an algebra morphism in C . Then mF(B) (F( f ) ⊗ F( f )) = F(mB )ϕ2,B,B (F( f ) ⊗ F( f )) = F(mB ( f ⊗ f ))ϕ2,A,A = F( f )F(mA )ϕ2,A,A = F( f )mF(A) , and F( f )η F(A) = F( f η A )ϕ0 = F(η B )ϕ0 = η F(B) , which completes the proof. A A

When A is an algebra in a monoidal category C we will denote mA by and A 1

the unit morphism η A by r . Then, in diagrammatic notation, the associativity and A

unit conditions for mA and η A take the form A A A

A A A



= and

A

A

A

r

=

A

A

= A

A

r

.



(2.1.2)

A

Note that, in this kind of computation, we will always assume that all the constraints of C are defined by identity morphisms, that is, that C is strict monoidal. Then the computations are still valid for an arbitrary monoidal category; see Remark 1.35. X

In general, if C is a monoidal category, by fhwe denoted a morphism f : X → Y Y

60

Algebras and Coalgebras in Monoidal Categories

in C . Thus, f : A → B is an algebra morphism in C if A A

A A

1

B

B

fh fh = and

fh B

1

r = r . fh B

When C is braided, one can associate to an algebra A in C another two (possibly different) algebra structures on A. Proposition 2.4 Let (C , c) be a braided category and (A, mA , η A ) an algebra in C . Then Aop+ := (A, mA ◦ cA,A , η A ) and Aop− := (A, mA ◦ c−1 A,A , η A ) are algebras in C . op+ op− −1 (resp. A ) is called the c-opposite (resp. c -opposite) algebra associated to A A in the category (C , c). Proof The multiplication mAop+ is associative since Proposition 1.37, (1.5.10) and the associativity of mA imply that A A A

A A A

A A A

A A A

A A A

=

A

=







A

.

= =





A

A

A

We also have that A

r

A

A

r

= =



A A

A

r

A

and

A

r

A

= = ,



A A

A

because of Proposition 1.49, and so Aop+ is an algebra in C . That Aop− is an algebra in C follows by applying the above arguments to A viewed as an algebra in C in . Definition 2.5 An algebra A in a braided category is called commutative (or braided commutative) if A = Aop+ or, equivalently, if A = Aop− , as algebras in C . We next see that the braiding c of C or its inverse c−1 gives rise to an algebra structure on the tensor product A ⊗ B of two algebras A, B. This is a consequence of the naturality of the braiding, as the cross product construction below shows. If A and B are algebras in a monoidal category C and ψ : B ⊗ A → A ⊗ B is a morphism in C then one can introduce on A ⊗ B a multiplication given, in the case when C is strict, by the formula m = (mA ⊗ mB ) ◦ (IdA ⊗ ψ ⊗ IdB ).

(2.1.3)

2.1 Algebras in Monoidal Categories

61

By A#ψ B we denote the object A ⊗ B endowed with the unit tensor product morphism and with the multiplication defined as in (2.1.3). Then m, written as mA#ψ B in what follows, takes the form A B A B

mA#ψ B =

B A

e

,

A

where ψ =

e .

A B

B

Definition 2.6 If A#ψ B is an algebra in C then we call it a cross product algebra of A and B. A#ψ B is a cross product algebra under the following conditions on ψ . Theorem 2.7 Let C be a monoidal category, A and B algebras in C and ψ : B ⊗ A → A ⊗ B a morphism in C . Then the following are equivalent: (i) A#ψ B is a cross product algebra; (ii) the following equalities hold: B B A

B B A

e

e

=

A B

e

,



A

B

B A A

B A A

B A e B A

r r e , r r . = = , = e e e

A B A B

A B

A

A B

B

A B

If this is the case, ψ is called a twisting morphism between A and B. Proof We first prove that η A ⊗ η B is a right (resp. left) unit for mA#ψ B defined in (2.1.3) if and only if the third (resp. fourth) equality in (ii) holds. Indeed, if η A ⊗ η B is a right unit for mA#ψ B we have A B

r r A B e = ,

A B A

B

so that

r r B e r = = . e

A B r

A B

B

B

r

A

B

Conversely, if the third equality in (ii) holds then A B

A B

A B r A B r e e = = = ,



A B A

r r

B

A

B

A

B

so η A ⊗ η B is a right unit for mA#ψ B . The left-handed case is similar.

62

Algebras and Coalgebras in Monoidal Categories (i) ⇔ (ii). We have mA#ψ B associative if and only if A B A B A B

A B A B A B

e

e

e

A

. e

=

(2.1.4)



B

A

B

In view of the above results it is enough to show that (2.1.4) is equivalent to the first two equalities in (ii). Indeed, if (2.1.4) holds then by (2.1.2) and the unit conditions proved above we have r

B B A

=

e

B

r e

B A

r

A

B

B A

r

e

r

B B A

= e

=

e

A B

B

r

A

e

e

.



A

B

B

Similarly, one can prove the second equality in (ii). Conversely, if the first two equalities in (ii) hold then A B A B A B

A B A B A B

A

A B A B A B A B

e

e

A B A B A B

e e e e



=

=

e







B

e

A

e

A B A B A B

e

e

=

, e

=



A

B

A

B

B

as required, where we used: in the first equality the first relation in (ii), in the second one the fact that the multiplication of A is associative, in the third one the second equality in (ii), and in the last one the fact that the multiplication of B is associative. This finishes the proof of the theorem.

2.1 Algebras in Monoidal Categories

63

Remark 2.8 If A#ψ B is a cross product algebra in C , the morphisms iA := IdA ⊗ η B : A → A#ψ B and iB := η A ⊗ IdB : B → A#ψ B are algebra morphisms in C . The cross product algebra has the following Universal Property. Proposition 2.9 Let A#ψ B be a cross product algebra in C . If (X, mX , η X ) is an algebra in C and u : A → X, v : B → X are morphisms of algebras in C such that mX (u ⊗ v)ψ = mX (v ⊗ u),

(2.1.5)

then there exists a unique morphism w : A#ψ B → X of algebras in C such that wiA = u and wiB = v. This morphism w is given explicitly by w = mX (u ⊗ v). Proof Observe first that the algebra morphisms iA and iB in Remark 2.8 satisfy the condition in the statement, namely that mA#ψ B (iB ⊗ iA ) = mA#ψ B (iA ⊗ iB )ψ = ψ . Suppose there is an algebra map w : A#ψ B → X such that wiA = u and wiB = v. Then w = wmA#ψ B (iA ⊗ iB ) = mX (w ⊗ w)(iA ⊗ iB ) = mX (wiA ⊗ wiB ) = mX (u ⊗ v), and this shows the uniqueness of w. Conversely, define w = mX (u ⊗ v). Then w is multiplicative since A B A B

e e uh vh

uh uh vh vh

vh = = = uh uh vh 

wh





X X X

A B A B

A B A B

e

A B A B

e

X A B A B

vh uh h vh

u

=





A B A B

A

h vh uh vh

u

= =

B A

B

  wh wh ,





X

X

X 1 A

B

1

X

r

1

r r 1 r r  r, uh vh= . We also have = = where we denoted w = 

wh

wh

X and this finishes the proof.

r

X

X

X

The braiding c of a braided category C and its inverse c−1 satisfy the conditions in Theorem 2.7 (ii), because of Proposition 1.49 and Proposition 1.37. Thus A#cB,A B and A#c−1 B are cross product algebras in C . A,B

64

Algebras and Coalgebras in Monoidal Categories

Definition 2.10 Let A, B be two algebras in a braided category (C , c). We call the c-tensor product (resp. c−1 -tensor product) algebra between A and B in C the cross product algebra A#cB,A B (resp. A#c−1 B), and we denote it by A ⊗+ B (resp. A ⊗− B). A,B

Note that A ⊗− B is nothing but A ⊗+ B considered in C in instead of C . Also, A ⊗+ B is the object A ⊗ B of C endowed with the algebra structure in C given, if C is strict, by (mA ⊗ mB ) ◦ (IdA ⊗ cB,A ⊗ IdB ) and tensor product unit morphism. Similarly, A ⊗− B is an algebra in C with the multiplication given, in the strict case, by (mA ⊗ mB ) ◦ (IdA ⊗ c−1 A,B ⊗ IdB ) and tensor product unit morphism. As a concrete example, let G be an abelian group and endow the category of Ggraded vector spaces VectG with a braided structure given by an abelian 3-cocycle (φ , R) on G as in Proposition 1.44. Example 2.11 If (φ , R) is an abelian 3-cocycle on an abelian group G and A, A are G-graded quasialgebras with reassociator φ , then the multiplication of the G-graded quasialgebra A ⊗+ A (with reassociator φ ) is given by (a ⊗ a )(b ⊗ b ) =

φ (| a |, | a |, | b |)φ (| a || b |, | a |, | b |) R(| a |, | b |)ab ⊗ a b , φ (| a || a |, | b |, | b |)φ (| a |, | b |, | a |)

for all homogeneous elements a, b ∈ A and a , b ∈ A . For instance, if VectZ−12 is the braided category of super vector spaces (see Example 1.47), then for any two algebras A, A in VectZ−12 the algebra structure of A ⊗+ A in VectZ−12 is determined by 

(a ⊗ a )(b ⊗ b ) = (−1)|a ||b| ab ⊗ a b , for all homogeneous elements a, b ∈ A and a , b ∈ A . The unit of A ⊗+ A is 1A ⊗ 1A . Proof Since the category VectG (φ ,R) is not in general strict monoidal, the multiplication of the tensor product G-graded quasialgebra A ⊗+ A is given by the following composition: mA⊗+ A : (A ⊗ A ) ⊗ (A ⊗ A )

a−1 A⊗A ,A,A

- ((A ⊗ A ) ⊗ A) ⊗ A

aA,A ,A ⊗IdA

(IdA ⊗cA ,A )⊗IdA

a−1 ⊗IdA A,A,A

aA⊗A,A ,A

- (A ⊗ (A ⊗ A)) ⊗ A - ((A ⊗ A) ⊗ A ) ⊗ A

- (A ⊗ (A ⊗ A )) ⊗ A

- (A ⊗ A) ⊗ (A ⊗ A )

mA ⊗mA

- A ⊗ A .

If we keep in mind the monoidal structure of VectG φ , a straightforward computation leads to the formula in the statement; we leave the verification to the reader. The second assertion follows by specializing this result for G = Z2 and (φ , R) the abelian 3-cocycle on Z2 defined in Example 1.47. Proposition 2.12 Let A, B,C be three algebras in a braided category (C , c). Then (A ⊗± B) ⊗± C and A ⊗± (B ⊗± C) are isomorphic as algebras in C .

2.2 Coalgebras in Monoidal Categories

65

Proof The isomorphism is produced by the associativity constraint of C . If we assume C strict monoidal then the multiplications of (A ⊗+ B) ⊗+ C and A ⊗+ (B ⊗+ A B C A B C

C) come out as

. For the minus case we have to replace in this

A

C

B

diagram the braiding c of C with its inverse c−1 .

2.2 Coalgebras in Monoidal Categories The concept of monoidal coalgebra, that is, of a coalgebra in a monoidal category C , can be easily obtained by reversing the sense of the arrows in the definition of a monoidal algebra. In other words, a coalgebra in C is precisely an algebra in C opp , the opposite monoidal category associated to C . Consequently, results and constructions for monoidal algebras that have diagrammatic proofs or definitions produce “dual” results and constructions for coalgebras: we simply have to reverse the sense of the arrows in diagrams or, equivalently, to turn upside down diagrammatic computations. Furthermore, if the category is rigid monoidal, we shall see in the next section that the algebra and coalgebra concepts are actually equivalent. For these reasons in the remainder of this section we will omit most of the proofs. Definition 2.13 Let (C , ⊗, 1, a, l, r) be a monoidal category. A coalgebra in C is an algebra in C opp , the opposite monoidal category associated to C . Explicitly, a coalgebra in C is a triple (C, ΔC , ε C ), where ΔC : C → C ⊗ C and ε C : C → 1 are morphisms in C such that • ΔC , called the comultiplication of C, is coassociative in the sense that the diagram (C ⊗C) ⊗C

ΔC ⊗IdC

C ⊗C

aC,C,C

C ⊗ (C ⊗C)

IdC ⊗ΔC

ΔC

C

ΔC

C ⊗C

is commutative; • ε C , called the counit of C, is a counit for ΔC ; this means that the diagrams 1 ⊗C lC−1

C are commutative.

ε C ⊗IdC

ΔC

C ⊗C and

C⊗1 rC−1

C

IdC ⊗ε C

ΔC

C ⊗C

66

Algebras and Coalgebras in Monoidal Categories

If (C, ΔC , ε C ) and (D, ΔD , ε D ) are coalgebras in C then a coalgebra morphism between C and D is a morphism f : C → D in C such that the diagrams C

f

ΔC

C ⊗C

D ΔD

f⊗f

D⊗D

and

C εC

f

D εD

1

are commutative (that is, f : D → C is an algebra morphism in C opp ). If, moreover, f is an isomorphism in C we say that C and D are isomorphic coalgebras in C . As in the algebra case, the meaning of a coalgebra in a monoidal category differs from one category to another. Examples 2.14 (1) The unit object of a monoidal category C is a coalgebra in C via Δ1 = l1−1 = r1−1 : 1 → 1 ⊗ 1 and ε 1 = Id1 . (2) Any set X has a unique coalgebra structure in Set. In this way any function f : X → Y becomes a coalgebra morphism in Set. To see this, suppose that X has a coalgebra structure in Set given by ΔX : X → X × X and ε X : X → {∗}. Then ε X (x) = ∗, for all x ∈ X. For x ∈ X write ΔX (x) = (y, z), for some y, z ∈ X. Since ε X is a counit for ΔX it follows that (∗, z) = (∗, x) and (y, ∗) = (x, ∗), so y = z = x and ΔX (x) = (x, x). Conversely, it is easy to see that (X, ΔX , ε X ) is a coalgebra in Set, where ΔX (x) = (x, x) and ε X (x) = ∗, for any x ∈ X. (3) Let C be a category. A coalgebra in the monoidal category of endo-functors [C , C ] is called a comonad in C . It consists of a triple (U, Δ, ε ), where U : C → C is a functor and Δ : U → U ◦U and ε : U → IdC are natural transformations such that U(ΔM ) ◦ ΔM = ΔU(M) ◦ ΔM ,

(2.2.1)

U(ε M ) ◦ ΔM = ε U(M) ◦ ΔM = IdU(M) ,

(2.2.2)

for any object M of C . Indeed, since a coalgebra in [C , C ] is an algebra in [C , C ]opp everything follows from Example 2.2 (3). (4) Let G be a group, φ an invertible normalized 3-cocycle with coefficients in a field k, and VectG φ the category of G-graded k-vector spaces endowed with the monoidal structure from Proposition 1.13. Then a coalgebra in VectG φ is a G-graded  vector space C = x∈G Cx together with k-linear maps Δ : C → C ⊗C and ε : C → k such that (IdC ⊗ ε )Δ = IdC = (ε ⊗ IdC )Δ, and 

• Δ(Cx ) ⊆ uv=x Cu ⊗Cv , for all x ∈ G; • ε (Cx ) = 0, for all x ∈ G, x = e; • for any c ∈ Cx , if we write (Δ ⊗ IdC )Δ(c) = ∑i ci ⊗ di ⊗ ei and (IdC ⊗ Δ)Δ(c) =

2.2 Coalgebras in Monoidal Categories

67

∑ j cj ⊗ d j ⊗ ej , with all ci s , di s, etc. homogeneous elements, then

∑ φ (|ci |, |di |, |ei |)ci ⊗ di ⊗ ei = ∑ cj ⊗ d j ⊗ ej . i

j

Indeed, all the conditions are reformulations of the following facts: ε is a counit for Δ, Δ and ε must preserve the degree of homogeneous elements, and Δ has to be coassociative up to the associativity constraint of VectG φ. G In what follows, we call a coalgebra C in Vectφ a G-graded quasicoalgebra with reassociator φ . When φ is trivial we simply say that C is a G-graded coalgebra. (5) Let k be a commutative ring and R a k-algebra. We call a coalgebra in R MR an R-coring. Explicitly, an R-coring is a triple (C, ΔC , εC ) consisting of an R-bimodule C and R-bimodule morphisms ΔC : C → C ⊗R C and εC : C → R such that ΔC is coassociative up to the associativity constraint of R MR and εC is a counit for it. If for c ∈ C we denote ΔC (c) = c1 ⊗R c2 (summation implicitly understood) then ΔC (rc) = rc1 ⊗R c2 , ΔC (cr) = c1 ⊗R c2 r, (c1 )1 ⊗R (c1 )2 ⊗R c2 = c1 ⊗R (c2 )1 ⊗R (c2 )2 ,

εC (rcr ) = rε (c)r

and

c1 εC (c2 ) = εC (c1 )c2 = c,

for all c ∈ C and r, r ∈ R. When R = k this reduces to the notion of k-coalgebra. Remark 2.15 For C a k-coalgebra denote Δ2 = (ΔC ⊗ IdC )ΔC = (IdC ⊗ ΔC )ΔC , and ⊗(n−1) ) ◦ Δn−1 , for all n ≥ 2. By induction on n one can show that Δn = (ΔC ⊗ IdC ⊗(n−p−1)

Δn = (IdC⊗p ⊗ ΔC ⊗ IdC

) ◦ Δn−1 ,

for all n ≥ 2 and 1 ≤ p ≤ n − 1. This equality allows us to introduce the so-called sigma notation for coalgebras, also known as the Sweedler notation or as the Heyneman–Sweedler notation for coalgebras. For (C, ΔC , εC ) a k-coalgebra and c ∈ C we usually have ΔC (c) = ∑i xi ⊗ yi , for some families of elements (xi )i and (yi )i in C. Instead, we denote ΔC (c) = c1 ⊗ c2 , summation understood. Then the coassociativity property of ΔC comes out as (c1 )1 ⊗ (c1 )2 ⊗ c2 = c1 ⊗ (c2 )1 ⊗ (c2 )2 := c1 ⊗ c2 ⊗ c3 , for all c ∈ C. Likewise, the fact that εC is a counit for ΔC can be written in sigma notation as ε (c1 )c2 = ε (c2 )c1 = c, for all c ∈ C. In general, for c ∈ C we denote Δn (c) = c1 ⊗ · · · ⊗ cn+1 and owing to the above formula the meaning of this notation is (c1 )1 ⊗(c1 )2 ⊗c2 ⊗· · ·⊗cn or c1 ⊗c2 ⊗((c3 )1 )1 ⊗((c3 )1 )2 ⊗(c3 )2 ⊗c4 ⊗· · ·⊗cn−1 , and so on. Similar notation will be used in what follows when we work with different types of coalgebras, if of course we are allowed to evaluate morphisms on elements. For instance see the R-coring case considered in Example 2.14 (5). The counterpart of Proposition 2.3 is the following.

68

Algebras and Coalgebras in Monoidal Categories

Proposition 2.16 Let (F, ψ0 , ψ2 ) : C → D be an opmonoidal functor and C a coalf - D is a coalgebra gebra in C . Then F(C) is a coalgebra in D. Moreover, if C - F(D) is a coalgebra morphism in D.

F( f )

morphism in C then F(C)

Proof One can easily see that (F, ψ0 , ψ2 ) : C opp → D opp is a monoidal functor. Thus, if C is a coalgebra in C then C is an algebra in C opp and so, by Proposition 2.3, we obtain that F(C) is an algebra in D opp , and therefore a coalgebra in D. For further use note that F(C) becomes a coalgebra in D with ψ2,C,C

F(Δ )

C F(C ⊗C) −→ F(C) ⊗ F(C), ΔF(C) : F(C) −→

F(ε )

ψ

0 C ε F(C) : F(C) −→ F(1) −→ 1.

- D is a coalgebra morphism in C then D f- C is an algebra morphism F( f ) in C opp , and therefore F(D) - F(C) is an algebra morphism in D opp ; see PropoIf C

f

sition 2.3. Hence F(C)

- F(D) is a coalgebra morphism in D, as needed.

F( f )

For a coalgebra (C, ΔC , ε C ) in a monoidal category C we denote the morphism ΔC by

C

C

C C

1

 and the counit ε C by r . Then the coassociativity of ΔC and the counit

property of ε C can be written as follows: C

  C C C

C



=



C

and

C C C

C

 C  = r , r = C

C

C

respectively, where again we used the convention that in diagrammatic computations the monoidal category C is always assumed strict monoidal. Note that, in general, X

is the diagrammatic notation for the identity morphism of an object X of C . X

Thus f : C → D is a coalgebra morphism in C if and only if C

C  fh = and fh fh 

C

D D

1

h C r . r =

f

1

D D

Proposition 2.17 If (C , c) is a braided category and (C, ΔC , ε C ) is a coalgebra in −1 ◦ ΔC , ε C )) is a coalgebra C then Ccop+ := (C, cC,C ◦ ΔC , ε C ) (resp. Ccop− := (C, cC,C in C called the c-coopposite (resp. c−1 -coopposite) coalgebra associated to C. Proof

Turn upside down the diagrammatic proof of Proposition 2.4.

2.2 Coalgebras in Monoidal Categories

69

Definition 2.18 A coalgebra C in a braided category C is called cocommutative if C = Ccop+ or equivalently C = Ccop− , as coalgebras in C . We also say that C is braided cocommutative. We end this section by presenting the cross product coalgebra construction. Once more, we will omit the proofs since they can be obtained from the corresponding proofs for algebras by turning the diagrams upside down. If C, D are coalgebras in a monoidal category C and ψ : C ⊗ D → D ⊗ C is a morphism in C , we define on C ⊗ D a comultiplication which, for C strict, reads C

D

C D  , where ψ = e . Δ = (IdC ⊗ ψ ⊗ IdD ) ◦ (ΔC ⊗ ΔD ) = e

(2.2.3)

D C

C D C D

We present necessary and sufficient conditions such that C ⊗ D with the comultiplication (2.2.3) and tensor product counit morphism is a coalgebra in C . If this is the case, we call C ⊗ D a cross product coalgebra between C and D, and denote it by C#ψ D. Moreover, Δ defined in (2.2.3) will be denoted by ΔC#ψ D . Theorem 2.19 Let C be a monoidal category, C and D coalgebras in C , and ψ : C ⊗ D → D ⊗C a morphism in C . Then the following are equivalent: (i) C#ψ D is a cross product coalgebra; (ii) the following equalities hold: C D

C

e

 D D C

=

D

 e , e

D D C

C D

C

D C C

D C C

D

C D C D  C D C D e e e , r r . = = , = r r  e

e

D

D

C

C

If (C , c) is a braided category and C, D are coalgebras in C then ψ := cC,D : C ⊗ D → D ⊗C satisfies the conditions in Theorem 2.19, and therefore C#cC,D D is a cross product coalgebra in C . It will be called the c-tensor product coalgebra between C and D in C , and will be denoted by C ⊗+ D. Thus C ⊗+ D is the object C ⊗ D of C viewed as a coalgebra in C with comultiplication defined, if C is strict, by (IdC ⊗ cC,D ⊗ IdD ) ◦ (ΔC ⊗ ΔD ) and tensor product counit morphism. Similarly, by C ⊗− D we denote the coalgebra in C with comultiplication given, if C is strict, by (IdC ⊗ c−1 D,C ⊗ IdD ) ◦ (ΔC ⊗ ΔD ) and tensor product counit morphism. −1 We will call it the c -tensor product coalgebra between C and D; it coincides with C ⊗+ D considered in C in instead of C . Example 2.20 Let (φ , R) be an abelian 3-cocycle on an abelian group G, and C, D two G-graded quasicoalgebras with reassociator φ . Then Δ(c ⊗ d) =

φ (|c1 |, |c2 |, |d1 |)φ (|c1 ||d1 |, |c2 |, |d2 |) φ (|c1 ||c2 |, |d1 |, |d2 |)φ (|c1 |, |d1 |, |c2 |) R(|c2 |, |d1 |)(c1 ⊗ d1 ) ⊗ (c2 ⊗ d2 )

70

Algebras and Coalgebras in Monoidal Categories

gives the comultiplication of the c-tensor product G-graded quasicoalgebra with reassociator φ . Here c ∈ C, d ∈ D are homogeneous elements and ΔC (c) = c1 ⊗ c2 and ΔD (d) = d1 ⊗ d2 (summations implicitly understood). Consequently, if C, D are coalgebras in the braided category of super vector spaces VectZ−12 , we have that their c-tensor product coalgebra is C ⊗ D endowed with the comultiplication given by ΔC⊗+ D (c ⊗ d) = (−1)|c2 ||d1 | (c1 ⊗ d1 ) ⊗ (c2 ⊗ d2 ), for all homogeneous elements c ∈ C, d ∈ D. The counit is defined by εC⊗+ D (c ⊗ d) = εC (c)εD (d), for all c ∈ C, d ∈ D. Here we assumed that ΔC (c) decomposes as a sum of homogeneous elements as c1 ⊗ c2 , and similarly for ΔD (d). Proof

The c-tensor product coalgebra structure on C ⊗ D is given by Δ :C⊗D

ΔC ⊗ΔD

- (C ⊗C) ⊗ (D ⊗ D)

−1 aC⊗C,D,D

- ((C ⊗C) ⊗ D) ⊗ D

aC,C,D ⊗IdD

(IdC ⊗cC,D )⊗IdD

−1 aC,D,C ⊗IdD

aC⊗D,C,D

- (C ⊗ (C ⊗ D)) ⊗ D - ((C ⊗ D) ⊗C) ⊗ D

- (C ⊗ (D ⊗C)) ⊗ D

- (C ⊗ D) ⊗ (C ⊗ D).

The braided structure on VectG (φ ,R) from Proposition 1.44 yields the formula for Δ in the statement. The particular situation follows from the above arguments and the braided structure on VectZ−12 described in Example 1.47. The next result can be regarded as the dual version of Proposition 2.12; the proof is left to the reader. Proposition 2.21 For C, D, E coalgebras in a braided category C we have that (C ⊗± D) ⊗± E ∼ = C ⊗± (D ⊗± E) as coalgebras, the isomorphism being defined by the associativity constraint of C .

2.3 The Dual Coalgebra/Algebra of an Algebra/Coalgebra Unless otherwise specified, throughout this section C is a monoidal category with left/right duality. The goal is to prove that in this case there exists an equivalence between the category of algebras and algebra morphisms in C and the category of coalgebras and coalgebra morphisms in C . To this end we first show that when C is a monoidal category and A is an algebra in C that admits a left or right dual object then the dual object admits a coalgebra structure in C . An analogous result holds if we consider coalgebras instead of algebras. Proposition 2.22 Let C be a monoidal category, A an algebra in C and C a coalgebra in C . Then the following assertions hold: (i) if A has a left/right dual object then the dual has a coalgebra structure in C ; (ii) if C has a left/right dual object then the dual has an algebra structure in C .

2.3 The Dual Coalgebra/Algebra of an Algebra/Coalgebra

71

Proof (i). We only prove the left-handed version, as the right-handed one is the reverse monoidal version of the left-handed one. To see how the coalgebra structure on A∗ appears from the algebra structure of A, assume for the moment that C is left rigid. Then by Proposition 1.71 we have that opp the left dual functor (−)∗ is a strong monoidal functor from C to C opp = C . Thus, opp according to Proposition 2.3 it carries algebras to algebras. Since an algebra in C is actually a coalgebra in C we get that A∗ has a coalgebra structure in C , and this can be also viewed as a coalgebra structure in C . Working out the details, the coalgebra structure on A∗ (resp. on ∗ A if C is right rigid) is obtained with the help of the isomorphism λ (resp. λ  ) constructed in Section 1.7. More precisely, if (A, mA , η A ) is an algebra in C with left dual A∗ , we have that A , which is A∗ equipped with −1 λA,A

m∗

r−1 ∗

IdA∗ ⊗η

ev

A A A ΔA : A∗ −→ (A ⊗ A)∗ −→ A∗ ⊗ A∗ and ε A : A∗ −→ A∗ ⊗ 1 −→ A A∗ ⊗ A −→ 1,

is a coalgebra in C , where m∗A is the transposed morphism associated to mA . In diagrammatic notation, by the definition of λ −1 and (1.6.6) we have that A∗

ΔA =

  A⊗A  =



A∗

 



A∗

and

εA  =

A⊗A





r ,

1



A∗ A∗

A∗ A∗

and so for defining ΔA and ε A we only need the existence of the left dual object of A, as we assumed in the statement of the proposition. So, when C is monoidal and left rigid, from the above comments it follows that A is a coalgebra in C . Since we assumed that only A (not all the objects of C ) admits a left dual object, for the completeness of the proof we need to include a direct check for the fact that A is indeed a coalgebra in C . From the definition, the coassociativity of ΔA is equivalent to A∗

 

A∗

   

 

= ,











A∗ A∗ A∗

A∗

A∗ A∗

72

Algebras and Coalgebras in Monoidal Categories

which follows because of (1.6.6) and the fact that mA is associative. Now, A∗

 





A∗



 A∗ r A∗  (1.6.6) =

=



(1.6.6)

=

r



A∗

A∗



A∗

A∗

shows that ε A is a left counit for ΔA . That ε A is a right counit for ΔA can be proved in a similar manner. (ii) We omit this as it is the dual version of (i). Note only that when C is a coalgebra in C such that C has a left dual object C∗ in C then C∗ is an algebra in C via the following structure: C∗ C∗

 1

mC∗ =



,



η C∗

 = r . C∗

C∗

From now on by C we denote the above algebra structure on C∗ in C . Remark 2.23 is given by

The coalgebra structure in C on a right dual ∗A of an algebra A in C ∗A

 •  • Δ∗ A =



and

ε ∗A =

.

• 1

• ∗A

∗A

r

∗A

In what follows it will be denoted by A. Likewise, if C is a coalgebra in C admitting a right dual object ∗C in C then ∗C is an algebra in C via the structure given by ∗C ∗C

 •

1



m∗ C =

,

• •

 • η ∗C = r , ∗C

∗C

where, as before, the evaluation and coevaluation morphisms with a black dot are those associated to a right dual object. This algebra will be denoted by C.

2.3 The Dual Coalgebra/Algebra of an Algebra/Coalgebra

73

Examples 2.24 (1) In Hopf algebra theory the coalgebra structure on A∗ presented above is the so-called co-opposite dual coalgebra of the algebra A. To obtain the dual coalgebra of the algebra A we have to switch the tensor components of A∗ ⊗ A∗ with the help of the (symmetric) braiding of k M . This is why, in general, we keep the same terminology for the coalgebra structure on A∗ from Proposition 2.22. Namely, we call A the left co-opposite dual coalgebra of the algebra A. When C is braided then Acop+ is a coalgebra in C as well; see Proposition 2.17. We simply denote it by A∗ and call it the left dual coalgebra of the algebra A. Hence A∗

ΔA ∗ =

 



A∗

and

εA ∗ =

r

.

1

A∗ A∗

In the case of right duality we call A the right co-opposite dual coalgebra of the algebra A, and if, moreover, C is braided we call ∗ A := ( A)cop+ the right dual coalgebra of the algebra A. (2) Assume again that C is the category of vector spaces over a field k, and that C is a finite-dimensional coalgebra in k M . Then the algebra structure on C∗ = ∗C is determined by c∗ d ∗ (c) = c∗ (c2 )d ∗ (c1 ),

1C∗ (c) = ε (c), ∀ c∗ , d ∗ ∈ C∗ , c ∈ C,

and remark that actually we do not need C to be finite dimensional. Furthermore, in Hopf algebra theory this is precisely the opposite dual algebra structure of the coalgebra C. This is why, in general, for a coalgebra C in a monoidal category we call the algebra structure on C∗ /∗C from Proposition 2.22 the left/right opposite dual algebra structure of the coalgebra C. As we have already mentioned it will be denoted by C /C. Furthermore, in the situation when C is braided we denote Cop− /(C)op− by C∗ /∗C and call it, simply, the left/right dual algebra structure of the coalgebra C. The dual (co)algebra process can be iterated. For instance, if C is a coalgebra then C is an algebra, and so we can consider either the coalgebra  (C ) or (C ) . We next see that the canonical isomorphisms between an object X and its “double” duals behave well with respect to the initial (co)algebra structure of X. More precisely, we have the following results. For simplicity, we assume from the beginning that our category is rigid. Proposition 2.25 Let C be a rigid monoidal category and C a coalgebra in C . Then the isomorphisms θC : C → (C) and θC : C →  (C ) considered in Proposition 1.73 are coalgebra isomorphisms in C . Proof

Let us start by noting that the coalgebra structure of (C) coming from the

74

Algebras and Coalgebras in Monoidal Categories

left co-opposite dual coalgebra structure of the algebra C is given by (C)

Δ(C) =

 (C  )

 



(C)

and

ε (C) =



(C)

• •

(C)

(C)

 (C  )

r



=

1 C

  

=



 (C  )

 •

 • . Thus θC is a coalgebra morphism if and only if

r 1

 •

C

 

•  



• = 

• •

• (C) (C)

C

and

C  • = r .



r • 1

1

(C)

(C)

The two relations above follow easily by applying first (1.6.6) and then (1.6.9), so θC : C → (C) is indeed a coalgebra isomorphism in C . The assertion concerning θC can be proved in a similar manner, so we leave it to the reader. We consider now the braided case. Proposition 2.26 Let C be a rigid braided category and C a coalgebra in C . Then C ∼ =  (C ) ∼ = = C, as algebras in C . Consequently, C := (C ) ∼ = (C) ∼ =C∼  (C) := C, as coalgebras in C . Proof We will see that the isomorphism ΘC : C → C defined in the proof of Corollary 1.76 is an algebra morphism, too. For this we have to show that C∗ C∗ C∗

C∗

 •

 

•  •  =

 



• ∗C •

∗C

 • 1

and

1  •  • r = r .

∗C

∗C

2.3 The Dual Coalgebra/Algebra of an Algebra/Coalgebra

75

Indeed, by the naturality of the braiding and Proposition 1.49 we have

X Y

(a)

∗X

X Y

=

• Y Y

(a) ∗X

∗X

X∗ Y X

,

•

(b)



Y

  • • = ∗X

=

Y Y

Y

Y X

X∗ Y X

Y Y

 ,

(b)

;

 .

= X Y X∗

Y X

(2.3.1)

(2.3.2)

X Y X∗

With the help of these relations we compute: C∗ C∗ C∗

C∗

  •  =



∗C

• C∗ C∗  •  

=



C∗ C∗

 •

 =



=

C∗



∗C

∗C



C∗

C∗





∗C

 •

 •

=



∗C

C∗ C∗



∗C

 •



 =



∗C

C∗ C∗

 =

C∗

= 

∗C

 •

C∗

 •

 •

 =



∗C





C∗ C∗

 •

 •

 =

∗C

C∗

C∗ C∗

C∗ C∗

 =

∗C





∗C

76

Algebras and Coalgebras in Monoidal Categories  •

C∗ C∗

C∗ C∗

 =



 • 

=

 •



C∗

 •

=

 •

 •



,



• •



C∗

 •

• •

∗C

∗C

∗C

as required. More precisely, we used (2.3.2a) in the first equality, and a similar relation in the tenth equality; (1.6.6) in the second equality; Proposition 1.37 in the third, seventh, eighth, ninth and twelfth equalities; the equation (2.3.1b) in the fourth equality; the identity (2.3.2a) in the fifth and sixth equality; (1.5.10) in the eleventh equality; (1.6.9) in the last but one equality and the definition of ΘC from the proof of Corollary 1.76 in the last one. That ΘC respects the counits follows from (2.3.2b), which implies 1

 = r

Y

Y

Y C∗

Y

 , and so r C∗

1

 • • (1.6.6)  r = = r

∗C

∗C

1

 • r , ∗C

as required. Hence our proof is complete. By working with algebras instead of coalgebras one can prove the following results. As these are the dual situations the details will be skipped. Proposition 2.27 Let A be an algebra in a rigid monoidal category C . Then the morphisms θA : A → ( A) and θA : A →  (A ) defined in the proof of Proposition 1.73 are algebra isomorphisms in C . Proposition 2.28 For A an algebra in a rigid braided category C , the isomorphism   Θ−1 A : A → A from the proof of Corollary 1.76 is a coalgebra isomorphism in C . Consequently, A := (A ) ∼ =  (A ) ∼ =  ( A) :=  A, = ( A) ∼ =A∼ as algebras in C . Proof This is the “upside down” version of Proposition 2.26. More precisely, it follows by turning upside down the diagrammatic computations performed in the proof of Proposition 2.26. This is because Θ−1 A is a coalgebra morphism if and only

2.3 The Dual Coalgebra/Algebra of an Algebra/Coalgebra

77

if the equalities below hold: ∗A

∗A

 •  •



•

∗A





•  = and

•





r

, r =

• •

1 1

•

A∗ A∗

A∗

∗A

A∗

and these are precisely the ones proved in Proposition 2.26, turned upside down. If C is a monoidal category then by Coalg(C ) we denote the subcategory of C whose objects are coalgebras and morphisms are coalgebra morphisms. Likewise, by Alg(C ) we denote the subcategory of algebra objects and algebra morphisms in C . The next result contributes to the duality formalism that we have already mentioned several times so far. Corollary 2.29 If C is a rigid monoidal category then Coalg(C ) and Alg(C ) are equivalent categories. (−)

Proof

We show that Coalg(C ) - Alg(C ) is a pair of functors that produces the  (−)

desired equivalence. For this, in view of the above results, it is enough to prove that θ = (θX : X → (∗ X)∗ )X∈C and θ  = (θX : X → ∗ (X ∗ ))X∈C are natural transformaf tions. Indeed, for instance, if X - Y is a morphism in C then (∗ X)∗ ∗

∗

( f) =

(∗ X)∗

 ∗fh =



  • h f ,

•

(∗ Y ) ∗

(∗ Y ) ∗

and so X

  • (1.6.6) ∗ ∗ h f = ( f ) ◦ θX = •

• (∗Y )∗

X

X  •  (1.6.9) fh  h f = = θY ◦ f ,

•

•

• (∗Y )∗ (∗Y )∗

as required. In a similar way one can show that θ  is a natural transformation; the details are left to the reader.

78

Algebras and Coalgebras in Monoidal Categories

2.4 Categories of Representations To any algebra in a monoidal category C we associate the so-called category of representations (or modules) over it. Of course, this is inspired by the classical representation theory for k-algebras. Let C be a monoidal category and A an algebra in C . Definition 2.30 A left A-module M in C is an object M ∈ C together with a morphism μ M : A ⊗ M → M in C which is associative and unital, in the sense that (A ⊗ A) ⊗ M

mA ⊗IdM

μM

A⊗M

A⊗M

M μM

aA,A,M IdA ⊗μ M

A ⊗ (A ⊗ M)

μM

η A ⊗IdM

M

lM

1 ⊗ M.

A ⊗ M,

If (M, μ M ) and (N, μ N ) are left A-modules then a morphism f : M → N in C is called left A-linear or a morphism of left A-modules if and only if f μ M = μ N (IdA ⊗ f ). We denote by A C the category of left A-modules and left A-linear morphisms in C . A

A

M

In diagrammatic computations,  will be the short form of M

M

A A M

 =   M

r

M

M

= 

and

M

M

 , the μh M M

left A-module structure morphism μ M . Then we have A A

M

.

(2.4.1)

M

In a similar manner one can define CA , the category of right A-modules and right M

A-linear morphisms in C . For simplicity, we denote by

A

the morphism  M

M

A

 h representing the right A-module structure morphism ν M : M ⊗ A → M of M νM M

in C . The equality below together with the unit property expresses that M is a right A-module in the monoidal category C : M

A A

 =  M

M A A

.  M

(2.4.2)

2.4 Categories of Representations

79

Actually, a right A-module in C is nothing but a left A-module in C , the reverse monoidal category associated to C . We now present some concrete examples of modules in monoidal categories. Examples 2.31 (1) Any algebra A in a monoidal category is a left and right Amodule via its multiplication mA . (2) A left module in Set is a left G-set with G a monoid. A module morphism in Set is a morphism of G-sets. (3) If C is a category and T = (T, m, η ) is a monad in C then a left T -module in [C , C ] is a functor M ∈ [C , C ] together with a natural transformation μ : T ◦ M → M such that μX ◦ T (μX ) = μX ◦ mT (X) and μX ◦ η M(X) = IdM(X) , for any object X of C . A morphism of left T -modules in [C , C ] between (M, μ ) and (M  , μ  ) is a natural transformation f : M → M  satisfying fX ◦ μX = μX ◦ T ( fX ), for any X ∈ C . (4) Let G be a group and φ ∈ H 3 (G, k∗ ). If A is a G-graded quasialgebra with reassociator φ then a left A-module in VectG φ is a G-graded vector space M together with a k-linear map A ⊗ M → M (a ⊗ m → am is the left action of A on M) such that Ax My ⊆ Mxy , for all x, y ∈ G, 1A m = m and (aa )m = φ (|a|, |a |, |m|)a(a m), for all homogeneous elements a, a ∈ A and m ∈ M. When C is a braided category we have the following result. Proposition 2.32 Let C be a braided category and A an algebra in C . Then A C and CAop+ are isomorphic, where Aop+ is the c-opposite algebra associated to A. A M

Proof

If M is a left A-module then it becomes a right Aop+ -module with

M

Indeed, by Proposition 1.37, M ∈ A C and (1.5.10) we have M A

A

M A A

M A A

M A A

M A A



 =

 M

. 

= = =



  M

.

   M

M

M

This, together with Proposition 1.49, implies that M ∈ CAop+ , as claimed. Also, by the naturality of the braiding it follows that a left A-module morphism becomes a right Aop+ -module morphism. Thus we have defined a functor from A C to CAop+ .

80

Algebras and Coalgebras in Monoidal Categories A M

Similarly, if M ∈ CAop+ then M is a left A-module with

, and in this way  M

a right Aop+ -module morphism turns into a left A-module morphism. Clearly, this functor is the inverse of the one defined above, so A C and CAop+ are isomorphic. By working with dual objects, we get another way to pass from left/right to right/ left modules. Proposition 2.33 Let C be a monoidal category and A an algebra in C . If M is a left A-module in C such that M admits a left dual object M ∗ in C then M ∗ becomes a right A-module in C via the structure morphism −1 rM ∗ ⊗A

ν M∗ : M ∗ ⊗ A −→ (M ∗ ⊗ A) ⊗ 1 aM ∗ ,A,M⊗M ∗

−→

IdM ∗ ⊗A ⊗coevM

−→

M ∗ ⊗ (A ⊗ (M ⊗ M ∗ ))

IdM ∗ ⊗(μ M ⊗IdM ∗ )

−→

(M ∗ ⊗ A) ⊗ (M ⊗ M ∗ )

IdM ∗ ⊗a−1 A,M,M ∗

−→

M ∗ ⊗ ((A ⊗ M) ⊗ M ∗ )

a−1 M ∗ ,M,M ∗

M ∗ ⊗ (M ⊗ M ∗ ) −→ (M ∗ ⊗ M) ⊗ M ∗

evM ⊗IdM ∗

−→

l ∗

M 1 ⊗ M ∗ −→ M∗,

(2.4.3) where μ M : A ⊗ M → M is the structure morphism of M as a left A-module in C . Similarly, if M is a right A-module and M has a right dual object ∗ M in C then ∗ M with the structure given by −1 lA⊗ ∗M

μ ∗ M : A ⊗ ∗ M −→ 1 ⊗ (A ⊗ ∗ M) a∗ M,M,A⊗∗ M

−→

∗

M ⊗ (M ⊗ ∗ M)

( M ⊗ M) ⊗ (A ⊗ ∗ M)

−→

M ⊗ (M ⊗ (A ⊗ ∗ M))

Id∗ M ⊗(ν M ⊗Id∗ M ) ∗

−→

coevM ⊗IdA⊗∗ M ∗ Id∗ M ⊗a−1 M,A,∗ M

−→

Id∗ M ⊗evM ∗

−→

∗

M ⊗ ((M ⊗ A) ⊗ ∗ M) r∗

M M ⊗ 1 −→ M∗

(2.4.4)

becomes a left A-module in C , where ν M : M ⊗ A → M is the morphism that defines on M a right A-module structure in C . Proof This time we prove only the second assertion; the first one follows in a similar manner (or it can be viewed as the reverse monoidal version of the second one). Without loss of generality, we can assume that C is strict monoidal. Then we have A ∗M

 •

M A

  .  , where ν M :=  

• M

μ ∗M = ∗M

2.4 Categories of Representations

81

Thus ∗ M is a left A-module in C via μ∗ M since A A ∗M

A

 •

A

 •

A A ∗M

 •

∗M

 •



 

    = 



• •

=   

•

∗M ∗M

 • r

∗M

  = 

•

and ∗M

  

• ∗M

∗M

 •

= Id∗ M , as required.

• ∗M

To any algebra morphism we can associate the functor called restriction of scalars; the proof of the next result is easy and left to the reader. Proposition 2.34 Let A, B be algebras in a monoidal category C , f : A → B an algebra morphism in C , and M a left B-module in C . Then M is a left A-module in A

C via the structure morphism

h

M

f

. In this way we have a functor F : B C → A C ,  M

called the restriction of scalars (F acts as identity on morphisms). We now show that in a braided category the tensor product of two modules is still a module. This result will be used in Section 2.6 when we will study when a category of representations is a monoidal category as well. Proposition 2.35 Let C be a braided category and A, B algebras in C . If M ∈ A C and N ∈ B C then M ⊗ N is a left A ⊗+ B-module in C with A

μ M⊗N :=

B M

N

.   M

N

If we replace the braiding in the above definition by its inverse then with this new structure M ⊗ N is a left A ⊗− B-module in C . Proof

Consider only the first situation; the other one is obtained by considering C in

82

Algebras and Coalgebras in Monoidal Categories

instead of C . We compute: A B A B M N

A B A B M N A B A B M N

=   M





N

=   



N

M

 

M N

A B A B

N B M N

M A B A



 =

=



   

N

M

N

M B M N

A B A

A B A

 =

B M

  =

  M

N

N

,   M

N

as required. We used (2.4.1) in the first and sixth equality and Proposition 1.37 in the second, third, fourth and fifth equalities. By using Proposition 1.49, one can easily check that η A ⊗ η B acts as identity on M ⊗ N.

2.5 Categories of Corepresentations In this section we deal with the dual situation of Section 2.4. As we have seen earlier, a coalgebra in a monoidal category C is nothing else than an algebra in C opp , the opposite category associated to C . Thus, by duality, we can obtain the notion of a comodule over a coalgebra. Namely, a comodule in C is a module in C opp . Definition 2.36 Let C be a monoidal category and (C, Δ, ε ) a coalgebra in C . We say that N ∈ C together with a morphism ρ N : N → N ⊗C in C is a right C-comodule

2.5 Categories of Corepresentations

83

(or corepresentation) if (IdN ⊗ ε C ) ◦ ρ N = IdN and the following diagram N

ρN

N ⊗C

ρ N ⊗IdC

(N ⊗C) ⊗C

ρN

aN,C,C IdN ⊗ΔC

N ⊗C

N ⊗ (C ⊗C)

is commutative. If (N, ρ N ) and (N  , ρ N  ) are right C-comodules in C then a morphism f : N → N  in C is called right C-colinear, or a morphism of right C-comodules, if ( f ⊗ IdC ) ◦ ρ N = ρ N  ◦ f . By C C we denote the category of right C-comodules and right C-comodule morphisms in C . N



For a right C-comodule N in C we denote its structure morphism ρ N by

N

C

N

h ρN instead of  . Then we have N

C N

N

  = N

C C

  N C C

N

and

 N r = .

(2.5.1)

N

N

Likewise, we can define left C-comodules in a monoidal category C . This time the structure morphism of a left C-comodule N is of the form λ N : N → C ⊗ N. For N

N

simplicity, it will be denoted by C

C C N

N

C

N

C

C

N

N

N

 and 

N

 = 

h  instead of  . Then λN

r

 N = . In what follows, by C C we denote the category of N N

left C-comodules and left C-comodule morphisms in C . Usually, we will work with left modules over algebras and right comodules over coalgebras. Any result that we obtain for right comodules has an analogue for left comodules, because a left C-comodule can be identified with a right C-comodule in C , the reverse monoidal category associated to C . Furthermore, if C is braided we have the following result. Proposition 2.37 Let C be a coalgebra in a braided category C . Then C C and cop+ are isomorphic, where Ccop+ is the c-coopposite coalgebra associated to C. CC

84

Algebras and Coalgebras in Monoidal Categories M

 Proof

If M ∈ C C then M becomes a right Ccop+ -comodule via

, and then M C

a left C-comodule morphism becomes a right Ccop+ -comodule morphism in C . Similarly, a right Ccop+ -comodule becomes a left C-comodule with the structure M

 ; in this way a right Ccop+ -comodule morphism turns into a left

morphism C M

C-comodule morphism. The two correspondences above define functors that are inverse to each other. Since the proof of the above assertions are the formal duals of the ones in Proposition 2.32 we leave the details to the reader. Examples 2.38 (1) Any coalgebra C in a monoidal category C is a left and right C-comodule via its comultiplication Δ. (2) From Examples 2.14 (2) we know that any object X in Set has a unique coalgebra structure in Set given by ΔX (x) = (x, x) and ε X (x) = ∗, for all x ∈ X. Thus, giving a right X-comodule structure on a set N reduces to giving a map from N to X, in the sense that any right X-comodule structure on N has the form ρN (n) = (n, f (n)), for some map f : N → X. If (N, f ) and (N  , f  ) are right X-comodules in Set then a map g : N → N  is right X-colinear if and only if f  ◦ g = f . (3) If (L, Δ, ε ) is a comonad on a category C then a right L-comodule in [C , C ] is a functor N : C → C together with a natural transformation ρ : N → N ◦ L such that ρL(X) ◦ ρX = N(ΔX ) ◦ ρX and N(ε X ) ◦ ρX = IdN(X) , for all X ∈ C . A morphism f : (N, ρ ) → (N  , ρ  ) between two right L-comodules in [C , C ] is a natural transformation f : N → N  satisfying ρX ◦ fX = fL(X) ◦ ρX , for all X ∈ C . (4) If k is a field and (C, ΔC , εC ) is a k-coalgebra then a right C-comodule in k M is a pair (N, ρN ), where N is a k-vector space and ρN : N → N ⊗C is a k-linear map such that (ρN ⊗ IdC ) ◦ ρN = (IdN ⊗ ΔC ) ◦ ρN and (IdN ⊗ εC ) ◦ ρN = IdN . The sigma notation for a right C-comodule N with structure map ρN : N → N ⊗C is ρ (n) = n(0) ⊗ n(1) , for all n ∈ N. Then the conditions in the definition of a right C-comodule can now be written as (n(0) )(0) ⊗ (n(0) )(1) ⊗ n(1) = n(0) ⊗ (n(1) )1 ⊗ (n(1) )2 := n(0) ⊗ n(1) ⊗ n(2) , and ε (n(1) )n(0) = n, for all n ∈ N. Also, f : (N, ρ ) → (N  , ρ  ) is a morphism of right C-comodules in k M if f is a k-linear map obeying ( f ⊗ IdC ) ◦ ρ = ρ  ◦ f . In sigma notation this means that f (n)(0) ⊗ f (n)(1) = f (n(0) ) ⊗ n(1) , ∀ n ∈ N.

2.5 Categories of Corepresentations

85

The functor corestiction of scalars can be constructed as follows (it is the formal dual of the functor constructed in Proposition 2.34). Proposition 2.39 Let C and D be coalgebras in a monoidal category and f : C → D a coalgebra morphism in C . If M is a right C-comodule then M becomes a right DM

 . In this way we have a functor F : C C → C D , which comodule in C with fh M

D

we call the functor corestriction of scalars (F acts as identity of morphisms). We leave it to the reader to prove the dual version of Proposition 2.35. Proposition 2.40 Let C, D be coalgebras in a braided category C , M a right Ccomodule and N a right D-comodule in C . Then M ⊗ N is a right C ⊗+ D-comodule N

M

  . If we consider the inverse of the braiding instead of the in C with N C

M

D

braiding we then obtain a C ⊗− D-comodule structure for M ⊗ N in C . If the category C is left or right rigid then working with modules is equivalent to working with comodules over the co-opposite dual coalgebra. Proposition 2.41 Let C be a monoidal category and A an algebra in C .  (i) If C is right rigid then the categories A C and A C are isomorphic.  (ii) If C is left rigid then the categories CA and C A are isomorphic. ∗A Consequently, if C is braided rigid then A C and C are isomorphic, and also CA ∗ and A C are isomorphic. Proof Since ∗ A = ( A)cop+ the last assertions are consequences of the first ones and of Proposition 2.37.   We will prove only the isomorphism A C ∼ = C A can = A C ; the isomorphism CA ∼ A be proved in a similar manner. For this, we define first a functor F : A C → C . F acts as identity on objects and morphisms, and if M is a left A-module in C then F(M) becomes a left  A-comodule in C with the structure morphism

 •

,  ∗A

A

M

where  is the left A-module structure of M in C . Indeed, we have M • r 

M (1.6.9)

=

 • M

r

M

=  M

M

, M

M

M

86

Algebras and Coalgebras in Monoidal Categories M

M

M •  • • •   • •   = ,



=  •

• ∗A ∗A

 ∗A ∗A

M

M

 ∗A ∗A

M

by (1.6.9) and the fact that M ∈ A C . One can easily check that in this way an A-linear morphism becomes a left  A-colinear morphism in C , and so F is indeed a functor. For the other way around, if M is a left  A-comodule in C then by G(M) we denote A

M

 . We have

the same object M endowed with the left A-action

• M A A

M



A A

A A

M

M

•   •

 

• = = 



•

• •

•

• 

• M M A A



M



which is

r

M

=

•

M

M



; see (1.6.9). Also,

•

M

M

r

 M = , and thereM M

fore G(M) is a left A-module. It can be easily checked that any morphism f : M → N  in A C turns into a morphism from G(M) to G(N) in A C . Thus we have a well defined functor G : A C → A C . By (1.6.9) it follows that F and G define a pair of  inverse functors, hence the categories A C and A C are isomorphic. The dual version of Proposition 2.41 is the following. Its proof is left to the reader. Proposition 2.42 Let C be a monoidal category and C a coalgebra in C . (i) If C is right rigid then the categories C C and CC are isomorphic. (ii) If C is left rigid then the categories C C and C C are isomorphic. Consequently, if C is braided rigid then C C and ∗C C are isomorphic, and also C C and C ∗ are isomorphic. C Another result involving dual objects that we can prove is the following. Proposition 2.43 Let C be a coalgebra in a monoidal category C . (i) Let (V, λ V ) be a left C-comodule admitting a right dual ∗V , with evaluation

2.6 Braided Bialgebras

87

and coevaluation morphisms evV and coevV . Then ∗V becomes a right C-comodule, with right C-coaction ∗

l∗−1

V V −→ 1 ⊗ ∗V

a−1 ∗ V,C,V ⊗Id∗ V

−→

coevV ⊗Id∗V ∗

((∗V ⊗C) ⊗V ) ⊗ ∗V

Id∗V ⊗C ⊗evV ∗

−→

(Id∗V ⊗λ V )⊗Id∗V ∗

( V ⊗V ) ⊗ ∗V

−→

( V ⊗ (C ⊗V )) ⊗ ∗V

−→

a∗V ⊗C,V,∗V

−→

(∗V ⊗C) ⊗ (V ⊗ ∗V )

r∗V ⊗C

( V ⊗C) ⊗ 1 −→ ∗V ⊗C .

(ii) In a similar way, if (V, ρ V ) is a right C-comodule admitting a left dual V ∗ , with evaluation and coevaluation morphisms evV and coevV , then V ∗ becomes a left C-comodule, with left C-coaction r−1 ∗

V V∗ ⊗1 V ∗ −→

IdV ∗ ⊗aV,C,V ∗

−→

evV ⊗IdC⊗V ∗

−→

IdV ∗ ⊗coevV

−→

V ∗ ⊗ (V ⊗V ∗ )

V ∗ ⊗ (V ⊗ (C ⊗V ∗ ))

IdV ∗ ⊗(ρ V ⊗IdV ∗ )

aV−1∗ ,V,C⊗V ∗

−→

−→

V ∗ ⊗ ((V ⊗C) ⊗V ∗ )

(V ∗ ⊗V ) ⊗ (C ⊗V ∗ )

lC⊗V ∗

1 ⊗ (C ⊗V ∗ ) −→ C ⊗V ∗ .

Proof Assume that C is strict monoidal, and then look through a mirror at the upside down version of the diagrammatic proof of Proposition 2.33.

2.6 Braided Bialgebras Let (C , ⊗, 1, a, l, r, c) be a pre-braided category and H = (H, m, η ) an algebra in C equipped with two algebra morphisms Δ : H → H ⊗+ H and ε : H → 1 in C . Here 1 has the algebra structure in C given by l1 = r1 : 1 ⊗ 1 → 1 and Id1 . If X,Y ∈ H C then, by Proposition 2.35, X ⊗Y ∈ H⊗+ H C . Since Δ and ε are algebra morphisms in C , by Proposition 2.34 we get that X ⊗Y and 1 are objects of H C via the structure morphisms given by H

X Y



H

and   X

ε := r ,

(2.6.1)

1

Y

H

respectively, where

 is, as usual, the notation for the morphism Δ : H → H ⊗ H. H H

Clearly, if f and g are morphisms in H C then f ⊗ g is left H-linear, too. So the tensor product of C induces a functor from H C × H C to H C . Proposition 2.44 With notation as above, (H C , ⊗, 1, a, l, r) is a monoidal category if and only if (H, Δ, ε ) is a coalgebra in C .

88

Algebras and Coalgebras in Monoidal Categories

Proof Without loss of generality we can assume that C is strict monoidal. We have that (H C , ⊗, 1, a, l, r) is a monoidal category if and only if a, l and r are defined by families of isomorphisms in H C . Now, according to (2.6.1), for X,Y, Z ∈ H C we have that aX,Y,Z is an H-linear morphism if and only if H

X Y

Z

 

H

Y

Z

=   

  X

X



Y

,

 

Z

X

Y

Z

if and only if H

H

X Y



Z



=    Y

Z





X

X Y

,   

Z X

Y

Z

because of Proposition 1.37. It is clear now that the above equality is satisfied if Δ is coassociative. Conversely, if the above equality holds for any X,Y, Z ∈ H C then take X = Y = Z = H and regard them as left H-modules via the multiplication m of H. Composing both sides of the resulting equality to the right with IdH ⊗ η H ⊗ η H ⊗ η H , by Proposition 1.49 we get that Δ is coassociative, as required. Now, for an object X ∈ H C we have that lX : 1 ⊗ X → X is left H-linear if and H

X

H X  =  . By similar arguments as above the latest equality is only if r  X

X H

 H = . In a similar manner one can show that rX : X ⊗ 1 → X is equivalent to r H

H H

 H left H-linear if and only if r = . Thus l and r of C induce unit constraints for HC

H

H

if and only if ε is a counit for Δ, and this finishes the proof.

We can also consider the dual situation. Namely, consider a coalgebra (H, Δ, ε ) in

2.6 Braided Bialgebras

89

a pre-braided category C and two coalgebra morphisms m : H ⊗+ H → H and η : + 1 → H. If X,Y ∈ C H then X ⊗Y ∈ C H⊗ H , by Proposition 2.40. By Proposition 2.39 we obtain that X ⊗Y ∈ C H , and so ⊗ induces a tensor product on C H . In fact, X ⊗Y X

Y

  is a right H-comodule via

X

Y

H H

, where is the standard notation H

H

1

1

for m. If η = r then 1 is a right H-comodule via r . If we turn the diagrams in the H

H

proof of Proposition 2.44 upside down we obtain a proof for the following result. Proposition 2.45 Let (H, Δ, ε ) be a coalgebra in a pre-braided category C and m : H ⊗+ H → H and η : 1 → H two coalgebra morphisms in C . Then the category C H equipped with the tensor product defined above, unit object 1 of C viewed as a right H-comodule via η , and with the constraints a, l, r of C , is a monoidal category if and only if (H, m, η ) is an algebra in C . The two results presented above can be unified by introducing the notion of bialgebra in a pre-braided category. Definition 2.46 A bialgebra H = (H, m, η , Δ, ε ) in a pre-braided category C is an algebra (H, m, η ) and a coalgebra (H, Δ, ε ) in C such that Δ : H → H ⊗+ H and ε : H → 1 are algebra morphisms. A morphism of bialgebras in C is a morphism in C that is both an algebra and a coalgebra morphism. In diagrammatic notation, the axioms for Δ and ε to be algebra morphisms in C read as H H

= H H

H

H

 ,

H

H

1

r

H H 1

= r r  H H

H H

and

H H = r r , r 1 1

1

r r 1

1

= 1

(2.6.2) respectively. Having in mind the c-tensor product coalgebra construction we obtain: Corollary 2.47 H = (H, m, η , Δ, ε ) is a bialgebra in (C , c) if and only if (H, m, η ) is an algebra in C , (H, Δ, ε ) is a coalgebra in C , and m : H ⊗+ H → H and η : 1 → H are coalgebra morphisms in C . Consequently, if H is a bialgebra in C then both H C and C H are monoidal categories. A bialgebra in a pre-braided category will also be called a braided bialgebra. If C is braided then to a braided bialgebra we can associate another four braided bialgebras. Proposition 2.48

Let (B, m, η , Δ, ε ) be a bialgebra in a braided category C .

90

Algebras and Coalgebras in Monoidal Categories

Denote by Bop− the object B endowed with the c−1 -opposite multiplication associated to m and with the same coalgebra structure as that of B, and by Bcop− the object B endowed with the algebra structure of B and with the c−1 -coopposite coalgebra structure associated to Δ. Similarly, denote by Bop+,cop− the object B with the algebra structure of Bop+ and coalgebra structure of Bcop− , and by Bop−,cop+ the object B equipped with the algebra structure of Bop− and coalgebra structure of Bcop+ . Then Bop− and Bcop− are bialgebras in C in , and Bop+,cop− and Bop−,cop+ are bialgebras in C . Proof We use the fact that B is a bialgebra in C , Proposition 1.37 and the definition of the multiplication of Bop− to compute B

B



B B

B

B



B

=  B B

 =

B



B

B

 =

=



B



B B

B

B



,

B

B

B

is a bialgebra in C in . If we turn the above computation and this proves that upside down we get a proof for the fact that Bcop− is a bialgebra in C in , too. We next show that Bop+,cop− is a bialgebra in C . Actually, using the fact that Bcop− is a bialgebra in C in and arguments similar to the ones above we get that Bop−

B

B



B

B



B B

B

B



B B

=

 =

B B

B

B

= 



=

.



B

B

Once again, the proof for the fact that Bop−,cop+

B

B

B

B

is a bialgebra in C follows by turning

the above diagrams upside down. We next supply a list of examples of braided bialgebras. Example 2.49 A bialgebra in Set is a monoid. Indeed, we know from Example 2.2 (2) that an algebra in Set is a monoid, and by Example 2.14 (2) that any set has a unique coalgebra structure in Set. Thus a bialgebra in Set must have the form

2.6 Braided Bialgebras

91

(H, m, η , Δ, ε ), with m defined by the multiplication of the monoid H, η defined by the neutral element e of H, and with Δ(h) = (h, h) and ε (h) = {∗}, for all h ∈ H. Now, H × H is an algebra in Set (i.e. a monoid) with multiplication (h, g)(h , g ) = (hh , gg ) and unit (e, e). From here we easily conclude that Δ and ε are algebra morphisms, and so (H, m, η , Δ, ε ) is a bialgebra in Set. Example 2.50 Let k be a field and k M the category of k-vector spaces. A bialgebra in k M , which we will call a k-bialgebra, is a k-algebra H which is also a k-coalgebra such that, for all h, g ∈ H, Δ(hg) = Δ(h)Δ(g) = h1 g1 ⊗ h2 g2

and

Δ(1H ) = 1H ⊗ 1H ,

ε (hg) = ε (h)ε (g) and ε (1H ) = 1. Example 2.51

Let k be a field and H =

k[X] (X 2 )

= k1 ⊕ kx, the k-algebra generated

by 1 and x with relation x2 = 0. Then H is a bialgebra within VectZ−12 , the braided category of super vector spaces defined in Example 1.47, with structure given by Δ(x) = x ⊗ 1 + 1 ⊗ x and ε (x) = 0, extended by linearity and as algebra morphisms from H to H ⊗+ H and from H to k in VectZ−12 , respectively. Since H0 = k1 and H1 = kx we have Δ(x)2 = (x ⊗ 1 + 1 ⊗ x)(x ⊗ 1 + 1 ⊗ x) = (−1)|1||x| x2 ⊗ 1 + (−1)|1||1| x ⊗ x + (−1)|x||x| x ⊗ x + (−1)|x||1| 1 ⊗ x2 = 0, and so Δ is well defined; obviously, ε is also well defined. It is easy to see that Δ is coassociative and that ε is a counit for it. Thus H is a bialgebra in VectZ−12 , as stated. The above computation shows that Δ is not well defined in k M , unless char(k) = 2, so in general H is not an ordinary k-bialgebra. We point out that a bialgebra in VectZ−12 is usually called a super bialgebra. A pre-braided monoidal functor carries bialgebras to bialgebras. Proposition 2.52 Let (F, ϕ0 , ϕ2 ) : C → D be a pre-braided monoidal functor between the pre-braided categories (C , c) and (D, d). If H is a bialgebra in C then F(H) is a bialgebra in D. Proof Let (H, mH , η H , ΔH , ε H ) be the bialgebra in the statement. By Propositions 2.3 and 2.16, F(H) has an algebra and a coalgebra structure in D. We show that with this structures F(H) is a bialgebra in D. To this end observe first that −1 −1 (ϕ2,H,H ⊗ ϕ2,H,H )(IdF(H) ⊗ cF(H),F(H) ⊗ IdF(H) )(ϕ2,H,H ⊗ ϕ2,H,H ) −1 = (ϕ2,H,H ⊗ ϕ2,H,H )(IdF(H) ⊗ ϕ2,H,H ⊗ IdF(H) )(F(IdH ) ⊗ F(cH,H ) ⊗ F(IdH )) −1 −1 (IdF(H) ⊗ ϕ2,H,H ⊗ IdF(H) )(ϕ2,H,H ⊗ ϕ2,H,H ) −1 −1 = (ϕ2,H,H ⊗ ϕ2,H,H )((IdF(H) ⊗ ϕ2,H,H )ϕ2,H,H⊗H ⊗ IdF(H) ) (F(IdH ⊗ cH,H ) −1 −1 ⊗ F(IdH )) (ϕ2,H,H⊗H (IdF(H) ⊗ ϕ2,H,H ) ⊗ IdF(H) )(ϕ2,H,H ⊗ ϕ2,H,H ) −1 −1 = (ϕ2,H,H ⊗ ϕ2,H,H )((ϕ2,H,H ⊗ IdF(H) )ϕ2,H⊗H,H ⊗ IdF(H) )(F(IdH ⊗ cH,H )

92

Algebras and Coalgebras in Monoidal Categories −1 −1 ⊗ F(IdH ))(ϕ2,H⊗H,H (ϕ2,H,H ⊗ IdF(H) ) ⊗ IdF(H) )(ϕ2,H,H ⊗ ϕ2,H,H ) −1 = (IdF(H⊗H) ⊗ ϕ2,H,H )(ϕ2,H⊗H,H ⊗ IdF(H) )(F(IdH ⊗ cH,H ) ⊗ F(IdH )) −1 (ϕ2,H⊗H,H ⊗ IdF(H) )(IdF(H⊗H) ⊗ ϕ2,H,H ) −1 −1 = (IdF(H⊗H) ⊗ ϕ2,H,H )(ϕ2,H⊗H,H ⊗ IdF(H) )ϕ2,H⊗H⊗H,H −1 F(IdH ⊗ cH,H ⊗ IdH )ϕ2,H⊗H⊗H,H (ϕ2,H⊗H,H ⊗ IdF(H) )(IdF(H⊗H) ⊗ ϕ2,H,H ) −1 = ϕ2,H⊗H,H⊗H F(IdH ⊗ cH,H ⊗ IdH )ϕ2,H⊗H,H⊗H ,

where, for simplicity, we assumed that C is strict monoidal. We used the fact that F is a pre-braided monoidal functor in the first equality, the naturality of ϕ2 in the second and the fifth equality, and the commutativity of the first diagram in Definition 1.22 for the third and the sixth equality. This fact allows us to compute: −1 F(ΔH mH )ϕ2,H,H ΔF(H) mF(H) = ϕ2,H,H −1 = ϕ2,H,H F(mH ⊗ mH )F(IdH ⊗ cH,H ⊗ IdH )F(ΔH ⊗ ΔH )ϕ2,H,H −1 = (F(mH ) ⊗ F(mH ))ϕ2,H⊗H,H⊗H F(IdH ⊗ cH,H ⊗ IdH )

ϕ2,H⊗H,H⊗H (F(ΔH ) ⊗ F(ΔH )) = (F(mH )ϕ2,H,H ⊗ F(mH )ϕ2,H,H )(IdF(H) ⊗ cF(H),F(H) ⊗ IdF(H) ) −1 −1 F(ΔH ) ⊗ ϕ2,H,H F(ΔH )) (ϕ2,H,H

= (mF(H) ⊗ mF(H) )(IdF(H) ⊗ cF(H),F(H) ⊗ IdF(H) )(ΔF(H) ⊗ ΔF(H) ), as required, where we applied the naturality of ϕ2 in the third equality and the relation obtained above in the fourth equality. We also have: −1 F(ΔH η H )ϕ0 ΔF(H) η F(H) = ϕ2,H,H −1 = ϕ2,H,H F(η H ⊗ η H )F(l1−1 )ϕ0 −1 = (F(η H ) ⊗ F(η H ))ϕ2,1,1 F(l1−1 )ϕ0 −1 = (F(η H ) ⊗ F(η H ))(ϕ0 ⊗ IdF(1) )lF(1) ϕ0

= (F(η H ) ⊗ F(η H ))(ϕ0 ⊗ ϕ0 )l1−1 , and so ΔF(H) is an algebra morphism. This time we used in the third equality the naturality of ϕ2 , in the fourth equality the commutativity of the first square diagram in Definition 1.22, and in the last equality the naturality of l. Similarly, one can prove that ε F(H) is an algebra morphism, for instance

ε F(H) mF(H) = ϕ0−1 F(l1 )F(ε H ⊗ ε H )ϕ2,H,H = ϕ0−1 F(l1 )ϕ2,1,1 (F(ε H ) ⊗ F(ε H )) = ϕ0−1 lF(1) (ϕ0−1 ⊗ IdF(1) )(F(ε H ) ⊗ F(ε H )) = l1 (ε F(H) ⊗ ε F(H) ). The remaining details are left to the reader. Let C be a rigid braided category and H an object of C that has both an algebra structure (H, mH , η H ), and a coalgebra structure (H, ΔH , ε H ) in C .

2.6 Braided Bialgebras

93

Proposition 2.53 With C , H as above we have that ΔH is multiplicative if and only if ΔH  is multiplicative, if and only if Δ H is multiplicative. Proof

We use the left dual structures of H to compute H∗ H∗

H∗ H∗

  H∗ H∗  (1.6.6)

 = =



H∗ H∗ 



 



.



H∗ H∗

H∗ H∗

Similarly, (mH ∗ ⊗ mH ∗ )(IdH ∗ ⊗ cH ∗ ,H ∗ ⊗ IdH ∗ )(ΔH ∗ ⊗ ΔH ∗ ) is equal to H∗ H∗ H∗



H∗

   







 









 







(1.6.6)

=



  







H



H

H∗ H∗ H∗ H∗

  



=

H∗ H∗





(1.6.6)

=















 











 H∗ H∗

H∗ H∗

94

Algebras and Coalgebras in Monoidal Categories H∗ H∗

 



=



H∗ H∗

 

  

(1.6.6)

=



.













H∗ H∗

H∗ H∗

It is now clear that ΔH  is multiplicative if and only if ΔH is. Since a right dual in C is a left dual in the reverse monoidal category associated to C , the second equivalence follows by applying the above arguments to C instead of C . From the previous result and the one below we can conclude that the bialgebra notion is selfdual, provided the category is rigid. Proposition 2.54 If C is a rigid braided category and H in C has both an algebra and a coalgebra structure, then H is a bialgebra if and only if H  is a bialgebra, if and only if  H is a bialgebra. Proof We have proved in Proposition 2.53 that ΔH is multiplicative if and only if ΔH  is so. Furthermore, H∗ H∗ H∗

= r 1



H∗ H∗ H∗

and r r = 1

r 

(1.6.6)

r



1

H∗

H∗ H∗



H∗

=



1

r

r . From here we conclude that ε H  is multiplicative if and 

1 1

1

 

1

r

only if

r 1 = r r . In a similar way we compute that =   H H r ∗ ∗

H H

H

H

H∗ H∗

and

2.7 Braided Hopf Algebras

95

1

 r r = r  , and therefore ΔH  = η  ⊗ η  if and only if ε is multiplicaH H H r ∗ ∗ 1

H

H

H∗ H∗

tive. The reader can check that ε H  ◦ η H  = Id1 if and only if ε H ◦ η H = Id1 . Definition 2.55 Let H be a bialgebra in a braided rigid category C . We call H  (resp.  H) the left (resp. right) op-cop dual braided bialgebra associated to H. Since C is braided, by Proposition 2.48 we have that H op−,cop+ and ( H)op−,cop+ are bialgebras in C , too. We will call H op−,cop+ the left dual braided bialgebra of H, and ( H)op−,cop+ the right dual braided bialgebra of H. These two will be simply denoted by H ∗ , and ∗ H, respectively. In the symmetric monoidal case the left and right duals are isomorphic as bialgebras in C . Proposition 2.56 If C is a symmetric category and H is a bialgebra in C then  H and H  , and so also ∗ H and H ∗ , are isomorphic as braided bialgebras. Proof From Proposition 2.26 we know that ΘH : H  →  H is an algebra isomorphism, and from Proposition 2.28 we have that ΘH : H  →  H is a coalgebra isomorphism. Moreover, ΘH is the morphism ΘH regarded in C in , so ΘH = ΘH when C is symmetric; so in this case ΘH is a bialgebra isomorphism.

2.7 Braided Hopf Algebras We can achieve now the main goal of this chapter, namely to introduce the concept of a braided Hopf algebra. We first need the following result. Lemma 2.57 Let C be a monoidal category, (A, m, η ) an algebra in C and (C, Δ, ε ) a coalgebra in C . Then HomC (C, A) becomes an algebra in Set, that is, a monoid; see Example 2.2 (2). The multiplication is defined by f ∗ g = mA ( f ⊗ g)ΔC , for all f , g ∈ HomC (C, A), and the unit is η A ε C . The multiplication ∗ is called the convolution product and the invertible elements in HomC (C, A) are called convolution invertible. Proof

The multiplication ∗ on HomC (C, A) is associative since C



C



C



hh   ( f ∗ g) ∗ h = fh gh = fh gh hh= fh gh hh= f ∗ (g ∗ h) ,







A

A

A

96

Algebras and Coalgebras in Monoidal Categories

for all f , g, h ∈ HomC (C, A). The following computations show that the morphism η A ε C is a unit for the multiplication ∗: C

C  C r gh r gh= gh ; = r

A

A

A

C

 C C fh r fh r fh. = = r

A

A A

Note that in the above computations we used the coassociativity of Δ, the associativity of m, and the unit and counit axioms, respectively. Definition 2.58 Let (C , c) be a pre-braided category. A bialgebra H in C is called a Hopf algebra in C if the identity morphism IdH is an invertible element in the monoid HomC (H, H) considered in Lemma 2.57. The inverse of IdH will be denoted by S and will be called the antipode of H. Remark 2.59 A Hopf algebra in a pre-braided category will be also called a braided Hopf algebra or a braided group. Thus, a braided bialgebra H is a braided Hopf algebra if and only if there exists a morphism S : H → H in C such that S ∗ IdH = IdH ∗ S = η H ε H , where ∗ was defined in Lemma 2.57. More precisely, S must satisfy H

H

 H  r Sh Sh. = r =



H

H

(2.7.1)

H

Note that the antipode of a braided Hopf algebra H is uniquely determined by the equalities in (2.7.1). We supply a list of examples of braided Hopf algebras. By using different techniques, more examples will be constructed later in this book, especially within the braided category of Yetter–Drinfeld modules over a quasi-Hopf algebra. Examples 2.60 (1) A Hopf algebra in Set is a group. (2) A Hopf algebra in a category of vector spaces k M is a k-bialgebra H together with a k-linear map S : H → H such that S(h1 )h2 = h1 S(h2 ) = ε (h)1H , ∀ h ∈ H. A Hopf algebra in k M will be called a Hopf k-algebra. (3) The k-super bialgebra described in Example 2.51 is a Hopf algebra in VectZ−12 with antipode defined by S(1) = 1 and S(x) = −x, extended by linearity and as an anti-algebra morphism of H. We call a Hopf algebra in VectZ−12 a super Hopf algebra. Definition 2.61

If H, K are Hopf algebras within a pre-braided category C then a

2.7 Braided Hopf Algebras

97

morphism of Hopf algebras between H and K is a bialgebra morphism between H and K in C . A Hopf algebra morphism automatically respects the antipodes. Proposition 2.62 If H, K are Hopf algebras in C with antipodes SH and SK , respectively, and f : H → K is a bialgebra morphism in C , then SK ◦ f = f ◦ SH . H

Proof

 fh   fh Sh Sh We will compute in two different ways, where, for simplicity, we fh



K

denoted SH by S and SK by S . More precisely, on the one hand, we use the fact that f is an algebra morphism and (2.7.1) to compute H

H

  fh fh     Sh fh Sh Sh Sh = = fh



fh

K



H

H   H fh r hr fh r = f .   r = Sh Sh  fh Sh

K

K

K

K

On the other hand, by using the fact that ΔH is coassociative, f is a coalgebra morphism, mK is associative and (2.7.1) we get that H

H

H

H     fh fh Sh fh Sh Sh   fh fh  fh Sh Sh fh fh fh= = = =   fh Sh Sh  Sh







K

K

K



H

 H  H fh h r Sh= r Sh= S , fh r fh fh



K

K

K

K

as desired. This finishes the proof. Lemma 2.63

Let (F, ϕ0 , ϕ2 ) : C → D be a strong monoidal functor between two

98

Algebras and Coalgebras in Monoidal Categories

monoidal categories, A an algebra in C and C a coalgebra in C . Then for f , g ∈ HomC (C, A), we have F( f ∗ g) = F( f ) ∗ F(g). Proof

From the naturality of ϕ2 , it follows that

F( f ⊗ g)ϕ2,C,C = ϕ2,A,A (F( f ) ⊗ F(g)) : F(C) ⊗ F(C) → F(A ⊗ A).

(2.7.2)

Then we easily compute that F( f ) ∗ F(g) = mF(A) ◦ (F( f ) ⊗ F(g)) ◦ ΔF(C) −1 = F(mA ) ◦ ϕ2,A,A ◦ (F( f ) ⊗ F(g)) ◦ ϕ2,C,C ◦ F(ΔC )

= F(mA ◦ ( f ⊗ g) ◦ ΔC ) = F( f ∗ g), as stated. A pre-braided monoidal functor behaves well with respect to the antipodes. Proposition 2.64 Let (F, ϕ0 , ϕ2 ) be a pre-braided monoidal functor between the pre-braided categories C and D. If H is a Hopf algebra in C , then F(H) is a Hopf algebra in D. Proof We have already seen that our result holds for bialgebras. If S is an antipode for H, then it follows from Lemma 2.63 that F(S) is an antipode for F(H). The next result says that the antipode of a braided Hopf algebra is an anti-algebra and an anti-coalgebra homomorphism of H. Proposition 2.65 Let (C , c) be a pre-braided category and H a Hopf algebra in C with antipode S. Then the following relations hold: H H

(a)

H H

1

H

r = Sh Sh, Sh= r ; Sh

H H

1

H H

(2.7.3)

1

1

H

H

H

 H H Sh (b) = Sh Sh , r = r .  h

S

H H

Proof We only prove the relations in (2.7.3a) (the ones in (2.7.3b) can be proved by turning the diagrams in the proof of (2.7.3a) upside down). To prove the first equality in (2.7.3a) we compute: H

H



H

H



H

H



H

H

H H   H H

r 

r Sh Sh 

Sh Sh r Sh Sh = = = = = Sh Sh.

h

S r h h Sh S S



Sh Sh

r



H



H

H

H H

H

2.7 Braided Hopf Algebras

99

We used the associativity of m in the first equality, (2.6.2) in the second and third equalities, and Proposition 1.49 in the penultimate equality. But, on the other hand,

H H

H

H



H



H



 Sh Sh

H



H

 

 h

S

 Sh Sh

 Sh

= = =





Sh





Sh



Sh





Sh H

H

H  Sh Sh

H H H



H





 h

S

H

h

S



= 

Sh

=

H







S











H

H H

H



= r = r Sh

H

H

H

 H H r

= ,

Sh h

S

H

H

H



 = Sh Sh





h S





Sh =

h

H



 Sh

S

H



h

H

H

100

Algebras and Coalgebras in Monoidal Categories

where this time we used: (2.6.2) in the first equality, the coassociativity of Δ in the second and fourth equalities, Proposition 1.37 in the third and seventh equalities, the associativity of m in the fifth equality, (2.7.1) in the sixth and eighth equalities, Proposition 1.49 in the ninth equality, and the counit property in the last equality. Comparing the two computations above we obtain the desired relation. The second relation in (2.7.3) follows, for instance, by “applying” the unit of H to the upper parts of the two members of the first equality in (2.7.1). If H is a Hopf algebra in a braided rigid category then its duals also have braided Hopf algebra structures. We denoted by H  (resp.  H) the left (resp. right) dual of H endowed with the left (resp. right) (co-)opposite (co)algebra structure dual to the underlying (co)algebra structure of H. A similar notation was used for H ∗ /∗ H. The result below can be seen as a completion of Proposition 2.54. Proposition 2.66 Let H be a Hopf algebra with antipode S in a braided rigid category C . Then H  and  H, and so also H ∗ and ∗ H, are Hopf algebras in C with antipodes S∗ and ∗ S, respectively. Proof As usual, we prove only the statement related to H  . By the dual structure of H  and the properties of S we have H∗ H∗

 ∗ Sh =



H∗

   Sh







H∗





 Sh

(1.6.6)

=





 H∗

=  , r



H∗

r

H∗ H∗

and similarly H∗ H∗

 ∗ Sh =

H∗

H∗

   Sh







 



=

 Sh





H∗

This finishes the proof of the proposition.

(1.6.6)

 H∗

r

= . r H∗

H∗

2.8 Notes

101

2.8 Notes The concept of monoidal algebra was introduced by Majid in [146] as a generalization to the monoidal setting of a reformulation at the level of commutative diagrams of the old notion of k-algebra. The cross product algebra construction was first considered in the particular case when C is a category of vector spaces. It was introduced independently and with different names in several papers, see for instance [69, 215, 67]. Later, Bespalov and Drabant generalized this construction to an arbitrary monoidal category; see [32, 33]. The notion of monoidal coalgebra is from [146], and the cross product coalgebra construction is from [32, 33]. As we have already explained, some of the results in this chapter are formal dualizations of properties valid for monoidal algebras. The results in the sections about (co)representations are considered folklore. Hopf algebras appeared in 1941 in a work in topology by H. Hopf, but Milnor and Moore founded the modern theory of Hopf algebras in [156]. For more information about Hopf algebra theory we recommend the books [1, 73, 157, 189, 209]. The concept of braided Hopf algebra was introduced by Majid in [146] as a natural generalization of the classical concept of Hopf algebra. The result stating that the notion of braided bialgebra is self-dual is taken from [212]. The observation that the dual (co)algebra structure makes sense in any monoidal category is taken from [148].

3 Quasi-bialgebras and Quasi-Hopf Algebras

We introduce the concepts of quasi-bialgebra and quasi-Hopf algebra by using a categorical point of view. We present the basic properties of these objects and study their invariance under a twist. We also introduce the dual notions, called dual quasi-bialgebra and dual quasi-Hopf algebra.

3.1 Quasi-bialgebras Throughout, from now on, by an algebra we mean a unital associative algebra over a field k, and any algebra or anti-algebra morphism is assumed to be unital. Usually we denote the multiplication of an algebra by mA : A ⊗ A a ⊗ b → ab ∈ A and the unit by 1A . If A is an algebra and M is a left (resp. right) A-module, unless otherwise specified, we will denote the action of A on M by A ⊗ M a ⊗ m → a · m ∈ M (resp. M ⊗ A m ⊗ a → m · a ∈ M). A k-linear map will be simply called a linear map. In this section we will construct non-strict monoidal structures on the category of representations of an ordinary k-algebra endowed with some additional structures. This will lead to the notion of quasi-bialgebra. Definition 3.1 Let C , D be monoidal categories and F : C → D a functor between them. We call F a quasi-monoidal functor if there exist a natural isomorphism ϕ2 = (ϕ2,X,Y : F(X)F(Y ) → F(X ⊗Y ))X,Y ∈C and an isomorphism ϕ0 : F(1) → I in D (without any further conditions). Here ⊗ and 1 stand for the tensor product and the unit object of C , while  and I are the tensor product and the unit object of D, respectively. It is immediate that any strong monoidal functor is quasi-monoidal. This justifies our terminology and also the concept of quasi-bialgebra that we will introduce soon. Throughout this section k is a field and H is a k-algebra with unit 1H . We next investigate the monoidal structures on H M for which the forgetful functor F : H M → k M is a quasi-monoidal functor. Lemma 3.2 Let H be a k-algebra. Then giving a monoidal structure on H M such that the forgetful functor F : H M → k M is a quasi-monoidal functor is equivalent

104

Quasi-bialgebras and Quasi-Hopf Algebras

to giving a monoidal structure on H M that comes by a restriction of the monoidal structure on k M to H M . More precisely, this means that (a) for any two left H-modules X,Y the tensor product X ⊗Y in k M admits a left H-module structure; (b) the tensor product in k M of two left H-module morphisms is a morphism in M , and so ⊗ induces a functor from H M × H M to H M ; H (c) k, the unit object of k M , admits a left H-module structure; (d) there exist functorial isomorphisms a = (aX,Y,Z : (X ⊗Y ) ⊗ Z → X ⊗ (Y ⊗ Z))X,Y,Z ∈ H M , l = (lX : k ⊗ X → X)X in

HM

∈ HM

and r = (rX : X ⊗ k → X)X

∈ HM

such that the Pentagon Axiom and the Triangle Axiom are satisfied.

Proof Everything follows from the definition of a quasi-monoidal functor, since in our case it acts as identity on objects and morphisms. The next result is a reconstruction type theorem for quasi-bialgebras. Proposition 3.3 Let k be a field and H a k-algebra. Then there exists a one-to-one correspondence between • monoidal structures on H M such that the forgetful functor F : H M → k M is a quasi-monoidal functor; • 5-tuples (Δ, ε , Φ, l, r) consisting of two k-algebra maps Δ : H → H ⊗ H and ε : H → k and invertible elements Φ ∈ H ⊗ H ⊗ H and l, r ∈ H such that, for all h ∈ H, the following relations hold:   (IdH ⊗ Δ)(Δ(h)) = Φ (Δ ⊗ IdH )(Δ(h)) Φ−1 , (ε ⊗ IdH )(Δ(h)) = l −1 hl,

(IdH ⊗ ε )(Δ(h)) = r−1 hr,

(IdH ⊗ IdH ⊗ Δ)(Φ)(Δ ⊗ IdH ⊗ IdH )(Φ) = (1H ⊗ Φ)(IdH ⊗ Δ ⊗ IdH )(Φ)(Φ ⊗ 1H ), (IdH ⊗ ε ⊗ IdH )(Φ) = r ⊗ l −1 . Proof Assume that the strict monoidal structure on k M induces a monoidal structure on H M , that is, the forgetful functor is a quasi-monoidal functor. In particular, this implies that we have a left H-module structure · : H ⊗ (H ⊗ H) → H ⊗ H on H ⊗ H. If we define Δ : H → H ⊗ H by Δ(h) = h · (1H ⊗ 1H ), for all h ∈ H, we claim that Δ is an algebra map. Indeed, it is clear that Δ(1H ) = 1H ⊗ 1H . To see that Δ is multiplicative we proceed as follows. Let X ∈ H M and fix x ∈ X. Then ϕx : H h → h·x ∈ X is a left H-module morphism. Similarly, for Y ∈ H M and y ∈ Y define ϕy : H → Y , a left H-module morphism. According to (a) and (b) in Lemma 3.2, we have that ϕx ⊗ ϕy : H ⊗ H → X ⊗Y is a left H-linear morphism, hence (ϕx ⊗ ϕy )(h · (h ⊗ h )) = h · (h · x ⊗ h · y), for all h, h , h ∈ H. If we take h = h = 1H and denote Δ(h) = h1 ⊗ h2 (summation implicitly understood), we get that h · (x ⊗ y) = h1 · x ⊗ h2 · y, for all h ∈ H, and so

3.1 Quasi-bialgebras

105

Δ determines completely the left H-module structure on the tensor product X ⊗ Y . From the condition h · (h · (1H ⊗ 1H )) = (hh ) · (1H ⊗ 1H ), applied to all h, h ∈ H, it follows immediately that Δ is multiplicative, assuming that H ⊗ H has the usual componentwise algebra structure. We look now at the condition (c) in Lemma 3.2. We claim that giving a left H-module structure on k is equivalent to giving an algebra map ε : H → k. Indeed, if · : H ⊗ k → k gives a left H-module structure on k then the map defined by ε (h) = h · 1k , for all h ∈ H, is an algebra morphism since ε (hg) = (hg) · 1k = h · (g · 1k ) = ε (g)h · 1k = ε (h)ε (g) and ε (1H ) = 1H · 1k = 1k . For the converse, if ε : H → k is an algebra map then clearly k is a left H-module via the structure defined by h · κ = ε (h)κ , for all h ∈ H and κ ∈ k. Thus (c) in Lemma 3.2 implies the existence of an algebra map ε : H → k such that h· κ = ε (h)κ , for all h ∈ H and κ ∈ k. Thus, the monoidal structure on H M is induced by a triple (H, Δ, ε ) as follows. If X and Y are left H-modules then so is X ⊗Y via the left diagonal H-action h · (x ⊗ y) = h1 · x ⊗ h2 · y,

(3.1.1)

where Δ(h) = h1 ⊗ h2 is the notation for the comultiplication Δ of H. The unit object is k considered as a left H-module via h · κ = ε (h)κ , for all h ∈ H, κ ∈ k. We now look at the associativity and unit constraints of H M . For a left H-module X and x ∈ X consider again ϕx : H → X given by ϕx (h) = h · x, for all h ∈ H. The map ϕx is left H-linear, so by the naturality of a the diagram (H ⊗ H) ⊗ H

aH,H,H

H ⊗ (H ⊗ H)

(ϕx ⊗ϕy )⊗ϕz

ϕx ⊗(ϕy ⊗ϕz )

(X ⊗Y ) ⊗ Z

aX,Y,Z

X ⊗ (Y ⊗ Z)

is commutative, for all X,Y, Z ∈ H M , x ∈ X, y ∈ Y and z ∈ Z. So, if we denote Φ := aH,H,H ((1H ⊗ 1H ) ⊗ 1H ) we have aX,Y,Z ((x ⊗ y) ⊗ z) = Φ · (x ⊗ (y ⊗ z)), ∀ x ∈ X, y ∈ Y, z ∈ Z.

(3.1.2)

Furthermore, since a is a natural isomorphism it follows that Φ is invertible, its inverse being defined by a−1 H,H,H (1H ⊗ (1H ⊗ 1H )). With this description of a we obtain that the left H-linearity of a is equivalent to the first equality in the statement, and that (1.1.1) holds if and only if the third relation in the statement is verified. The naturality of the unit constraints produces the commutative diagrams: k⊗H

lH

ϕx

Idk ⊗ϕx

k⊗X

lX

H ⊗k

H

X

and

rH

ϕx

ϕx ⊗Idk

X ⊗k

H

rX

X.

If we denote l := lH (1k ⊗ 1H ) then, by writing the first diagram for X = H, with the

106

Quasi-bialgebras and Quasi-Hopf Algebras

left regular H-module structure, one can see that l is invertible with l −1 = lH−1 (1H ). Also, the first diagram for arbitrary X ∈ H M shows that lX (κ ⊗ x) = κ l · x, for all κ ∈ k and x ∈ X. Thus lX is left H-linear if and only if ε (h1 )lh2 · x = hl · x, for all h ∈ H and x ∈ X. By taking X = H and x = 1H we get that lX is left H-linear if and only if ε (h1 )lh2 = hl, for all h ∈ H. Similarly, using the commutativity of the second square diagram we obtain that rX is left H-linear if and only if rε (h2 )h1 = hr, for −1 (1H ). Thus the left all h ∈ H, where r := rH (1H ⊗ 1k ) is invertible with r−1 = rH and right unit constraints are left H-linear maps if and only if the second set of relations in the statement hold. Finally, one can easily see that (1.1.2) is equivalent to the fourth equality in the statement. Conversely, if there exist Φ, l, r as above, then H M is monoidal with the tensor product and unit object as in (3.1.1), with associativity constraint as in (3.1.2), left unit constraint defined by lX (κ ⊗ x) = κ l · x, and right unit constraint defined by rX (x ⊗ κ ) = κ r · x, for all X ∈ H M , x ∈ X and κ ∈ k. The verification of this fact is straightforward, so we leave it to the reader. Definition 3.4 Let H be a k-algebra, Δ : H → H ⊗ H and ε : H → k two algebra maps. We call (H, Δ, ε ) a quasi-bialgebra if there exist invertible elements Φ ∈ H ⊗ H ⊗ H and l, r ∈ H satisfying the conditions in Proposition 3.3. Hence, (H, Δ, ε ) is a quasi-bialgebra if and only if its category of left modules is monoidal with tensor product given by Δ and unit object k viewed as a left H-module via ε . The quasibialgebra with its entire structure is denoted by (H, Δ, ε , Φ, l, r). A k-bialgebra is a quasi-bialgebra for which Φ = 1H ⊗ 1H ⊗ 1H and l = r = 1H . As before, for k-bialgebras we denote Δ(h) = h1 ⊗ h2 and (Δ ⊗ IdH )(Δ(h)) = (IdH ⊗ Δ)(Δ(h)) = h1 ⊗ h2 ⊗ h3 , etc. For quasi-bialgebras, since Δ is only quasi-coassociative we adopt the further convention (as above, the summation is implicitly understood): (Δ ⊗ IdH )(Δ(h)) = h(1,1) ⊗ h(1,2) ⊗ h2 ,

(IdH ⊗ Δ)(Δ(h)) = h1 ⊗ h(2,1) ⊗ h(2,2) ,

for all h ∈ H. Furthermore, we will denote the tensor components of Φ by capital letters, and those of Φ−1 by lower case letters, namely Φ = X 1 ⊗ X 2 ⊗ X 3 = T 1 ⊗ T 2 ⊗ T 3 = V 1 ⊗V 2 ⊗V 3 = · · · Φ−1 = x1 ⊗ x2 ⊗ x3 = t 1 ⊗ t 2 ⊗ t 3 = v1 ⊗ v2 ⊗ v3 = · · · We next show that any quasi-bialgebra is equivalent to a quasi-bialgebra for which l = r = 1H . We first need to prove the following result. Proposition 3.5 Let H be a quasi-bialgebra, F ∈ H ⊗ H an invertible element. Let ΔF : H → H ⊗ H, lF = ε (G )lG , 1

2

ΔF (h) = FΔ(h)F −1 , rF = ε (G )rG , 2

ΦF = (1H ⊗ F)(IdH ⊗ Δ)(F)Φ(Δ ⊗ IdH )(F

−1

1

)(F

−1

(3.1.3) (3.1.4)

⊗ 1H ),

(3.1.5)

3.1 Quasi-bialgebras

107

where F = F 1 ⊗ F 2 , F −1 = G1 ⊗ G2 is the formal notation for the tensor components of F and F −1 , respectively. Then HF := (H, ΔF , ε , ΦF , lF , rF ) is a quasi-bialgebra as well. Moreover, the categories H M and HF M are monoidally isomorphic; more precisely, the identity functor from H M to HF M is a strong monoidal functor. Proof Denote by F 1 ⊗ F 2 = F1 ⊗ F2 and G 1 ⊗ G 2 = G1 ⊗ G2 some more copies of F and F −1 , respectively. We then have ΦF = F1 X 1 G11 G1 ⊗ F 1 F21 X 2 G12 G2 ⊗ F 2 F22 X 3 G2 , and so we compute: ΦF (ΔF ⊗ IdHF )(ΔF (h)) = ΦF (ΔF (F 1 h1 G 1 ) ⊗ F 2 h2 G 2 ) = (F1 X 1 G11 ⊗ F 1 F21 X 2 G12 ⊗ F 2 F22 X 3 G2 )((G1 ⊗ G2 )ΔF (F 1 h1 G 1 ) ⊗ F 2 h2 G 2 ) = (F1 X 1 G11 ⊗ F 1 F21 X 2 G12 ⊗ F 2 F22 X 3 G2 )(Δ(F 1 h1 G 1 )(G1 ⊗ G2 ) ⊗ F 2 h2 G 2 ) = (F1 X 1 ⊗ F 1 F21 X 2 ⊗ F 2 F22 X 3 )(Δ(h1 G 1 )(G1 ⊗ G2 ) ⊗ h2 G 2 ) = F1 h1 X 1 G11 G1 ⊗ F 1 F21 h(2,1) X 2 G21 G2 ⊗ F 2 F22 h(2,2) X 3 G 2 = F1 h1 X 1 G11 G1 ⊗ ΔF (F2 h2 )(F 1 X 2 G21 G2 ⊗ F 2 X 3 G 2 ) = (IdHF ⊗ ΔF )(ΔF (h))(F1 X 1 G11 G1 ⊗ F 1 F21 X 2 G21 G2 ⊗ F 2 F22 X 3 G 2 ) = (IdHF ⊗ ΔF )(ΔF (h)) ΦF , as required. Clearly ε (F 1 )F 2 is invertible with inverse ε (G1 )G2 , and ε (F 2 )F 1 is invertible with inverse ε (G2 )G1 . Therefore lF−1 = ε (F 1 )F 2 l −1 and rF−1 = ε (F 2 )F 1 r−1 , and from here we compute (ε ⊗ IdHF )(ΔF (h)) = ε (F 1 h1 G1 )F 2 h2 G2 = ε (F 1 )F 2 l −1 hl ε (G1 )G2 = lF−1 hlF and similarly (IdHF ⊗ ε )(ΔF (h)) = rF−1 hrF , for all h ∈ H. We also have (IdHF ⊗ ε ⊗ IdHF )(ΦF ) = ε (F 1 F21 X 2 G12 G2 )F1 X 1 G11 G1 ⊗ F 2 F22 X 3 G2 = ε (F 1 X 2 G2 )F1 X 1 r−1 G1 rG1 ⊗ F 2 l −1 F2 lX 3 G2 = ε (F 1 G2 )F1 G1 rG1 ⊗ F 2 l −1 F2 G2 = ε (F 1 )ε (G2 )rG1 ⊗ F 2 l −1 = rF ⊗ lF−1 . By the above formula for ΦF we have (IdHF ⊗ ΔF ⊗ IdHF )(ΦF ) = F1 X 1 G11 G1 ⊗ F 1 F11 F2(1,1) X12 G1(2,1) G21 G 1 ⊗ F 2 F21 F2(1,2) X22 G1(2,2) G22 G 2 ⊗ F 2 F22 X 3 G2 , and this implies (IdHF ⊗ ΔF ⊗ IdHF )(ΦF )(ΦF ⊗ 1HF ) = F1 X 1 G11Y 1 G11 G 1 ⊗ F 1 F11 F2(1,1) X12 G1(2,1)Y 2 G12 G 2 ⊗ F 2 F21 F2(1,2) X22 G1(2,2)Y 3 G2 ⊗ F 2 F22 X 3 G2 .

108

Quasi-bialgebras and Quasi-Hopf Algebras

We then compute: (1HF ⊗ ΦF )(IdHF ⊗ ΔF ⊗ IdHF )(ΦF )(ΦF ⊗ 1HF ) = F1 X 1 G11Y 1 G11 G 1 ⊗ F 1 Z 1 F2(1,1) X12 G1(2,1)Y 2 G12 G 2 ⊗ F 1 F12 Z 2 F2(1,2) X22 G1(2,2)Y 3 G2 ⊗ F 2 F22 Z 3 F22 X 3 G2 = F1 X 1Y 1 G1(1,1) G11 G 1 ⊗ F 1 F21 Z 1 X12Y 2 G1(1,2) G12 G 2 ⊗ F 1 F12 F2(2,1) Z 2 X22Y 3 G12 G2 ⊗ F 2 F22 F2(2,2) Z 3 X 3 G2 = F1 X 1Y11 G1(1,1) G11 G 1 ⊗ F 1 F21 X 2Y21 G1(1,2) G12 G 2 ⊗ F 1 F12 F2(2,1) X13Y 2 G12 G2 ⊗ F 2 F22 F2(2,2) X23Y 3 G2

 = (IdHF ⊗ IdHF ⊗ ΔF )(ΦF ) F1 F11Y11 G1(1,1) G11 G 1 ⊗ F2 F21Y21 G1(1,2) G12 G 2  ⊗ F 1 F12Y 2 G12 G2 ⊗ F 2 F22Y 3 G2 = (IdHF ⊗ IdHF ⊗ ΔF )(ΦF )(ΔF ⊗ IdHF ⊗ IdHF )(ΦF ). Hence HF is indeed a quasi-bialgebra. We next show that the identity functor, viewed as a functor from H M to has a strong monoidal structure. For X,Y ∈ H M define ϕ2,X,Y : X ⊗Y → X ⊗Y by

ϕ2,X,Y (x ⊗ y) = G1 · x ⊗ G2 · y,

HF M ,

(3.1.6)

for all x ∈ X and y ∈ Y . We have

ϕ2,X,Y (h · (x ⊗ y)) = ϕ2,X,Y (F 1 h1 G1 · x ⊗ F 2 h2 G2 · y) = h1 G1 · x ⊗ h2 G2 · y = h · ϕ2,X,Y (x ⊗ y), for all h ∈ H, and this shows that ϕ2,X,Y is left HF -linear. It can be easily checked that ϕ2 = (ϕ2,X,Y )X,Y ∈ H M is a family of natural isomorphisms and that the commutativity of the first diagram in Definition 1.22 reduces to the definition of ΦF . Also, if we take ϕ0 = Idk then the commutativity of the two square diagrams in Definition 1.22 follows from the definitions of lF and rF , respectively. If F ∈ H ⊗ H is an invertible element, we call the quasi-bialgebra HF the twisting of H by F. Motivated by the above results we introduce the following concepts. Definition 3.6 For (H, Δ, ε , Φ, l, r) and (H  , Δ , ε  , Φ , l  , r ) two quasi-bialgebras, a morphism between them is an algebra map χ : H → H  such that (χ ⊗ χ ) ◦ Δ = Δ ◦ χ , (χ ⊗ χ ⊗ χ )(Φ) = Φ ,

ε ◦ χ = ε,

χ (l) = l 

and χ (r) = r .

If, in addition, χ is an isomorphism we call it a quasi-bialgebra isomorphism. Two quasi-bialgebras H and H  are called equivalent (or twist equivalent) if there exists an invertible element F ∈ H  ⊗ H  such that H and HF are isomorphic as quasibialgebras.

3.1 Quasi-bialgebras

109

Remark 3.7 If H is a quasi-bialgebra and F, F  ∈ H ⊗ H are invertible elements, one can easily see that (HF )F  = HFF  , and so (HF )F −1 = (HF −1 )F = H. Consequently, the twist equivalence between quasi-bialgebras is an equivalence relation. Remark 3.8 If H and H  are isomorphic quasi-bialgebras, then H M and H  M are monoidally isomorphic (via the restriction of scalars functors). Corollary 3.9 If H and H  are twist equivalent quasi-bialgebras then H M and H  M are monoidally isomorphic. Lemma 3.10 For any quasi-bialgebra H there exists an invertible element F ∈ H ⊗ H for which lF = rF = 1H . Proof Let us start by proving that ε (l) = ε (r). For this, observe first that by applying ε ⊗ ε to both sides of (IdH ⊗ ε ⊗ IdH )(Φ) = r ⊗ l −1 we get ε (X 1 )ε (X 2 )ε (X 3 ) = ε (r)ε (l −1 ). On the other hand, it follows from ε (h1 )h2 = l −1 hl that ε (h1 )ε (h2 ) = ε (h), for any h ∈ H. Therefore, by applying ε ⊗ ε ⊗ ε ⊗ ε to both sides of (IdH ⊗ IdH ⊗ Δ)(Φ)(Δ ⊗ IdH ⊗ IdH )(Φ) = (1H ⊗ Φ)(IdH ⊗ Δ ⊗ IdH )(Φ)(Φ ⊗ 1H ) and by using that ε (X 1 )ε (X 2 )ε (X 3 ) is invertible in k we obtain ε (X 1 )ε (X 2 )ε (X 3 ) = 1. Together with ε (X 1 )ε (X 2 )ε (X 3 ) = ε (r)ε (l −1 ) this implies ε (l) = ε (r), as stated. Let us denote c := ε (l) = ε (r). We show that F = c−1 r ⊗ l is the twist that we need. Note first that c−1 = ε (l −1 ) = ε (r−1 ). Now, since F −1 = cr−1 ⊗ l −1 we have lF = cε (r−1 )ll −1 = cc−1 1H = 1H

and rF = cε (l −1 )rr−1 = cc−1 1H = 1H .

So our proof is complete. Corollary 3.11 Any quasi-bialgebra is twist equivalent to a quasi-bialgebra of the form (H, Δ, ε , Φ, l, r) for which l = r = 1H . Motivated by these facts, we will modify Definition 3.4 and, from now on, by a quasi-bialgebra H we mean a 4-tuple (H, Δ, ε , Φ), where H is an associative algebra with unit 1H , Φ is an invertible element in H ⊗ H ⊗ H (called in what follows the reassociator of H) and Δ : H → H ⊗ H (called the comultiplication) and ε : H → k (called the counit) are algebra homomorphisms satisfying the identities (IdH ⊗ Δ)(Δ(h)) = Φ(Δ ⊗ IdH )(Δ(h))Φ−1 ,

(3.1.7)

(IdH ⊗ ε )(Δ(h)) = h,

(3.1.8)

(ε ⊗ IdH )(Δ(h)) = h,

for all h ∈ H, and Φ is a normalized 3-cocycle, in the sense that (1H ⊗ Φ)(IdH ⊗ Δ ⊗ IdH )(Φ)(Φ ⊗ 1H ) = (IdH ⊗ IdH ⊗ Δ)(Φ)(Δ ⊗ IdH ⊗ IdH )(Φ), (IdH ⊗ ε ⊗ IdH )(Φ) = 1H ⊗ 1H .

(3.1.9) (3.1.10)

One can easily see that the identities (3.1.8), (3.1.9) and (3.1.10) also imply that (ε ⊗ IdH ⊗ IdH )(Φ) = (IdH ⊗ IdH ⊗ ε )(Φ) = 1H ⊗ 1H .

(3.1.11)

110

Quasi-bialgebras and Quasi-Hopf Algebras

Since we have assumed l = r = 1H in the definition of a quasi-bialgebra H, we need to assume also that ε (F 1 )F 2 = ε (F 2 )F 1 = 1H when we twist H by an invertible element F = F 1 ⊗ F 2 ∈ H ⊗ H (in order to obtain lF = rF = 1H ). We will call such an F a twist or gauge transformation on H. Examples 3.12 (1) Any k-bialgebra H is a quasi-bialgebra with Φ = 1H ⊗ 1H ⊗ 1H . In particular, if F is a twist on H then in general HF is no longer a k-bialgebra but it is a quasi-bialgebra. (2) Together with a quasi-bialgebra H = (H, Δ, ε , Φ) we also have H op , H cop and op,cop as quasi-bialgebras, where “op” means opposite multiplication and “cop” H means opposite comultiplication (i.e. Δcop (h) = h2 ⊗ h1 , for all h ∈ H). The quasibialgebra structures are obtained by putting Φop = Φ−1 , Φcop = (Φ−1 )321 := x3 ⊗ x2 ⊗ x1 and Φop,cop = Φ321 := X 3 ⊗ X 2 ⊗ X 1 , respectively. More generally, if t denotes a permutation of {1, 2, . . . , n} with inverse t −1 , for an element Ω1 ⊗ · · · ⊗ Ωn ∈ H ⊗n we set Ωt(1)t(2)···t(n) = Ωt

−1 (1)

⊗ · · · ⊗ Ωt

−1 (n)

.

Non-trivial examples of quasi-bialgebras will be given in the next sections.

3.2 Quasi-Hopf Algebras The categorical meaning of the next concept will be explained later on in this chapter. Definition 3.13 A quasi-bialgebra H is called a quasi-Hopf algebra if there exist an anti-algebra endomorphism S of H, called an antipode, and elements α , β ∈ H such that, for all h ∈ H, we have S(h1 )α h2 = ε (h)α and h1 β S(h2 ) = ε (h)β ,

(3.2.1)

X β S(X )α X = 1H and S(x )α x β S(x ) = 1H .

(3.2.2)

1

2

3

1

2

3

Recall that a k-bialgebra is called a Hopf algebra if there exists a k-linear map S : H → H such that S(h1 )h2 = ε (h)1H = h1 S(h2 ), for all h ∈ H. Note that the quasiHopf notion is more general than the Hopf algebra notion, as the next result explains. Proposition 3.14 If H is a Hopf algebra with antipode S then S is an anti-algebra and an anti-coalgebra morphism. Proof

This follows from the more general result in Proposition 2.65.

Remark 3.15 Let H be a quasi-Hopf algebra and Alg(H, k) the set of algebra maps from H to k, endowed with the multiplication νξ = (ν ⊗ ξ ) ◦ Δ. One can easily see that this product is associative with unit ε . By (3.2.2) we obtain that for any ν ∈ Alg(H, k) we have ν (α ) = 0 = ν (β ). By using this and (3.2.1), it follows that Alg(H, k) is a group, the inverse of an element ν ∈ Alg(H, k) being the element ν ◦ S.

3.2 Quasi-Hopf Algebras

111

The observations below point out some differences that exist between Hopf algebras and quasi-Hopf algebras (proofs are straightforward and left to the reader). Remarks 3.16 (1) The axioms for a quasi-Hopf algebra immediately imply that ε ◦ S = ε and ε (α )ε (β ) = 1, so, by rescaling α and β , we may (and will from now on) assume without loss of generality that ε (α ) = ε (β ) = 1. (2) If S, α , β satisfy the conditions (3.2.1) and (3.2.2) then for any invertible element u ∈ H the same conditions are satisfied by S, α and β , where S(h) = uS(h)u−1 , ∀ h ∈ H, α = uα and β = β u−1 .

(3.2.3)

So the antipode of a quasi-Hopf algebra is not unique. (3) If H is a k-bialgebra that admits a quasi-Hopf algebra structure then by (3.2.2) we obtain that α and β are inverses of each other. If we take u = α −1 = β we then have S(h) = α S(h)α −1 , for all h ∈ H, and α = β = 1H , so H is a Hopf algebra with antipode S. Consequently, a k-bialgebra is a Hopf algebra if and only if it is a quasi-Hopf algebra. (4) The quasi-Hopf notion is invariant under a twist. More precisely, if F is a twist on H then the quasi-bialgebra HF is, moreover, a quasi-Hopf algebra with SF = S,

αF = S(G1 )α G2

and βF = F 1 β S(F 2 ),

(3.2.4)

where F = F 1 ⊗ F 2 and F −1 = G1 ⊗ G2 . (5) If H = (H, Δ, ε , Φ, S, α , β ) is a quasi-Hopf algebra then H op,cop is a quasi-Hopf algebra with Sop,cop = S, αop,cop = β and βop,cop = α . If S is bijective then H op and H cop are quasi-Hopf algebras as well. This time the structures are obtained by putting Sop = Scop = S−1 , αop = S−1 (β ), βop = S−1 (α ), αcop = S−1 (α ) and βcop = S−1 (β ), respectively. The second remark above has also a converse. Proposition 3.17 If two triples (S, α , β ) and (S, α , β ) satisfy (3.2.1) and (3.2.2) then there exists a unique invertible element u ∈ H such that (3.2.3) holds. Proof

If there exists u satisfying (3.2.3) then u = uS(x1 )α x2 β S(x3 ) = S(x1 )uα x2 β S(x3 ) = S(x1 )α x2 β S(x3 ),

and this proves the uniqueness of u. Set now u = S(x1 )α x2 β S(x3 ). If we write (3.1.7) under the form h(1,1) x1 ⊗ h(1,2) x2 ⊗ h2 x3 = x1 h1 ⊗ x2 h(2,1) ⊗ x3 h(2,2) , ∀ h ∈ H, then we obtain S(h(1,1) x1 )α h(1,2) x2 β S(h2 x3 ) = S(x1 h1 )α x2 h(2,1) β S(x3 h(2,2) ), and this implies uS(h) = S(h)u, for all h ∈ H. Likewise, if we write (3.1.9) under the equivalent form X 1Y11 x1 ⊗ X 2Y21 x2 ⊗ X13Y 2 x3 ⊗ X23Y 3 = Y 1 ⊗ X 1Y12 ⊗ X 2Y22 ⊗ X 3Y 3 ,

112

Quasi-bialgebras and Quasi-Hopf Algebras

we then have S(X 1Y11 x1 )α X 2Y21 x2 β S(X13Y 2 x3 )α X23Y 3 = S(Y 1 )α X 1Y12 β S(X 2Y22 )α X 3Y 3 , which can be rewritten as S(x1 )α x2 β S(x3 )α = α X 1 β S(X 2 )α X 3 , or as uα = α . This formula for H op,cop instead of H becomes β S(x1 )α x2 β S(x3 ) = β . So to end the proof it is enough to show that u−1 := S(x1 )α x2 β S(x3 ) is the inverse of u. On the one hand we have uS(x1 )α x2 β S(x3 ) = S(x1 )uα x2 β S(x3 ) = S(x1 )α x2 β S(x3 ) = 1H . On the other hand, by using again (3.1.9), we compute:

β u = β S(x1 )α x2 β S(x3 ) = Y 1 X 1 y11 x1 β S(Y12 X 2 y12 x2 )α Y22 X 3 y2 x13 β S(Y 3 y3 x23 ) = X 1 β S(X 2 )α X 3 β = β , and this allows us to show that S(x1 )α x2 β S(x3 )u = S(x1 )α x2 β uS(x3 ) = S(x1 )α x2 β S(x3 ) = 1H , as desired. This completes the proof. Definition 3.18 Let H = (H, Δ, ε , Φ, α , β , S) be a quasi-Hopf algebra. If u ∈ H is invertible, we denote by Hu the quasi-Hopf algebra (H, Δ, ε , Φ, uα , β u−1 , Su ), where Su (h) = uS(h)u−1 , for all h ∈ H. For a quasi-Hopf algebra H we show that the antipode is, up to conjugation by a twist, an anti-coalgebra morphism. In particular, we recover the result that the antipode is an ordinary anti-coalgebra morphism in the case when H is a Hopf algebra. To this end define γ , δ ∈ H ⊗ H by

γ

S(X 2 x21 )α X 3 x2 ⊗ S(X 1 x11 )α x3

= (3.1.9),(3.1.10),(3.2.1)

=

δ

Lemma 3.19

S(x1 X 2 )α x2 X13 ⊗ S(X 1 )α x3 X23 ,

= X11 x1 β S(X 3 ) ⊗ X21 x2 β S(X 2 x3 ) (3.1.9),(3.1.10),(3.2.1) 1 = x β S(x23 X 3 ) ⊗ x2 X 1 β S(x13 X 2 ).

(3.2.5) (3.2.6)

In a quasi-Hopf algebra H the following relations hold: (S ⊗ S)(Δcop (h1 ))γ Δ(h2 ) = ε (h)γ ,

(3.2.7)

Δ(h1 )δ (S ⊗ S)(Δ

(3.2.8)

cop

(h2 )) = ε (h)δ ,

for all h ∈ H, where Δcop is the opposite comultiplication of H, and Δ(X 1 )δ (S ⊗ S)(Δcop (X 2 ))γ Δ(X 3 ) = 1H ⊗ 1H , (S ⊗ S)(Δ

cop

(x ))γ Δ(x )δ (S ⊗ S)(Δ 1

2

cop

(x )) = 1H ⊗ 1H . 3

(3.2.9) (3.2.10)

3.2 Quasi-Hopf Algebras Proof

113

Set γ = γ 1 ⊗ γ 2 and δ = δ 1 ⊗ δ 2 . To check (3.2.7) we compute:

(S ⊗ S)(Δcop (h1 ))γ Δ(h2 ) = S(h(1,2) )γ 1 (h2 )1 ⊗ S(h(1,1) )γ 2 (h2 )2 = S(x1 X 2 h(1,2) )α x2 (X 3 h2 )1 ⊗ S(X 1 h(1,1) )α x3 (X 3 h2 )2 (3.1.7)

= S(x1 (h2 )1 X 2 )α x2 (h2 )(2,1) X13 ⊗ S(h1 X 1 )α x3 (h2 )(2,2) X23

(3.1.7)

= S((h2 )(1,1) x1 X 2 )α (h2 )(1,2) x2 X13 ⊗ S(h1 X 1 )α (h2 )2 x3 X23 (3.2.1)

(3.2.1)

= S(x1 X 2 )α x2 X13 ⊗ S(h1 X 1 )α h2 x3 X23 = ε (h)γ .

The formula in (3.2.8) can be proved in a similar manner, so we leave the details to the reader. Now (3.2.10) follows since (S ⊗ S)(Δcop (x1 ))γ Δ(x2 )δ (S ⊗ S)(Δcop (x3 )) =

S(x21 )γ 1 x12 δ 1 S(x23 ) ⊗ S(x11 )γ 2 x22 δ 2 S(x13 )

=

S(y1 X 2 x21 )α y2 X13 x12 δ 1 S(x23 ) ⊗ S(X 1 x11 )α y3 X23 x22 δ 2 S(x13 )

(3.1.9)

3 3 S(y1 z21 x1 X 2 )α y2 z2(2,1) x12 X(1,1) δ 1 S(z32 x23 X(2,2) )

=

3 3 δ 2 S(z31 x13 X(2,1) ) ⊗ S(z1 X 1 )α y3 z2(2,2) x22 X(1,2) (3.2.8),(3.1.7)

=

(3.2.1)

S(z2(1,1) y1 x1 )α z2(1,2) y2 x12 δ 1 S(z32 x23 ) ⊗ S(z1 )α z22 y3 x22 δ 2 S(z31 x13 )

=

S(y1 x1 )α y2 x12 δ 1 S(z32 x23 ) ⊗ S(z1 )α z2 y3 x22 δ 2 S(z31 x13 )

=

S(y1 x1 )α y2 x12t 1 β S(z32 x23t23 X 3 ) ⊗ S(z1 )α z2 y3 x22t 2 X 1 β S(z31 x13t13 X 2 )

(3.1.9),(3.1.11),(3.2.1)

=

(3.1.7),(3.2.1)

=

3 3 S(x1 )α x2 β S(z32 x(2,2) X 3 ) ⊗ S(z1 )α z2 x13 X 1 β S(z31 x(2,1) X 2)

S(x1 )α x2 β S(z32 X 3 x3 ) ⊗ S(z1 )α z2 X 1 β S(z31 X 2 )

(3.2.2)

S(z32 X 3 ) ⊗ S(z1 )α z2 X 1 β S(z31 X 2 )

(3.1.9)

S(Y 3 y3 ) ⊗ S(Y11 z1 y1 )α Y21 z2 y21 β S(Y 2 z3 y22 )

= =

(3.2.1),(3.1.11),(3.1.10)

=

(3.2.2)

1H ⊗ S(z1 )α z2 β S(z3 ) = 1H ⊗ 1H ,

as desired. The formula in (3.2.9) can be proved in a similar way. Another preliminary result that we need is the following. Lemma 3.20 Let H be a quasi-Hopf algebra and A a k-algebra. Suppose that there exist an algebra map f : H → A, an anti-algebra map g : H → A and elements ρ , σ ∈ A such that g(h1 )ρ f (h2 ) = ε (h)ρ ,

f (h1 )σ g(h2 ) = ε (h)σ , ∀ h ∈ H,

f (X )σ g(X )ρ f (X ) = 1A , 1

2

3

g(x )ρ f (x )σ g(x ) = 1A . 1

2

3

(3.2.11) (3.2.12)

If g : H → A is another anti-algebra map and ρ , σ ∈ A are such that (3.2.11) and (3.2.12) hold for f , g, σ and ρ as well, then there exists a unique invertible element F ∈ A such that ρ = F ρ , σ = σ F −1 and g(h) = Fg(h)F −1 , for all h ∈ H.

114

Quasi-bialgebras and Quasi-Hopf Algebras

Proof

If it exists then F is unique since

F = Fg(x1 )ρ f (x2 )σ g(x3 ) = g(x1 )F ρ f (x2 )σ g(x3 ) = g(x1 )ρ f (x2 )σ g(x3 ). Now take F = g(x1 )ρ f (x2 )σ g(x3 ). By (3.1.7) we have g(h(1,1) x1 )ρ f (h(1,2) x2 )σ g(h2 x3 ) = g(x1 h1 )ρ f (x2 h(2,1) )σ g(x3 h(2,2) ), and since f is an algebra morphism and g, g are anti-algebra morphisms, by (3.2.11) we obtain g(x1 )ρ f (x2 )σ g(x3 )g(h) = g(h) g(x1 )ρ f (x2 )σ g(x3 ), that is, Fg(h) = g(h)F, for all h ∈ H. Moreover, by (3.1.9) we have g(X 1Y11 x1 )ρ f (X 2Y21 x2 )σ g(X13Y 2 x3 )ρ f (X23Y 3 ) = g(Y 1 )ρ f (X 1Y12 )σ g(X 2Y22 )ρ f (X 3Y 3 ), which implies g(x1 )ρ f (x2 )σ g(x3 )ρ = ρ f (X 1 )σ g(X 2 )ρ f (X 3 ), by (3.2.11), (3.1.10) and (3.1.11). So by (3.2.12) we deduce that F ρ = ρ . By using (3.1.9) again we have

σ g(x1 )ρ f (x2 )σ g(x3 ) = f (Y 1 X 1 y11 x1 )σ g(Y12 X 2 y12 x2 ) ρ f (Y22 X 3 y2 x13 )σ g(Y 3 y3 x23 ), and so σ g(x1 )ρ f (x2 )σ g(x3 ) = f (X 1 )σ g(X 2 )ρ f (X 3 )σ , again because of (3.2.11), (3.1.10) and (3.1.11). Thus σ F = σ , by (3.2.12). We show that F is invertible with inverse F −1 := g(x1 )ρ f (x2 )σ g(x3 ). Indeed, FF −1 = Fg(x1 )ρ f (x2 )σ g(x3 ) = g(x1 )F ρ f (x2 )σ g(x3 ) (3.2.12)

= g(x1 )ρ f (x2 )σ g(x3 ) = 1A , F −1 F = g(x1 )ρ f (x2 )σ g(x3 )F = g(x1 )ρ f (x2 )σ Fg(x3 ) (3.2.12)

= g(x1 )ρ f (x2 )σ g(x3 ) = 1A , finishing the proof. We can prove now the main result of this section. Theorem 3.21 For a quasi-Hopf algebra H there exists a twist F on H such that FΔ(S(h))F −1 = (S ⊗ S)(Δcop (h)),

(3.2.13)

for all h ∈ H. Furthermore,

γ = FΔ(α ) and δ = Δ(β )F −1 .

(3.2.14)

Proof We apply Lemma 3.20 to A = H ⊗ H, f = Δ, g = Δ ◦ S : H → H ⊗ H, ρ = Δ(α ), σ = Δ(β ), g = (S ⊗ S) ◦ Δcop : H → H ⊗ H, ρ = γ and σ = δ . The conditions in (3.2.11) for f , g, σ , ρ as above come out as Δ(S(h1 )α h2 ) = ε (h)Δ(α )

and

Δ(h1 β S(h2 )) = ε (h)Δ(β ), ∀ h ∈ H,

3.2 Quasi-Hopf Algebras

115

which are true because of (3.2.1). The relations (3.2.12) become Δ(X 1 β S(X 2 )α X 3 ) = 1H ⊗ 1H and Δ(S(x1 )α x2 β S(x3 )) = 1H ⊗ 1H , which hold because of (3.2.2). Note now that the relations (3.2.11) for f , g, σ , ρ as above turn into (3.2.7) and (3.2.8), respectively, while the relations (3.2.12) for this setting turn into (3.2.9) and (3.2.10). Thus we are in the position to apply the above lemma. We obtain an invertible element F = g(x1 )ρ f (x2 )σ g(x3 ) = (S ⊗ S)(Δcop (x1 ))γ Δ(x2 β S(x3 )) ∈ H ⊗ H,

(3.2.15)

with inverse F −1 = g(x1 )ρ f (x2 )σ g(x3 ) = Δ(S(x1 )α x2 )δ (S ⊗ S)(Δcop (x3 )),

(3.2.16)

and satisfying the relations

ρ = F ρ ⇔ γ = FΔ(α ), g(h) = Fg(h)F

−1

σ = σ F −1 ⇔ δ = Δ(β )F −1 and

⇔ (S ⊗ S)(Δcop (h)) = FΔ(S(h))F −1 , ∀ h ∈ H.

These are precisely the desired relations. The equalities ε (F 1 )F 2 = ε (F 2 )F 1 = 1H are an immediate consequence of the axioms of a quasi-Hopf algebra. The twist defined in (3.2.15) will be called the Drinfeld twist associated to H, and will be usually denoted by f , F, F , etc. Definition 3.22 If (H, Δ, ε , Φ, S, α , β ) and (H  , Δ , ε  , Φ , S , α  , β  ) are quasi-Hopf algebras then χ : H → H  is a quasi-Hopf algebra morphism if it is a quasi-bialgebra morphism such that χ (α ) = α  , χ (β ) = β  and S ◦ χ = χ ◦ S. We say that χ is an isomorphism of quasi-Hopf algebras if, in addition, it is a bijective map (in which case its inverse χ −1 is also a quasi-Hopf algebra morphism). For a Hopf algebra H its antipode can be viewed as a Hopf algebra morphism from to H. More generally, we have the following result.

H op,cop

Proposition 3.23 If H is a quasi-Hopf algebra with antipode S, then S : H op,cop → H f is a quasi-Hopf algebra morphism, where H f is the twisting of the quasi-Hopf algebra H by the Drinfeld twist f defined in (3.2.15). Proof We have S(αop,cop ) = S(β ) and α f = S(g1 )α g2 , where f −1 = g1 ⊗ g2 is the inverse of f = f 1 ⊗ f 2 as in (3.2.16). It follows that α f = S(δ 1 )αδ 2 , and using the definition of δ = δ 1 ⊗ δ 2 from (3.2.6) we conclude that

αf

= (3.2.1),(3.1.11)

=

S(X11 x1 β S(X 3 ))α X21 x2 β S(X 2 x3 ) (3.2.2)

S(x1 β )α x2 β S(x3 ) = S(β ) = S(αop,cop ),

as required. Likewise, one can prove that S(βop,cop ) = S(α ) = β f := f 1 β S( f 2 ). The map S respects the multiplications and comultiplications of H op,cop and H f because it is, by definition, an anti-algebra endomorphism of H, and, moreover, the

116

Quasi-bialgebras and Quasi-Hopf Algebras

condition (S ⊗ S) ◦ ΔH op,cop = ΔH f ◦ S is exactly the relation (3.2.13). Thus, we only have to check that S respects the two reassociators, that is, (1H ⊗ f )(IdH ⊗ Δ)( f )Φ(Δ ⊗ IdH )( f −1 )( f −1 ⊗ 1H ) = (S ⊗ S ⊗ S)(X 3 ⊗ X 2 ⊗ X 1 ). (3.2.17) To prove this equality, with the help of Theorem 3.21, we compute: (S ⊗ S ⊗ S)(Φ321 )( f ⊗ 1H )(Δ ⊗ IdH )( f ) (S ⊗ S ⊗ S)(Φ321 )( f ⊗ 1H )(Δ ◦ S ⊗ S)(Δcop (x1 ))

=

(Δ ⊗ IdH )(γ )(Δ ⊗ IdH )(Δ(x2 β S(x3 )))   (3.2.13) = (S ⊗ S ⊗ S)(Φ321 )(S ⊗ S ⊗ S) (Δcop ⊗ IdH )(Δcop (x1 )) (3.1.7)

=

( f ⊗ 1H )(Δ ⊗ IdH )(γ )(Δ ⊗ IdH )(Δ(x2 β S(x3 )))   (S ⊗ S ⊗ S) (IdH ⊗ Δcop )(Δcop (x1 )) (S ⊗ S ⊗ S)(Φ321 ) ( f ⊗ 1H )(Δ ⊗ IdH )(γ )(Δ ⊗ IdH )(Δ(x2 β S(x3 ))),

and similarly (1H ⊗ f )(IdH ⊗ Δ)( f )Φ (1H ⊗ f )(S ⊗ Δ ◦ S)(Δcop (x1 ))(IdH ⊗ Δ)(γ )(IdH ⊗ Δ)(Δ(x2 β S(x3 )))Φ   (3.2.13) = (S ⊗ S ⊗ S) (IdH ⊗ Δcop )(Δcop (x1 )) (1H ⊗ f ) =

(3.1.7)

(IdH ⊗ Δ)(γ )Φ(Δ ⊗ IdH )(Δ(x2 β S(x3 ))). So, to prove (3.2.17), it suffices to show that (S ⊗ S ⊗ S)(Φ321 )( f ⊗ 1H )(Δ ⊗ IdH )(γ ) = (1H ⊗ f )(IdH ⊗ Δ)(γ )Φ.

(3.2.18)

Indeed, we have: (S ⊗ S ⊗ S)(Φ321 )( f ⊗ 1H )(Δ ⊗ IdH )(γ )  3 3 ⊗ f 2 S(x1 X 2 )2 α2 x22 X(1,2) = (S ⊗ S ⊗ S)(Φ321 ) f 1 S(x1 X 2 )1 α1 x12 X(1,1)  ⊗ S(X 1 )α x3 X23 (3.2.13),(3.2.14)

=

3 3 S(x21 X22Y 3 )γ 1 x12 X(1,1) ⊗ S(x11 X12Y 2 )γ 2 x22 X(1,2) ⊗ S(X 1Y 1 )α x3 X23

=

3 3 S(Z 2 z12 x21 X22Y 3 )α Z 3 z2 x12 X(1,1) ⊗ S(Z 1 z11 x11 X12Y 2 )α z3 x22 X(1,2)

⊗ S(X 1Y 1 )α x3 X23 (3.1.9)

=

1 3 S(Z 2 x(1,2) z12 X22Y 3 )α Z 3 x21 z2 T 1 X(1,1) 1 3 ⊗ S(Z 1 x(1,1) z11 X12Y 2 )α x2 z31 T 2 X(1,2) ⊗ S(X 1Y 1 )α x3 z32 T 3 X23

(3.1.7)

=

(3.2.1) (3.1.9)

=

S(Z 2 (z1 X 2 )2Y 3 )α Z 3 z2 X13 T 1 ⊗ S(x1 Z 1 (z1 X 2 )1Y 2 )α x2 (z3 X23 )1 T 2 ⊗ S(X 1Y 1 )α x3 (z3 X23 )2 T 3 S(Z 2 (X12V 2 z12 )2Y 3 )α Z 3 X22V 3 z2 T 1 ⊗ S(x1 Z 1 (X12V 2 z12 )1Y 2 )α

3.2 Quasi-Hopf Algebras

117

x2 (X 3 z3 )1 T 2 ⊗ S(X 1V 1 z11Y 1 )α x3 (X 3 z3 )2 T 3 (3.1.7)

S(Z 2V22Y 3 z12 )α Z 3V 3 z2 T 1 ⊗ S(x1 X 2 Z 1V12Y 2 z1(1,2) )α x2 X13 z31 T 2

=

(3.2.1)

⊗ S(X 1V 1Y 1 z1(1,1) )α x3 X23 z32 T 3

(3.1.9)

S(Z 2 z12 )α Z 3 z2 T 1 ⊗ S(x1 X 2 Z21 z1(1,2) )α x2 X13 z31 T 2

=

(3.2.1)

⊗ S(X 1 Z11 z1(1,1) )α x3 X23 z32 T 3 S(Z 2 z12 )α Z 3 z2 T 1 ⊗ S(Z21 z1(1,2) )γ 1 z31 T 2 ⊗ S(Z11 z1(1,1) )γ 2 z32 T 3

= (3.2.13),(3.2.14)

=

S(Z 2 z12 )α Z 3 z2 T 1 ⊗ f Δ(S(Z 1 z11 )α z3 )(T 2 ⊗ T 3 )

=

(1H ⊗ f )(γ 1 ⊗ Δ(γ 2 ))Φ = (1H ⊗ f )(IdH ⊗ Δ)(γ )Φ,

as desired. So our proof is complete. We next point out that one of the two conditions in (3.2.2) is redundant. Proposition 3.24 Let H be a quasi-bialgebra and (S, α , β ) a triple consisting of an anti-algebra endomorphism S of H and elements α , β ∈ H such that the conditions in (3.2.1) are satisfied. Denote a := X 1 β S(X 2 )α X 3 and b := S(x1 )α x2 β S(x3 ). Then a is a central element of H and b commutes with the elements in the image of S. Consequently, if b = 1H then a = 1H . The converse is also true when S is surjective. Proof

For all h ∈ H we have (3.2.1)

(3.1.7)

(3.2.1)

ha = h1 X 1 β S(h(2,1) X 2 )α h(2,2) X 3 = X 1 h(1,1) β S(X 2 h(1,2) )α X 3 h2 = ah. Similarly, we compute: S(h)b = S(x1 h)α x2 β S(x3 ) = S(x1 h1 )α x2 h(2,1) β S(x3 h(2,2) ) = S(h(1,1) x1 )α h(1,2) x2 β S(h2 x3 ) = bS(h), for all h ∈ H. On the other hand, by using (3.1.9), we compute:

α a = α X 1 β S(X 2 )α X 3 = S(X 1Y11 z1 y1 )α X 2Y21 z2 y21 β S(X13Y 2 z3 y22 )α X23Y 3 y3 = S(z1 )α z2 β S(z3 )α = bα . Thus, for all h, h ∈ H we have (S(h)α h )a = S(h)α h a = S(h)α ah = S(h)bα h = b(S(h)α h ). Since b has the form ∑i S(hi )α hi for some suitable families of elements hi , hi ∈ H, we deduce that ba = b2 , and so if b = 1H then clearly a = 1H , too. If S is surjective then (hα h )a = b(hα h ), for all h, h ∈ H. Both a and b can be written as ∑i hi α hi , for some hi , hi ∈ H. Consequently, a and b are central elements of H such that a2 = b2 = ab. It is clear now that a = 1H if and only if b = 1H . We end this section by recording some formulas that will be intensively used from now on. They involve the following four elements of H ⊗ H: pR = x1 ⊗ x2 β S(x3 ), pL

qR = X 1 ⊗ S−1 (α X 3 )X 2 ,

= X 2 S−1 (X 1 β ) ⊗ X 3 ,

qL = S(x )α x ⊗ x . 1

2

3

(3.2.19) (3.2.20)

118

Quasi-bialgebras and Quasi-Hopf Algebras

In what follows we will denote these elements by pR = p1 ⊗ p2 = P1 ⊗ P2 = · · · , qR = q1 ⊗ q2 = Q1 ⊗ Q2 = · · · , pL = p˜1 ⊗ p˜2 = P˜ 1 ⊗ P˜ 2 = · · · , qL = q˜1 ⊗ q˜2 = Q˜ 1 ⊗ Q˜ 2 = · · · . Note that in the definitions of qR and pL it is implicitly understood that the antipode S of H is assumed to be bijective. Note also that pR and qR in H cop are exactly (pL )21 := p˜2 ⊗ p˜1 and (qL )21 := 2 q˜ ⊗ q˜1 in H, respectively, while in H op they are exactly qR and pR in H, respectively. The proof of the equations below is postponed until Section 4.3 where we will show that more general formulas hold. As before we denote by f = f 1 ⊗ f 2 and f −1 = g1 ⊗ g2 the Drinfeld twist and its inverse, defined in (3.2.15) and (3.2.16). Proposition 3.25 If H is a quasi-Hopf algebra with bijective antipode, then the following relations hold: [1H ⊗ S−1 (h2 )]qR Δ(h1 ) = [h ⊗ 1H ]qR ,(3.2.21)

Δ(h1 )pR [1H ⊗ S(h2 )] = pR [h ⊗ 1H ], Δ(h2 )pL [S−1 (h1 ) ⊗ 1H ] = pL [1H ⊗ h],

[S(h1 ) ⊗ 1H ]qL Δ(h2 ) = [1H ⊗ h]qL , (3.2.22)

Δ(q )pR [1H ⊗ S(q )] = 1H ⊗ 1H ,

[1H ⊗ S−1 (p2 )]qR Δ(p1 ) = 1H ⊗ 1H ,

[S( p˜ ) ⊗ 1H ]qL Δ( p˜ ) = 1H ⊗ 1H ,

Δ(q˜

1

1

2

2

2

)pL [S−1 (q˜1 ) ⊗ 1H ] = 1H

⊗ 1H ,

X 1 p11 P1 ⊗ X 2 p12 P2 ⊗ X 3 p2 1 1 = x11 p1 ⊗ x(1,2) p21 g1 S(x3 ) ⊗ x(2,2) p22 g2 S(x2 ), 1

q

Q11 x1 ⊗ q2 Q12 x2 ⊗ Q2 x3 1 1 = q1 X11 ⊗ S−1 ( f 2 X 3 )q21 X(2,1) ⊗ S−1 ( f 1 X 2 )q22 X(2,2) ,

(3.2.23) (3.2.24) (3.2.25) (3.2.26)

x1 p˜1 ⊗ x2 p˜21 P˜ 1 ⊗ x3 p˜22 P˜ 2 3 3 = X(1,1) p˜11 S−1 (X 2 g2 ) ⊗ X(1,2) p˜12 S−1 (X 1 g1 ) ⊗ X23 p˜2 ,

(3.2.27)

Q˜ 1 X 1 ⊗ q˜1 Q˜ 21 X 2 ⊗ q˜2 Q˜ 22 X 3 3 3 = S(x2 ) f 1 q˜11 x(1,1) ⊗ S(x1 ) f 2 q˜12 x(1,2) ⊗ q˜2 x23 .

(3.2.28)

Let H be a quasi-Hopf algebra and F ∈ H ⊗ H a gauge transformation. We denote by pFR , qFR , pFL , qFL the elements defined by (3.2.19) and (3.2.20) corresponding to the quasi-Hopf algebra HF . Then, by using the explicit formulas for the structure of HF presented above, one can easily see that these elements are given by the following formulas (we denote F = F 1 ⊗ F 2 and F −1 = G1 ⊗ G2 ): pFR = FΔ(F 1 )pR (1H ⊗ S(F 2 )), qFR pFL qFL

= (1H ⊗ S

−1

2

(G ))qR Δ(G )F 2

= FΔ(F )pL (S

−1

1

(3.2.29) ,

(3.2.30)

(F ) ⊗ 1H ),

(3.2.31)

1

= (S(G ) ⊗ 1H )qL Δ(G )F 1

−1

2

−1

.

(3.2.32)

3.3 Examples of Quasi-bialgebras and Quasi-Hopf Algebras

119

3.3 Examples of Quasi-bialgebras and Quasi-Hopf Algebras The aim of this section is to present some examples of quasi-Hopf algebras that are not twist equivalent to a Hopf algebra. A first candidate can be built out of a commutative bialgebra H, viewed as a quasi-bialgebra via a reassociator Φ which is not of the form (1H ⊗ F)(IdH ⊗ Δ)(F)(Δ ⊗ IdH )(F −1 )(F −1 ⊗ 1H ), for some gauge transformation F on H. Concretely, we have the following simple example. Example 3.26 For k a field of characteristic different from 2 let H(2) be the twodimensional Hopf group algebra generated by the grouplike element g such that g2 = 1. It can be also viewed as a quasi-Hopf algebra with reassociator Φ = 1 ⊗ 1 ⊗ 1 − 2p− ⊗ p− ⊗ p− , and antipode defined by S(g) = g and distinguished elements α = g and β = 1. Here p− = 12 (1 − g). Proof As a Hopf algebra H(2) is k[C2 ], the group Hopf algebra associated to the cyclic group of order 2, C2 . If we assume C2 = g then k[C2 ] is the k-algebra generated by g with relation g2 = 1, while the coalgebra structure is given by stating that g is a grouplike element, that is, Δ(g) = g ⊗ g

and ε (g) = 1.

Since H(2) is commutative it can be also considered as a quasi-bialgebra via any invertible element Φ ∈ H(2)⊗3 satisfying (3.1.9) and (3.1.10). Thus, to conclude that H(2) is a quasi-bialgebra it is enough to check that Φ defined in the statement obeys these two conditions. In order to prove (3.1.9) first denote p+ = 12 (1 + g) and observe that 1 Δ(p− ) = (1 ⊗ 1 − g ⊗ g) 2 1 = ((p+ + p− ) ⊗ (p+ + p− ) − (p+ − p− ) ⊗ (p+ − p− )) 2 = p+ ⊗ p− + p− ⊗ p+ . It can be easily checked that {p+ , p− } is a pair of orthogonal idempotents in H(2), that is, p2± = p± and p− p+ = p+ p− = 0. These remarks allow us to compute (for simplicity we denote 1 ⊗ 1 ⊗ 1 ⊗ 1 by 1): (1 ⊗ Φ)(Id ⊗ Δ ⊗ Id)(Φ)(Φ ⊗ 1) = (1 − 2 ⊗ p− ⊗ p− ⊗ p− )(1 − 2p− ⊗ p+ ⊗ p− ⊗ p− − 2p− ⊗ p− ⊗ p+ ⊗ p− ) (1 − 2p− ⊗ p− ⊗ p− ⊗ 1) = (1 − 2p− ⊗ p+ ⊗ p− ⊗ p− − 2p− ⊗ p− ⊗ p+ ⊗ p− − 2 ⊗ p− ⊗ p− ⊗ p− ) (1 − 2p− ⊗ p− ⊗ p− ⊗ 1) = 1 − 2p− ⊗ p+ ⊗ p− ⊗ p− − 2p− ⊗ p− ⊗ p+ ⊗ p− − 2 ⊗ p− ⊗ p− ⊗ p− − 2p− ⊗ p− ⊗ p− ⊗ 1 + 4p− ⊗ p− ⊗ p− ⊗ p−

120

Quasi-bialgebras and Quasi-Hopf Algebras = 1 − 2p− ⊗ g ⊗ p− ⊗ p− − 2p− ⊗ p− ⊗ g ⊗ p− − 2 ⊗ p− ⊗ p− ⊗ p− − 2p− ⊗ p− ⊗ p− ⊗ 1 = 1 − 2p+ ⊗ p− ⊗ p− ⊗ p− − 2p− ⊗ p+ ⊗ p− ⊗ p− − 2p− ⊗ p− ⊗ p− ⊗ p+ − 2p− ⊗ p− ⊗ p+ ⊗ p− = (1 − 2p− ⊗ p− ⊗ p+ ⊗ p− − 2p− ⊗ p− ⊗ p− ⊗ p+ ) (1 − 2p+ ⊗ p− ⊗ p− ⊗ p− − 2p− ⊗ p+ ⊗ p− ⊗ p− ) = (Id ⊗ Id ⊗ Δ)(Φ)(Δ ⊗ Id ⊗ Id)(Φ),

as required. Since ε (p− ) = 0 it follows that (3.1.10) is satisfied as well. Thus H(2) is a quasi-bialgebra. It is, moreover, a quasi-Hopf algebra, since S(g)gg = g = ε (g)g, gS(g) = 1 = ε (g)1, X 1 β S(X 2 )α X 3 = α − 2p2− α p− = g − 2p− gp− = g + 2p− = 1 and S(x1 )α x2 β S(x3 ) = α − 2p− α p2− = g − 2p− gp− = g + 2p− = 1, where we used the fact that Φ−1 = Φ. Assume now that there is a gauge transformation F = F 1 ⊗ F 2 on H(2) such that Φ = (1 ⊗ F)(Id ⊗ Δ)(F)(Δ ⊗ Id)(F −1 )(F −1 ⊗ 1). Since {p+ , p− } is a basis of H(2) as well we can write F in the form F = ap+ ⊗ p+ + bp+ ⊗ p− + cp− ⊗ p+ + d p− ⊗ p− , for some scalars a, b, c, d ∈ k. Since we have ε (p− ) = 0, ε (p+ ) = 1 and ε (F 1 )F 2 = 1 = ε (F 2 )F 1 we get a = b = c = 1, and so F = p+ ⊗ p+ + p+ ⊗ p− + p− ⊗ p+ + d p− ⊗ p− = p+ ⊗ 1 + p− ⊗ p+ + d p− ⊗ p− , for some d ∈ k. One can easily see that F as above is invertible in H(2) ⊗ H(2) if and only if d is non-zero, and in this case F −1 = p+ ⊗ 1 + p− ⊗ p+ + d −1 p− ⊗ p− . By similar computations we get Δ(p+ ) = p+ ⊗ p+ + p− ⊗ p− , and therefore (1 ⊗ F)(Id ⊗ Δ)(F) = (1 ⊗ p+ ⊗ 1 + 1 ⊗ p− ⊗ p+ + d ⊗ p− ⊗ p− )(p+ ⊗ 1 ⊗ 1 + p− ⊗ p+ ⊗ p+ + p− ⊗ p− ⊗ p− + d p− ⊗ p+ ⊗ p− + d p− ⊗ p− ⊗ p+ ) = p+ ⊗ p+ ⊗ 1 + p− ⊗ p+ ⊗ p+ + d p− ⊗ p+ ⊗ p− + p+ ⊗ p− ⊗ p+ + d p− ⊗ p− ⊗ p+ + d p+ ⊗ p− ⊗ p− + d p− ⊗ p− ⊗ p− . Likewise, we compute: (Δ ⊗ Id)(F −1 )(F −1 ⊗ 1)

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121

= (p+ ⊗ p+ ⊗ 1 + p− ⊗ p− ⊗ 1 + p+ ⊗ p− ⊗ p+ + p− ⊗ p+ ⊗ p+ + d −1 p+ ⊗ p− ⊗ p− + d −1 p− ⊗ p+ ⊗ p− ) (p+ ⊗ 1 ⊗ 1 + p− ⊗ p+ ⊗ 1 + d −1 p− ⊗ p− ⊗ 1) = p+ ⊗ p+ ⊗ 1 + d −1 p− ⊗ p− ⊗ 1 + p+ ⊗ p− ⊗ p+ + p− ⊗ p+ ⊗ p+ + d −1 p+ ⊗ p− ⊗ p− + d −1 p− ⊗ p+ ⊗ p− . Hence Φ = p+ ⊗ p+ ⊗ 1 + p− ⊗ p+ ⊗ p+ + p− ⊗ p+ ⊗ p− + p+ ⊗ p− ⊗ p+ + p− ⊗ p− ⊗ p+ + p+ ⊗ p− ⊗ p− + p− ⊗ p− ⊗ p− = p+ ⊗ p+ ⊗ 1 + p− ⊗ p+ ⊗ 1 + p+ ⊗ p− ⊗ 1 + p− ⊗ p− ⊗ 1 = p+ ⊗ 1 ⊗ 1 + p− ⊗ 1 ⊗ 1 = 1 ⊗ 1 ⊗ 1, which is clearly a contradiction. Thus H(2) is a quasi-Hopf algebra that does not come from a Hopf algebra twisted by a gauge transformation. Infinite-dimensional quasi-Hopf algebras can be constructed as follows. Example 3.27 Let k be a field that contains a primitive fourth root of unity i (in particular the characteristic of k is not 2), and consider H± (∞) as being the k-algebra generated by g, x with relations g2 = 1 and xg = −gx. Then H± (∞) are quasi-Hopf algebras with comultiplication defined by Δ(g) = g ⊗ g, ε (g) = 1, Δ(x) = x ⊗ (p+ ± ip− ) + 1 ⊗ p+ x + g ⊗ p− x, ε (x) = 0, reassociator given by Φ = 1⊗1⊗1−2p− ⊗ p− ⊗ p− , antipode determined by S(g) = g, S(x) = −x(p+ ± ip− ), and distinguished elements α = g and β = 1. We denoted as before p± = 12 (1 ± g). Proof Observe that H± (∞) contain H(2), which is a quasi-Hopf algebra via the structure defined above. So to prove that H± (∞) are quasi-bialgebras it is sufficient to show that Φ(Δ ⊗ Id)(Δ(x)) = (Id ⊗ Δ)(Δ(x))Φ, and that Δ, extended as an algebra morphism from H± (∞) to H± (∞) ⊗ H± (∞), behaves well with respect to the relation xg = −gx (the other details are immediate). It is easy to see that Δ(p+ x) = p+ x ⊗ p+ + p+ ⊗ p+ x ± ip− x ⊗ p− − p− ⊗ p− x, Δ(p− x) = ±ip+ x ⊗ p− + p+ ⊗ p− x + p− x ⊗ p+ + p− ⊗ p+ x, Δ(p+ ± ip− ) = p+ ⊗ (p+ ± ip− ) + p− ⊗ (p− ± ip+ ), and so (Id ⊗ Δ)(Δ(x)) = x ⊗ (p+ ± ip− ) ⊗ p+ ± ix ⊗ p+ ⊗ p− + x ⊗ p− ⊗ p−

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+ 1 ⊗ p+ x ⊗ p+ + 1 ⊗ p+ ⊗ p+ x ± i1 ⊗ p− x ⊗ p− − 1 ⊗ p− ⊗ p− x ± ig ⊗ p+ x ⊗ p− + g ⊗ p+ ⊗ p− x + g ⊗ p− x ⊗ p+ + g ⊗ p− ⊗ p+ x. From here we compute: (Id ⊗ Δ)(Δ(x))Φ = (Id ⊗ Δ)(Δ(x)) − 2(Id ⊗ Δ)(Δ(x))p− ⊗ p− ⊗ p− = (Id ⊗ Δ)(Δ(x)) − 2(p+ x ⊗ p− ⊗ p− ∓ ip− ⊗ p+ x ⊗ p− − p− ⊗ p− ⊗ p+ x) = x ⊗ (p+ ± ip− ) ⊗ p+ ± ix ⊗ p+ ⊗ p− − gx ⊗ p− ⊗ p− ± i1 ⊗ x ⊗ p− + 1 ⊗ p− ⊗ gx + 1 ⊗ p+ x ⊗ p+ + 1 ⊗ p+ ⊗ p+ x + g ⊗ p+ ⊗ p− x + g ⊗ p− x ⊗ p+ = x ⊗ (p+ ± ip− ) ⊗ p+ ± ix ⊗ p+ ⊗ p− − gx ⊗ p− ⊗ p− ± i1 ⊗ x ⊗ p− + 1 ⊗ p+ x ⊗ p+ + 1 ⊗ 1 ⊗ p+ x + g ⊗ p+ ⊗ p− x − 1 ⊗ p− ⊗ p− x + g ⊗ p− x ⊗ p+ . Now, we have (Δ ⊗ Id)(Δ(x)) = x ⊗ (p+ ± ip− ) ⊗ (p+ ± ip− ) + 1 ⊗ p+ x ⊗ (p+ ± ip− ) + g ⊗ p− x ⊗ (p+ ± ip− ) + 1 ⊗ 1 ⊗ p+ x + g ⊗ g ⊗ p− x, which implies Φ(Δ ⊗ Id)(Δ(x)) = (Δ ⊗ Id)(Δ(x)) − 2(p− ⊗ p− ⊗ p− )(Δ ⊗ Id)(Δ(x)) = (Δ ⊗ Id)(Δ(x)) + 2(p− x ⊗ p− ⊗ p− ± ip− ⊗ p− x ⊗ p− − p− ⊗ p− ⊗ p− x) = x ⊗ (p+ ± ip− ) ⊗ p+ ± ix ⊗ p+ ⊗ p− + 1 ⊗ p+ x ⊗ (p+ ± ip− ) + g ⊗ p− x ⊗ p+ + 1 ⊗ 1 ⊗ p+ x + g ⊗ p+ ⊗ p− x − gx ⊗ p− ⊗ p− ± i1 ⊗ p− x ⊗ p− − 1 ⊗ p− ⊗ p− x = x ⊗ (p+ ± ip− ) ⊗ p+ ± ix ⊗ p+ ⊗ p− + 1 ⊗ p+ x ⊗ p+ + g ⊗ p− x ⊗ p+ + 1 ⊗ 1 ⊗ p+ x + g ⊗ p+ ⊗ p− x − gx ⊗ p− ⊗ p− ± i1 ⊗ x ⊗ p− − 1 ⊗ p− ⊗ p− x. By the two equalities above we conclude that Φ(Δ ⊗ Id)(Δ(x)) = (Id ⊗ Δ)(Δ(x))Φ, so the first assertion is proved. To prove the second one we have to verify that (g ⊗ g)Δ(x) = −Δ(x)(g ⊗ g). This follows easily from the equalities gx = −xg, gp+ = p+ and gp− = −p− , we leave the details to the reader. Finally, H± (∞) are quasi-Hopf algebras since S(x1 )gx2 = S(x)g(p+ ± ip− ) + gp+ x + S(g)gp− x = −x(p+ ± ip− )g(p+ ± ip− ) + p+ x + p− x = −x(p+ ∓ ip− )(p+ ± ip− ) + x = −x(p+ + p− ) + x = −x + x = 0 = ε (x)g, and similarly x1 S(x2 ) = xS(p+ ± ip− ) + S(x)S(p+ ) + gS(x)S(p− ) = x(p+ ± ip− ) − x(p+ ± ip− )p+ − gx(p+ ± ip− )p−

3.3 Examples of Quasi-bialgebras and Quasi-Hopf Algebras

123

= x(p+ ± ip− ) − xp+ ∓ igxp− = ±ixp− ± ixgp− = ±ixp− ∓ ixp− = 0 = ε (x)1. The remaining relations hold because they hold in H(2). Eight-dimensional quasi-Hopf algebras can be obtained from H± (∞) by factorizing it through a quasi-Hopf ideal. Definition 3.28 If H is a quasi-Hopf algebra then an ideal I of H is called a quasiHopf ideal if Δ(I) ⊆ I ⊗ H + H ⊗ I, ε (I) = 0 and S(I) ⊆ I. Proposition 3.29 If I is a quasi-Hopf ideal of a quasi-Hopf algebra H then the quotient algebra H/I has a unique quasi-Hopf algebra structure such that the canonical surjection p : H → H/I becomes a quasi-Hopf algebra morphism. Proof From Δ(I) ⊆ I ⊗ H + H ⊗ I it follows that (p ⊗ p)Δ(I) = 0, and so, by the Universal Property of a quotient algebra, there exists a unique algebra morphism  : H/I → H/I ⊗ H/I such that Δ  ◦ p = (p ⊗ p) ◦ Δ. Similarly, from ε (I) = 0 it Δ ε : H/I → k such that  ε ◦ p = ε, follows that there is a unique algebra morphism  and S(I) ⊆ I guarantees the existence and uniqueness of the anti-algebra morphism   ε , (p ⊗ p ⊗ S : H/I → H/I obeying S◦ p = p ◦ S. It is clear at this point that (H/I, Δ,  p(α ), p(β )) is the unique quasi-Hopf algebra structure on H/I for which p)(Φ), S, p : H → H/I becomes a quasi-Hopf algebra morphism. Example 3.30 Consider k a field that contains a primitive fourth root of unity i and let H± (8) be the unital algebras generated by g, x with relations g2 = 1, x4 = 0 and gx = −xg, and endowed with the (non-coassociative) comultiplication given by Δ(g) = g ⊗ g, ε (g) = 1, Δ(x) = x ⊗ (p+ ± ip− ) + 1 ⊗ p+ x + g ⊗ p− x, ε (x) = 0, where p± = 12 (1 ± g). Then H± (8) are eight-dimensional quasi-Hopf algebras with reassociator Φ = 1⊗1⊗1−2p− ⊗ p− ⊗ p− , antipode defined by S(g) = g and S(x) = −x(p+ ± ip− ), and distinguished elements α = g and β = 1. Proof By the above results all we have to check is the fact that the ideal generated by x4 in H± (∞) is a quasi-Hopf ideal. Then H± (8) are nothing else than the quotient quasi-Hopf algebras H± (∞)/(x4 ). Therefore, we must check that Δ(x4 ) ∈ (x4 ) ⊗ H± (∞) + H± (∞) ⊗ (x4 ), ε (x4 ) = 0 and S(x4 ) ∈ (x4 ). Indeed, a straightforward computation ensures that Δ(x2 ) = x2 ⊗ g + (1 ± i)(p+ x ⊗ p+ x + p− x ⊗ p+ x) + (1 ∓ i)(p+ x ⊗ p− x − p− x ⊗ p− x) + g ⊗ x2 , from which we obtain by calculation that Δ(x4 ) = x4 ⊗ 1 + 1 ⊗ x4 . Since ε is extended to the whole H± (∞) as an algebra map and ε (x) = 0 we get that ε (x4 ) = 0, and since S(x2 ) = x(p+ ± ip− )x(p+ ± ip− )

124

Quasi-bialgebras and Quasi-Hopf Algebras = x2 (p− ± ip+ )(p+ ± ip− ) = x2 (±ip− ± ip+ ) = ±ix2

it follows that S(x4 ) = S(x2 )2 = −x4 . Thus our claim is proved. It is easy to see that {gi x j | 0 ≤ i ≤ 1, 0 ≤ j ≤ 3} is a basis for H± (8), so H± (8) are eight-dimensional quasi-Hopf algebras. A 32-dimensional quasi-Hopf algebra can be constructed by amalgamating the quasi-Hopf algebras H+ (8) and H− (8), along the quasi-Hopf algebra H(2). To this end we first construct from H+ (8) and H− (8) an infinite-dimensional quasi-Hopf algebra H+− (∞) as follows. The proof of the result below is based on the computations performed in Example 3.27, and we leave it to the reader. Example 3.31 Denote by H+− (∞) the k-algebra generated by g, x, y with relations g2 = 1, x4 = y4 = 0, gx = −xg and gy = −yg. Then H+− (∞) is an infinite-dimensional quasi-Hopf algebra with structure determined by Δ(g) = g ⊗ g, ε (g) = 1, Δ(x) = x ⊗ (p+ + ip− ) + 1 ⊗ p+ x + g ⊗ p− x, ε (x) = 0, Δ(y) = y ⊗ (p+ − ip− ) + 1 ⊗ p+ y + g ⊗ p− y, ε (y) = 0, S(g) = g,

S(x) = −x(p+ + ip− ),

S(y) = −y(p+ − ip− ).

Its reassociator is Φ = 1⊗1⊗1−2p− ⊗ p− ⊗ p− , and the two distinguished elements of H+− (∞) are α = g and β = 1. Factorizing H+− (∞) through a certain quasi-Hopf ideal we obtain the following 32-dimensional quasi-Hopf algebra. Example 3.32 Assume again that k is a field which contains a primitive fourth root of unity i. By H(32) denote the unital k-algebra generated by g, x and y with relations g2 = 1, x4 = y4 = 0, gx = −xg, gy = −yg and yx = ixy. If we endow H(32) with the comultiplication and counit defined by the formulas Δ(g) = g ⊗ g, ε (g) = 1, Δ(x) = x ⊗ (p+ + ip− ) + 1 ⊗ p+ x + g ⊗ p− x, ε (x) = 0, Δ(y) = y ⊗ (p+ − ip− ) + 1 ⊗ p+ y + g ⊗ p− y, ε (y) = 0, where p± = 12 (1 ± g), then with these structures H(32) is a 32-dimensional quasiHopf algebra with reassociator Φ = 1 ⊗ 1 ⊗ 1 − 2p− ⊗ p− ⊗ p− . Its antipode is determined by S(g) = g, S(x) = −x(p+ + ip− ) and S(y) = −y(p+ − ip− ), and the distinguished elements are α = g and β = 1.

3.4 The Rigid Monoidal Structure of H M fd and MHfd

125

Proof We will prove that the ideal generated by z := xy + iyx in H+− (∞) is a quasiHopf ideal. Then the proof ends because of the identification of H(32) with the quotient quasi-Hopf algebra H+− (∞)/(z). To this end we compute: Δ(xy) = [x ⊗ (p+ + ip− ) + 1 ⊗ p+ x + g ⊗ p− x][y ⊗ (p+ − ip− ) + 1 ⊗ p+ y + g ⊗ p− y] = xy ⊗ 1 + x ⊗ p+ y + ixg ⊗ p− y − iy ⊗ p+ x + g ⊗ p+ xy + gy ⊗ p− x + g ⊗ p− xy, and similarly Δ(yx) = yx ⊗ 1 + y ⊗ p+ x − iyg ⊗ p− x + ix ⊗ p+ y + g ⊗ p+ yx + gx ⊗ p− y + g ⊗ p− yx. We conclude that Δ(z) = Δ(xy) + iΔ(yx) = xy ⊗ 1 + g ⊗ p+ xy + g ⊗ p− xy + iyx ⊗ 1 + ig ⊗ p+ yx + ig ⊗ p− yx = z ⊗ 1 + g ⊗ p− z + g ⊗ p+ z = z ⊗ 1 + g ⊗ z ∈ (z) ⊗ H + H ⊗ (z), as required. By ε (x) = ε (y) = 0 we deduce ε (z) = 0, and since S(z) = S(y)S(x) + iS(x)S(y) = y(p+ − ip− )x(p+ + ip− ) + ix(p+ + ip− )y(p+ − ip− ) = yx(p− − ip+ )(p+ + ip− ) + ixy(p− + ip+ )(p+ − ip− ) = yx(ip− − ip+ ) − ixy(ip− − ip+ ) = z(p− − p+ ) = −zg ∈ (z), we obtain that (z) is a quasi-Hopf ideal in H+− (∞), as stated. Finally, it can be easily checked that {g j xs yt | 0 ≤ j ≤ 1, 0 ≤ s ≤ 3, 0 ≤ t ≤ 3} is a basis in H(32), and so indeed H(32) is 32-dimensional. Other classes of quasi-bialgebras and quasi-Hopf algebras will be presented in the forthcoming chapters.

3.4 The Rigid Monoidal Structure of H M fd and MHfd We will describe the rigid structures of the monoidal categories H M fd and MHfd , the categories of finite-dimensional left and right representations, respectively, over the quasi-Hopf algebra H. Then we compute the canonical isomorphisms defined in Section 1.6 for the particular case when C is either H M fd or MHfd . In what follows, if V is a finite-dimesional vector space, if v∗ ∈ V ∗ and v ∈ V , we will sometimes denote v∗ (v) by v∗ , v.

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Proposition 3.33 Let H be a quasi-Hopf algebra with antipode S and distinguished elements α , β ∈ H. Then H M fd is left rigid and MHfd is right rigid. When S is bijective, both H M fd and MHfd are rigid monoidal categories. Proof Let V be a finite-dimensional left H-module and consider V ∗ = Homk (V, k) with the left H-action (h · v∗ )(v) = v∗ (S(h) · v), for all v∗ ∈ V ∗ , h ∈ H and v ∈ V . By definition S is an anti-algebra homomorphism of H, so V ∗ is a left H-module via this action. If we define the linear maps evV : V ∗ ⊗V → k,

evV (v∗ ⊗ v) = v∗ (α · v), ∀ v∗ ∈ V ∗ , v ∈ V,

coevV : k → V ⊗V ∗ ,

coevV (1) = ∑ β · vi ⊗ vi , i

and V ∗ , we claim that (coevV , evV ) : V ∗

where {vi }i and {vi }i

are dual bases in V is an adjunction in H M fd . Indeed, evV is H-linear since

V

evV (h · (v∗ ⊗ v)) = evV (h1 · v∗ ⊗ h2 · v) = (h1 · v∗ )(α h2 · v) (3.2.1)

= v∗ (S(h1 )α h2 · v) = ε (h)v∗ (α · v) = ε (h)evV (v∗ ⊗ v), for all v∗ ∈ V ∗ , h ∈ H and v ∈ V . Similarly, using the second equality in (3.2.1), one can easily see that coevV is left H-linear, too. Also, the left-hand side in (1.6.4) becomes, for all v ∈ V , rV ◦ (IdV ⊗ evV ) ◦ aV,V ∗ ,V ◦ (coevV ⊗ IdV ) ◦ lV−1 (v) = ∑ rV ◦ (IdV ⊗ evV ) ◦ aV,V ∗ ,V ((β · vi ⊗ vi ) ⊗ v) i

= ∑ rV ◦ (IdV ⊗ evV )(X 1 β · vi ⊗ (X 2 · vi ⊗ X 3 · v)) i

= ∑(X 2 · vi )(α X 3 · v)X 1 β · vi = ∑ vi (S(X 2 )α X 3 · v)X 1 β · vi i

i

(3.2.2)

= X 1 β S(X 2 )α X 3 · v = 1H · v = v, as required. The second equality in (3.2.2) implies (1.6.5), so we are done. Similarly, MHfd is right rigid. For any object V of MHfd consider ∗V as being Homk (V, k) equipped with the right H-module structure (∗ v · h)(v) = ∗ v(v · S(h)), for all ∗ v ∈ ∗V , h ∈ H and v ∈ V . Then ∗V and the linear maps evV : V ⊗ ∗V → k,

evV (v ⊗ ∗ v) = ∗ v(v · β ), ∀ ∗ v ∈ ∗V, v ∈ V,

coevV : k → ∗V ⊗V,

coevV (1) = ∑ vi ⊗ vi · α , i

define a right dual for V in MHfd ; we leave the verification of the details to the reader. When S is bijective, any V ∈ H M fd has a right dual as well. It is defined by ∗V := Hom (V, k) viewed as a left H-module via (h · ∗ v)(v) = ∗ v(S−1 (h) · v), for k all ∗ v ∈ ∗V , h ∈ H and v ∈ V , and the linear maps evV : V ⊗ ∗V → k, coevV : k → ∗V ⊗V,

evV (v ⊗ ∗ v) = ∗ v(S−1 (α ) · v), ∀ ∗ v ∈ ∗V, v ∈ V, coevV (1) = ∑ vi ⊗ S−1 (β ) · vi . i

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127

Actually, (3.2.1) implies that evV and coevV are left H-linear, while (3.2.2) implies (1.6.4) and (1.6.5). In a similar manner it can be shown that any V ∈ MHfd has a left dual, if S is bijective. This time the left dual of V is the same object V ∗ = Homk (V, k), viewed as a right H-module via (v∗ · h)(v) = v∗ (v · S−1 (h)), for all v∗ ∈ V ∗ , h ∈ H, v ∈ V , with the two morphisms in MH , evV : V ∗ ⊗V → k and coevV : k → V ⊗V ∗ , defined by evV (v∗ ⊗ v) = v∗ (v · S−1 (β )) and coevV (1) = ∑ vi · S−1 (α ) ⊗ vi . i

Once more we leave the details to the reader. The transpose of a morphism f : V → Y in H M fd or MHfd (in the sense of Proposition 1.68) coincides with the usual transpose of f in the category of k-vector spaces. Proposition 3.34 When C is the category of finite-dimensional left modules over a quasi-Hopf algebra, the maps f ∗ and, if the antipode is bijective, ∗ f coincide with the usual transpose map of f in k M , that is, f ∗ (y∗ ) = y∗ ◦ f and ∗ f (∗ y) = ∗ y ◦ f , for all y∗ ∈ Y ∗ , ∗ y ∈ ∗Y , respectively. A similar result holds in the situation when C is the category of finite-dimensional right modules over a quasi-Hopf algebra. Proof

Indeed, by the definition of f ∗ and (3.2.2), we have that f ∗ (y∗ ) = ∑x1 · y∗ , α x2 β · f (vi )x3 · vi i

= ∑y∗ , S(x1 )α x2 β S(x3 ) · f (vi )vi = y∗ ◦ f , i

where f,

∗y ◦

{vi , vi }i are dual bases in V and V ∗ . Similarly one for all ∗ y ∈ ∗Y . The right-handed case can be proved

can show that ∗ f (∗ y) = in a similar manner.

For H M fd we compute explicitly the isomorphism λ defined in (1.7.1) and its right-handed version λ  as follows. Proposition 3.35 Let H be a quasi-Hopf algebra and X,Y ∈ H M fd . Consider {xi }i and {xi }i dual bases in X and X ∗ , and {y j } j and {y j } j dual bases in Y and Y ∗ . Then −1 are given by the formulas the maps λX,Y and λX,Y

λX,Y (μ ) = ∑μ , g1 · xi ⊗ g2 · y j y j ⊗ xi , i, j

−1 ∗ λX,Y (y ⊗ x∗ )(x ⊗ y) = x∗ , f 1 · xy∗ , f 2 · y,

(3.4.1)

for all μ ∈ (X ⊗Y )∗ , x∗ ∈ X ∗ , y∗ ∈ Y ∗ , x ∈ X and y ∈ Y . −1  and λX,Y are given by If the antipode of H is bijective then the maps λX,Y  (ν ) = ν , S−1 (g2 ) · x ⊗ S−1 (g1 ) · y y j ⊗ xi , λX,Y ∑ i j i, j

−1 ∗ λX,Y ( y ⊗ ∗ x)(x ⊗ y) = ∗ x, S−1 ( f 2 ) · x∗ y, S−1 ( f 1 ) · y,

for all ν ∈ ∗ (X ⊗ Y ), ∗ x ∈ ∗ X, ∗ y ∈ ∗Y , x ∈ X and y ∈ Y . Here f = f 1 ⊗ f 2 is the Drinfeld twist from (3.2.15) and g = g1 ⊗ g2 is its inverse defined in (3.2.16).

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Quasi-bialgebras and Quasi-Hopf Algebras

Proof Let pR = p1 ⊗ p2 be the element defined in (3.2.19). We compute, for all y∗ ∈ Y ∗ , x∗ ∈ X ∗ , x ∈ X and y ∈ Y : −1 ∗ λX,Y (y ⊗ x∗ )(x ⊗ y) (1.7.2)

=

x∗ , S(x1 X 2 y12 )α x2 (X 3 y2 β S(y3 ))1 · x y∗ , S(X 1 y11 )α x3 (X 3 y2 β S(y3 ))2 · y

(3.2.5)

x∗ , S(y12 )γ 1 (y2 β S(y3 ))1 · xy∗ , S(y11 )γ 2 (y2 β S(y3 ))2 · y

(3.2.19)

=

x∗ , f 1 S(p1 )1 g1 γ 1 p21 · xy∗ , f 2 S(p1 )2 g2 γ 2 p22 · y

=

x∗ , f 1 (S(p1 )α p2 )1 · xy∗ , f 2 (S(p1 )α p2 )2 · y

=

(3.2.19),(3.2.2)

=

x∗ , f 1 · xy∗ , f 2 · y,

where we also used the relations in Theorem 3.21 in the third and fourth equality. By using (1.7.1) or by a direct computation it is easy to see that

λX,Y (μ ) = ∑μ , g1 · xi ⊗ g2 · y j y j ⊗ xi , for all μ ∈ (X ⊗Y )∗ . i, j

 The assertion concerning the morphism λX,Y can be proved in a similar way; we leave the details to the reader.

Note that the map θM defined in (1.7.3) specializes for C = H M fd as n

θM (m) = ∑ i m, m∗i m

(3.4.2)

i=1

for all m ∈ M, where, if (i m)i=1,n is a basis of M ∈ H M fd with dual basis (i m)i=1,n in M ∗ , then ∗i m is the image of i m under the canonical map M m → (m∗ → m∗ (m)) ∈  is defined by the same formula M ∗∗ . Furthermore, in this case the morphism θM (3.4.2) as θM . Similar formulas can be obtained for C = MHfd .

3.5 The Reconstruction Theorem for Quasi-Hopf Algebras We proved in Proposition 3.3 a reconstruction type theorem for quasi-bialgebras. The purpose of this section is to provide a similar result but now for quasi-Hopf algebras. For this, we must endow a quasi-monoidal functor with an extra property. Keeping in mind the statement of Proposition 3.33, this new property comes out naturally as follows: Definition 3.36 Let F : C → D be a quasi-monoidal functor between two left rigid monoidal categories. We√ say that F is a left rigid quasi-monoidal √functor if there exist isomorphisms F(X ) ∼ = F(X)∗ , natural in X ∈ Ob(C ). Here X is the left dual ∗ of X in C and F(X) is the left dual of F(X) in D. Let H be a quasi-bialgebra and denote by H M fd the category of finite-dimensional left representations over H. If k M fd is the category of finite-dimensional k-vector

3.5 The Reconstruction Theorem for Quasi-Hopf Algebras

129

spaces it follows that the forgetful functor F : H M fd → k M fd is a quasi-monoidal functor. We next see when F is a left rigid quasi-monoidal functor. Lemma 3.37 Let H be a quasi-bialgebra over k. Then the forgetful functor F : fd fd H M → k M is a left rigid quasi-monoidal functor if and only if: (LR1) For any V ∈ H M fd , the left dual V ∗ of V in k M fd admits a left H-module structure with respect to which there exists an adjunction (coevVH , evVH ) : V ∗  V in H M fd , that is, V ∗ with a suitable structure becomes a left dual for V in fd HM . √ √ (LR2) For any morphism f : V → W in H M fd we have f = f ∗ , where f is the left transpose of f in H M fd and f ∗ is the left transpose of f in k M fd . √

Proof The functor F acts√as identity on objects and morphisms. So F(V ) ∼ = F(V )∗ ∗ reduces to the choice of√V as being V , with additional structures as in (LR1) that come from those of V (recall that the left dual of an object is unique up to an isomorphism; see Proposition 1.66). Furthermore, given this choice we must require √ f √ = f ∗ , for any f : V → W in H M fd , and this is because the k-linear isomorphisms V ∼ = V ∗ are natural in V ∈ H M fd . Hence our proof is complete. We now present the reconstruction theorem for quasi-Hopf algebras. Note that in the proof we need the k-algebra H to be an object in H M fd , so we have to assume from the beginning that H is finite dimensional. Theorem 3.38 Let H be a finite-dimensional algebra over a field k. Then there exists a bijective correspondence between • quasi-Hopf algebra structures on H; • left rigid monoidal stuctures on H M fd for which the functor F : H M fd → k M fd that forgets the left H-action is a left rigid quasi-monoidal functor. Proof By Proposition 3.33 and Proposition 3.34 it follows that for any quasi-Hopf algebra H (not necessarily finite dimensional) the forgetful functor F : H M fd → fd k M is a left rigid quasi-monoidal functor. Thus by Proposition 3.3, Corollary 3.11 and Proposition 3.5 we only have to show that a finite-dimensional quasi-bialgebra H is a quasi-Hopf algebra, provided that the forgetful functor F : H M fd → k M fd is a left rigid quasi-monoidal functor. For this, let H be a finite-dimensional quasi-bialgebra and {hi }i a basis in H with dual basis {hi }i in H ∗ . Regard H ∈ H M fd via its multiplication and denote by · : H ⊗ H ∗ → H ∗ the left H-module structure of H ∗ , and by (ι , ε ) : H ∗  H the adjunction in fd H M as in (LR1). We claim that this data defines completely the left rigid monoidal structure on H M fd as well as the quasi-Hopf algebra structure on H, provided that the forgetful functor F : H M fd → k M fd is left rigid quasi-monoidal. Indeed, consider S : H → H given by S(h) = ∑(h · hi )(1H )hi , ∀ h ∈ H. i

(3.5.1)

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Quasi-bialgebras and Quasi-Hopf Algebras

Furthermore, for V ∈ H M fd and a fixed element v ∈ V , define ϕ : H → V by ϕv (h) = √ v ∗ h · v, for all h ∈ H, a left H-linear morphism. Then ϕv = ϕv : V ∗ → H ∗ is a left Hlinear morphism, that is, ϕv∗ (h · v∗ ) = h · ϕv∗ (v∗ ), for all h ∈ H and v∗ ∈ V ∗ . This is clearly equivalent to (h · v∗ ) ◦ ϕv = h · ϕv∗ (v∗ ) = h · ∑ ϕv∗ (v∗ )(hi )hi = ∑ h · v∗ (ϕv (hi ))hi = ∑ v∗ (hi · v)h · hi , i

i

i

as elements in H ∗ , for all h ∈ H. Evaluating both sides on 1H we get (h · v∗ )(v) = ∑ v∗ (hi · v)(h · hi )(1H ) = v∗ (S(h) · v), ∀ h ∈ H,

(3.5.2)

i

and this describes completely the left H-module structure of V ∗ , for any V ∈ H M fd . Since the definition of S depends only on H and the structure in (3.5.2) turns any V ∗ into a left H-module, it follows that S : H → H is an anti-algebra morphism. It remains to define the distinguished elements α , β ∈ H that together with S defined by (3.5.1) obey the relations (3.2.1)–(3.2.2). To this end, we write ι (1H ) = ∑ j x j ⊗ x j ∈ H ⊗ H ∗ and define

α = ∑ ε (hi ⊗ 1H )hi

β = ∑ x j (1H )x j .

and

i

(3.5.3)

j

We next see that α , β determine completely the adjunction (ιV , εV ) : V ∗  V in (LR1) that behaves well with respect to the left H-module structures in (3.5.2). This claim follows from (LR2) since for any morphism f : V → W in H M fd we have W∗

W∗ V

W∗ V

k

k

  ∗ h fh V W  V = f ∗ , and this implies = and fh = , ∗ fh W V

W

f

k

V∗

k

W V∗

(3.5.4)

W V∗

where, as the notation suggests, the evaluation and coevaluation morphisms with the letter V or W nearby is our diagrammatic notation for the evaluation and coevaluation morphisms corresponding to V and W , respectively, in H M fd . For V ∈ H M fd and fixed arbitrary v ∈ V consider again ϕv : H → V , the left Hlinear morphism defined above. By the equality in (3.5.4) involving the evaluation morphisms, specialized for ϕv , we get that εV (v∗ ⊗ h · v) = ε (v∗ ◦ ϕv ⊗ h), for all v∗ ∈ V ∗ and h ∈ H. By taking h = 1H we obtain that

εV (v∗ ⊗ v) = ε (v∗ ◦ ϕv ⊗ 1H ) =

∑ v∗ (ϕv (hi ))ε (hi ⊗ 1H )

i (3.5.3) ∗

= v (α · v),

for all v∗ ∈ V ∗ and v ∈ V , so εV is completely determined by α . Similarly, by taking f = ϕv in the equality of (3.5.4) that involves the coevaluation

3.6 Sovereign Quasi-Hopf Algebras

131

morphisms we deduce that (ϕv ⊗ IdH ∗ )ι = (IdV ⊗ ϕv∗ )ιV , as morphisms from k to H ⊗ H ∗ . If we write ιV (1k ) := ∑ yl ⊗ yl ∈ V ⊗V ∗ we get l

∑ ϕv (xi ) ⊗ xi = ∑ yl ⊗ yl ◦ ϕv ∈ V ⊗ H ∗ , i

l

and therefore

∑ yl (v)yl = ∑ xi (1H )xi · v

(3.5.3)

= β · v,

i

l

for all v ∈ V . It follows that ιV (1k ) = ∑s β · vs ⊗ vs , where {vs , vs }s are dual bases in V and V ∗ . So β describes completely the coevaluation morphisms ιV , V ∈ H M fd . It is clear at this moment that εV and ιV are H-linear morphisms, for all V ∈ H M fd , if and only if (3.2.1) are satisfied, and that (ιV , εV ) obeys (1.6.6) if and only if (3.2.2) are fulfilled. Otherwise stated, the triple (S, α , β ) defines an antipode for the quasibialgebra H, and therefore H is a quasi-Hopf algebra.

3.6 Sovereign Quasi-Hopf Algebras Let H be a quasi-Hopf algebra. In this section we investigate when H M fd is a sovereign monoidal category, in the sense of Definition 1.72. It will come up that sovereign structures on H M fd are in a one-to-one correspondence with the pivotal structures, and that the latter are completely determined by certain elements of H. If C is a monoidal category with left duality, then the functor (−)∗∗ := ((−)∗ )∗ : C → C is strong monoidal: we have an isomorphism φ0 : 1 → 1∗∗ , and for V,W ∈ C , we have the following family of isomorphisms in C :

φV,W : V ∗∗ ⊗W ∗∗

−1 λW ∗ ,V ∗

−→

(W ∗ ⊗V ∗ )∗

(λV,W )∗

−→

(V ⊗W )∗∗ .

λV,W : (V ⊗W )∗ → W ∗ ⊗V ∗ is the isomorphism described in (1.7.1), (λV,W )∗ is the transpose in C of λV,W , and λW−1∗ ,V ∗ is the inverse of λW ∗ ,V ∗ . Definition 3.39 Let C be a left rigid monoidal category. A pivotal structure on C is a monoidal natural isomorphism i between the strong monoidal functors IdC and (−)∗∗ . This means that i is a natural isomorphism satisfying the coherence conditions

φV,W ◦ (iV ⊗ iW ) = iV ⊗W ,

φ0 = i 1 .

(3.6.1)

Such a pair (C , i) is called pivotal category. Theorem 3.40 Let C be a left rigid monoidal category. Then C admits a pivotal structure if and only if it is sovereign. Proof Let C be a sovereign category (in particular, it is also right rigid). By Proposition 1.73, for all X ∈ C we have an isomorphism θX : X → (∗ X)∗ . Since C is sovereign we have (−)∗ = ∗ (−), so (∗ X)∗ = X ∗∗ . Thus we have θX : X → X ∗∗ . One

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Quasi-bialgebras and Quasi-Hopf Algebras

can easily see that θ = (θX )X∈Ob(C ) : IdC → (−)∗∗ is a natural isomorphism. It is, moreover, a natural monoidal transformation, and so provides a pivotal structure on C . To see this, recall that θX = (evX ⊗ IdX ∗∗ )(IdX ⊗ coevX ∗ ), for all X ∈ C , and thus V W

θV ⊗W =

V W

 (V ⊗W )∗

λV,W

=

V ⊗W

 (V ⊗W )∗ , ∀ V, W ∈ C ,

• •

• (V ⊗W )∗∗

(V ⊗W )∗∗

because λ = λ  since C is sovereign. On the other hand, V ∗∗ W ∗∗

φV,W

  V ∗∗ W ∗∗ (V ⊗W )∗ W ∗ ⊗V ∗  (V ⊗W )∗ λV,W

λV,W =  = ,



W ∗ ⊗V ∗

(3.6.2)

(V ⊗W )∗∗

 (V ⊗W )∗∗

and therefore V W

  (V ⊗W )∗  λ V,W

• φV,W (θV ⊗ θW ) = = •



V W

 (V ⊗W )∗ .

V ⊗W

• (V ⊗W )∗∗

(V ⊗W )∗∗

So θ is a pivotal structure on C . Conversely, let i be a pivotal structure on C , and for all V ∈ C define iV ⊗Id ∗

ev ∗

V 1, evV : V ⊗V ∗ −→V V ∗∗ ⊗V ∗ −→

IdV ∗ ⊗i−1

coev ∗

V coevV : 1 −→ V ∗ ⊗V ∗∗ −→V V ∗ ⊗V.

It is immediate that (V ∗ , evV , coevV ) is a right dual for V in C , and so C is right rigid, too. With respect to this right duality we have ∗ (V

 λV,W =

•  • X⊗Y

⊗W ) (3.6.1)

 

=

φV ⊗W



• ∗W ∗V

(V ⊗W )∗ (3.6.2)

= λV,W ,

(1.6.6)

(V ⊗W )∗ 

W∗ V∗

for all V,W ∈ C . So we have shown that the left and right duality functors coincide as monoidal functors, and therefore C is sovereign. This ends the proof.

3.6 Sovereign Quasi-Hopf Algebras

133

We can give now a description of all pivotal/sovereign structures on H M fd . Proposition 3.41 Let H be quasi-Hopf algebra with bijective antipode over a field k. Then we have a bijective correspondence between (i) pivotal structures i on C = H M fd ; (ii) sovereign structures on C = H M fd ; (iii) invertible elements gi ∈ H satisfying S2 (h) = g−1 i hgi , ∀ h ∈ H,

(3.6.3)

−1 )f, Δ(gi ) = (gi ⊗ gi )(S ⊗ S)( f21

(3.6.4)

where f = f 1 ⊗ f 2 is the Drinfeld twist defined in (3.2.15) and f21 = f 2 ⊗ f 1 . Proof The bijection between (i) and (ii) is established by Theorem 3.40. Let V ∈ H M fd , with H-action denoted by H ⊗ V h ⊗ v → hv ∈ V . We have a k-linear isomorphism V → V ∗∗ , and V ∗∗ can be regarded as V with newly defined left H-action h · v = S2 (h)v. Let us check that there is a bijective correspondence between natural isomorphisms between the functors IdC and (−)∗∗ and invertible elements gi ∈ H satisfying (3.6.3). Indeed, let i : IdC → (−)∗∗ be a natural isomorphism, and let gi = −1 2 S2 (i−1 H (1H )), ki = iH (1H ). For all h ∈ H, we have iH (h) = S (h)ki and iH (h) = −1 −2 −2 −2 S (hgi ). In particular, 1H = iH (ki ) = S (ki gi ) and 1H = iH (S (gi )) = gi ki , hence fd ki = g−1 i . Now take V ∈ H M , fix v ∈ V and define f : H h → hv ∈ V , a morfd phism in H M . From the naturality of i, we deduce that iV (v) = (iV ◦ f )(1H ) = ( f ◦ iH )(1H ) = g−1 i v. This means that i is completely determined by gi : iV (v) = g−1 i v.

(3.6.5)

−1 Now take V = H and v = h. Equation (3.6.5) tells us that S2 (h)g−1 i = iH (h) = gi h, and it follows that (3.6.3) is satisfied. For H a quasi-Hopf algebra and C = H M fd the isomorphisms λV,W were computed in Proposition 3.35, namely

λV,W (ψ ) = ∑ ψ (g1 vi ⊗ g2 w j )w j ⊗ vi , i, j

for all ψ ∈ (V ⊗ W )∗ , where {vi }i and {vi }i are dual bases of V and V ∗ , {w j } j and {w j } j are dual bases of W and W ∗ , and f −1 = g1 ⊗ g2 is the inverse of the Drinfeld twist. Also by Proposition 3.35 we know that −1 λV,W (w∗ ⊗ v∗ )(v ⊗ w) = v∗ , f 1 vw∗ , f 2 w,

for all v ∈ V , w ∈ W , v∗ ∈ V ∗ and w∗ ∈ W ∗ , where f = f 1 ⊗ f 2 is the Drinfeld twist defined in (3.2.15). It is now easy to establish that the first condition from (3.6.1) is equivalent to the following equivalent conditions:

φV,W (iV (v) ⊗ iW (w)) = iV ⊗W (v ⊗ w)

134

Quasi-bialgebras and Quasi-Hopf Algebras ⇔ λW−1∗ ,V ∗ (iV (v) ⊗ iW (w)) ◦ λV,W = iV ⊗W (v ⊗ w) −1 (w∗ ⊗ v∗ )) ⇔ λW−1∗ ,V ∗ (iV (v) ⊗ iW (w))(w∗ ⊗ v∗ ) = iV ⊗W (v ⊗ w)(λV,W −1 (w∗ ⊗ v∗ )(g−1 ⇔ iW (w)( f 1 · w∗ )iV (v)( f 2 · v∗ ) = λV,W i (v ⊗ w)) ∗ 1 −1 ∗ 1 −1 ∗ 2 −1 ⇔ v∗ (S( f 2 )g−1 i v)w (S( f )gi w) = v ( f (gi )1 v)w ( f (gi )2 w),

for all V,W ∈ C , v ∈ V , w ∈ W , v∗ ∈ V ∗ and w∗ ∈ W ∗ . Since H is an object of C , this last condition is equivalent to −1 −1 (S ⊗ S)( f21 )(g−1 Δ(g−1 i )= f i ⊗ gi ),

which is equivalent to (3.6.4). Now take x ∈ k. It follows from (3.6.5) that ik (x) = ∗∗ is the identity, we see that the second condition from ε (g−1 i )x. Since φ0 : k → k (3.6.1) is equivalent to ε (gi ) = 1. If (3.6.4) is satisfied, then ε (gi )2 = ε (gi ) (apply ε ⊗ ε to (3.6.4)), and it follows that ε (gi ) = 1. This completes our proof. Definition 3.42 A sovereign quasi-Hopf algebra is a quasi-Hopf algebra H for which there exists an invertible element g ∈ H satisfying (3.6.3) and (3.6.4). Remark 3.43 If H is a sovereign quasi-Hopf algebra via an invertible element g of H obeying (3.6.3) and (3.6.4), then H M fd is sovereign with the following rigid structure. If V ∈ H M fd , with H-action denoted by H ⊗ V h ⊗ v → hv ∈ V , then the left dual of V is V ∗ with H-module structure and evV and coevV as in Proposition 3.33. The right dual of V is again V ∗ considered as a left H-module via the antipode S of H, and with the evaluation and coevaluation morphisms given by evV : V ⊗V ∗ v ⊗ v∗ → v∗ (g−1 S−1 (α )v) = v∗ (S(α )g−1 v) ∈ k, coevV : k 1k → vi ⊗ S−1 (β )gvi = vi ⊗ gS(β )vi ∈ V ∗ ⊗V, summation implicitly understood, where {vi , vi }i are dual bases in V and V ∗ . Example 3.44 The two-dimensional quasi-Hopf algebra H(2) is sovereign via g = g, the generator of C2 . Proof From the structure of H(2) in Example 3.26 one can see that the Drinfeld twist of it is f = g ⊗ p− + 1 ⊗ p+ . Since f 2 = 1, it follows that f −1 = f . But f21 = p− ⊗ g + p+ ⊗ 1 1 = (1 ⊗ g − g ⊗ g + 1 ⊗ 1 + g ⊗ 1) 2 = g ⊗ p− + 1 ⊗ p+ = f , so f −1 (S ⊗ S)( f21 ) = 1, since S is the identity of H(2). Now everything follows from the commutativity of H(2) and the fact that g−1 = g is a grouplike element.

3.7 Dual Quasi-Hopf Algebras

135

3.7 Dual Quasi-Hopf Algebras The notion of dual quasi-bialgebra can be regarded as the formal dual of the notion of quasi-bialgebra. So most of the results below will be presented without proof. In what follows, by a k-coalgebra we mean a coassociative counital coalgebra C with counit ε : C → k and comultiplication Δ : C c → c1 ⊗ c2 ∈ C ⊗C. From a categorical point of view, dual quasi-bialgebras can be introduced as follows. Consider H a k-coalgebra and denote by M H the category of right H-comodules and right H-colinear maps in k M , the monoidal category of k-vector spaces. Definition 3.45 A dual quasi-bialgebra over a field k is a k-coalgebra H for which the monoidal structure of k M induces a monoidal structure on M H , that is: (a) for any two right H-comodules X,Y the tensor product X ⊗Y in k M admits a right H-comodule structure; (b) the tensor product in k M of two right H-comodule morphisms is a morphism in M H , and so ⊗ induces a functor from M H × M H to M H ; (c) k, the unit object of k M , admits a right H-comodule structure; (d) there exist functorial isomorphisms a = (aX,Y,Z : (X ⊗Y ) ⊗ Z → X ⊗ (Y ⊗ Z))X,Y,Z∈M H , l = (lX : k ⊗ X → X)X∈M H and r = (rX : X ⊗ k → X)X∈M H in M H such that the Pentagon Axiom and the Triangle Axiom are satisfied. After a reduction dual to the one in Corollary 3.11 we arrive at the following explicit characterization of a dual quasi-bialgebra. Proposition 3.46 A dual quasi-bialgebra is a 4-tuple (H, m, η , ϕ ) consisting of a k-coalgebra (H, Δ, ε ), two coalgebra morphisms m : H ⊗ H → H (called multiplication) and η : k → H (called unit), with notation η (1k ) = 1, and an invertible element ϕ : H ⊗ H ⊗ H → k in the convolution algebra Homk (H ⊗3 , k), called reassociator, such that the following conditions are satisfied: h1 (g1 l1 )ϕ (h2 , g2 , l2 ) = ϕ (h1 , g1 , l1 )(h2 g2 )l2 ,

(3.7.1)

1h = h1 = h,

(3.7.2)

ϕ (h1 , g1 , l1 m1 )ϕ (h2 g2 , l2 , m2 ) = ϕ (g1 , l1 , m1 )ϕ (h1 , g2 l2 , m2 )ϕ (h2 , g3 , l3 ), (3.7.3) ϕ (h, 1, g) = ε (h)ε (g),

(3.7.4)

for all h, g, l, m ∈ H, where we denoted m(h ⊗ g) by hg. It is easy to see that (3.7.3) and (3.7.4) also imply the identities:

ϕ (1, h, g) = ϕ (h, g, 1) = ε (h)ε (g), ∀ h, g ∈ H.

(3.7.5)

Let H be a dual quasi-bialgebra. For M ∈ M H we denote by ρM : M m → m(0) ⊗ m(1) ∈ M ⊗ H the right H-comodule structure of M. The category M H is monoidal.

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Quasi-bialgebras and Quasi-Hopf Algebras

The tensor product is given via m, that is, for any M, N ∈ M H , M ⊗ N ∈ M H via the structure map

ρM⊗N (m ⊗ n) = m(0) ⊗ n(0) ⊗ m(1) n(1) .

(3.7.6)

The associativity constraint aX,Y,Z : (X ⊗Y ) ⊗ Z → X ⊗ (Y ⊗ Z) of M H is aX,Y,Z ((x ⊗ y) ⊗ z) = ϕ (x(1) , y(1) , z(1) )x(0) ⊗ (y(0) ⊗ z(0) ),

(3.7.7)

for all X,Y, Z ∈ M H , x ∈ X, y ∈ Y and z ∈ Z. The unit is k as a trivial right Hcomodule, and the left and right unit constraints are the usual ones. If H is a dual quasi-bialgebra then a twist or a gauge transformation on H is an invertible element τ in the convolution algebra Homk (H ⊗2 , k), satisfying τ (1, h) = τ (h, 1) = ε (h), for all h ∈ H. The notion of dual quasi-bialgebra is invariant under a twist as follows. Proposition 3.47 Let H be a dual quasi-bialgebra and τ a twist on H with convolution inverse τ −1 . Then H with the same comultiplication, counit and unit as those of H, and with the new product and new reassociator given by h ·τ g = τ (h1 , g1 )h2 g2 τ −1 (h3 , g3 ),

ϕτ (h, g, l) = τ (g1 , l1 )τ (h1 , g2 l2 )ϕ (h2 , g3 , l3 )τ −1 (h3 g4 , l4 )τ −1 (h4 , g5 ), for all h, g, l ∈ H, is a dual quasi-bialgebra as well. It will be denoted by Hτ and called the twisting of H by τ . A dual quasi-bialgebra H is called a dual quasi-Hopf algebra if, in addition, there exist an anti-coalgebra morphism S : H → H and α , β ∈ H ∗ such that, for all h ∈ H: S(h1 )α (h2 )h3 = α (h)1, h1 β (h2 )S(h3 ) = β (h)1,

ϕ (h1 β (h2 ), S(h3 ), α (h4 )h5 ) = ϕ −1 (S(h1 ), α (h2 )h3 , β (h4 )S(h5 )) = ε (h). It follows from the axioms that S(1) = 1 and α (1)β (1) = 1, so we can assume that α (1) = β (1) = 1. Also, if τ is a twist on H then Hτ is a dual quasi-Hopf algebra with the same antipode as H and distinguished elements

ατ (h) = τ −1 (S(h1 ), h3 )α (h2 ) and βτ (h) = τ (h1 , S(h3 ))β (h2 ), ∀ h ∈ H. We now justify why we called such objects dual quasi-Hopf algebras. Let H be a finite-dimensional quasi-bialgebra and let {ei }i be a basis of H and {ei }i the corresponding dual basis of H ∗ , the linear dual of H. Since H is a k-algebra we get that H ∗ is a k-coalgebra with comultiplication ΔH ∗ (h∗ ) = ∑ h∗ (ei e j )ei ⊗ e j , ∀ h∗ ∈ H ∗ , i, j

and counit εH ∗ (h∗ ) = h∗ (1H ). H ∗ is also an H-bimodule, by h  h∗ , h  = h∗ (h h) and h∗  h, h  = h∗ (hh ), ∀ h∗ ∈ H ∗ , h ∈ H. Furthermore, the comultiplication of H defines a multiplication on H ∗ . Namely,

3.7 Dual Quasi-Hopf Algebras

137

h∗ g∗ (h) := h∗ (h1 )g∗ (h2 ), for all h∗ , g∗ ∈ H ∗ and h ∈ H. It is not associative, but since Δ is coassociative up to conjugation by Φ we obtain, for all h∗ , g∗ , l ∗ ∈ H ∗ , (h∗ g∗ )l ∗ = (X 1  h∗  x1 )[(X 2  g∗  x2 )(X 3  l ∗  x3 )]. Proposition 3.48 Let H be a finite-dimensional quasi-bialgebra with reassociator Φ. Then H ∗ with the dual algebra and coalgebra structures of H is a dual quasi-bialgebra with reassociator ϕ defined for all h∗ , g∗ , l ∗ ∈ H ∗ by ϕ (h∗ , g∗ , l ∗ ) = h∗ (X 1 )g∗ (X 2 )l ∗ (X 3 ). Furthermore, if H is a quasi-Hopf algebra with antipode S and distinguished elements α , β ∈ H then H ∗ is a dual quasi-Hopf algebra with antipode S∗ and distinguished elements α ∗ , β ∗ ∈ H ∗∗ , where α ∗ (h∗ ) = h∗ (α ) and β ∗ (h∗ ) = h∗ (β ), for all h∗ ∈ H ∗ . Proof

Let us start by noting that ΔH ∗ (h∗ ) = h∗1 ⊗ h∗2 ⇐⇒ h∗ (hg) = h∗1 (h)h∗2 (g), ∀ h, g ∈ H.

Thus h∗1 (g∗1 l1∗ )ϕ (h∗2 , g∗2 , l2∗ ) = ϕ (h∗1 , g∗1 , l1∗ )(h∗2 g∗2 )l2∗ ⇔ h∗1 (h1 )g∗1 (h(2,1) )l1∗ (h(2,2) )h∗2 (X 1 )g∗2 (X 2 )l2∗ (X 3 ) = h∗1 (X 1 )g∗1 (X 2 )l1∗ (X 3 )h∗1 (h(1,1) )g∗1 (h(1,2) )l1∗ (h2 ) ⇔ h∗ (h1 X 1 )g∗ (h(2,1) X 2 )l ∗ (h(2,2) X 3 ) = h∗ (X 1 h(1,1) )g∗ (X 2 h(1,2) )l ∗ (X 3 h2 ), for all h ∈ H, and the last equality is true because of (3.1.7). Δ is multiplicative and therefore ΔH ∗ is multiplicative, too. Since Δ respects the units we get that εH ∗ is multiplicative, and since ε is multiplicative it follows that ΔH ∗ respects the units. Finally, one can easily see that εH ∗ (1H ∗ ) = 1H ∗ (1H ) = ε (1H ) = 1k , thus H ∗ is a dual quasi-bialgebra. If H is a quasi-Hopf algebra then arguments similar to the ones above imply that H ∗ is a dual quasi-Hopf algebra. For instance, S∗ (h∗1 )α ∗ (h∗2 )h∗3 , h = h∗2 (α )h∗1 (S(h1 ))h∗3 (h2 ) = h∗ (S(h1 )α h2 ) = ε (h)h∗ (α ) = α ∗ (h∗ )1H ∗ , h, for all h∗ ∈ H ∗ , as required. The remaining details are left to the reader. Note that if H is a finite-dimensional dual quasi-bialgebra with reassociator ϕ then H ∗ is a quasi-Hopf algebra with the dual structure of H and reassociator Φ=

∑ ϕ (ei , e j , es )ei ⊗ e j ⊗ es ∈ H ∗ ⊗ H ∗ ⊗ H ∗ .

i, j,s

If, moreover, H is a dual quasi-Hopf algebra with antipode S and distinguished elements α , β ∈ H ∗ then H ∗ is a quasi-Hopf algebra with antipode S∗ and distinguished elements α ∗ , β ∗ ∈ H ∗ defined by α ∗ = α and β ∗ = β . If we iterate the procedure we then obtain that H ∼ = H ∗∗ as (dual) quasi-bialgebras

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or (dual) quasi-Hopf algebras. In both cases the isomorphism is produced by θH : H → H ∗∗ defined by θH (h)(h∗ ) = h∗ (h), for all h ∈ H and h∗ ∈ H ∗ . As the reader probably expects, the antipode of a dual-quasi Hopf algebra is an anti-algebra morphism in the following sense. Proposition 3.49 For a dual quasi-Hopf algebra H the antipode is an anti-algebra morphism, up to a conjugation by a twist. More precisely, define γ , δ ∈ (H ⊗ H)∗ by

γ (h, g) = ϕ (S(g2 ), S(h2 ), h4 )α (h3 )ϕ −1 (S(g1 )S(h1 ), h5 , g4 )α (g3 ), δ (h, g) = ϕ (h1 g1 , S(g5 ), S(h4 ))β (h3 )ϕ −1 (h2 , g2 , S(g4 ))β (g3 ), for all h, g ∈ H. If we define f , f −1 ∈ (H ⊗ H)∗ by f (h, g) = ϕ −1 (S(g1 )S(h1 ), h3 g3 , S(h5 g5 ))β (h4 g4 )γ (h2 , g2 ), f −1 (h, g) = ϕ −1 (S(h1 g1 ), h3 g3 , S(g5 )S(h5 ))α (h2 g2 )δ (h4 , g4 ), then f and f −1 are inverses in the convolution algebra Homk (H ⊗2 , k) and f (h1 , g1 )S(h2 g2 ) f −1 (h3 , g3 ) = S(g)S(h), for all h, g ∈ H. Moreover, the following relations hold:

γ (h, g) = f (h1 , g1 )α (h2 g2 )

and δ (h, g) = β (h1 g1 ) f −1 (h2 , g2 ).

As we have seen, examples of dual quasi-Hopf algebras can be obtained from finite-dimensional quasi-Hopf algebras by dualizing. Another class of examples can be constructed as follows. Notice that H(2) considered in Example 3.26 is the dual of kφ [C2 ] below, where C2 = g is the cyclic group of order 2 and φ is the unique non-trivial 3-cocycle on C2 determined by φ (gu , gv , gs ) = (−1)uvs , 0 ≤ u, v, s ≤ 1. Example 3.50 Let G be a group and H = k[G] the group algebra associated to G. With the coproduct and counit Δ(g) = g ⊗ g and ε (g) = 1, ∀ g ∈ G, extended by linearity, H has a Hopf algebra structure, its antipode S being defined by S(g) = g−1 , for all g ∈ G. However, since H is cocommutative (i.e. Δcop = Δ), for any normalized 3-cocycle φ on G, that is, for any map φ : G × G × G → k∗ such that (1.2.2) holds and φ (x, e, y) = 1, for all x, y ∈ G, the group Hopf algebra k[G] can be also viewed as a dual quasi-Hopf algebra with reassociator ϕ = φ , extended by linearity, antipode S as above and distinguished elements

α = ε and β (g) = φ (g, g−1 , g)−1 , ∀ g ∈ G. We denote this dual quasi-Hopf algebra structure on k[G] by kφ [G]. Proof

This is straightforward, so we leave it to the reader.

In the next result we show that a monoidal category of graded vector spaces is nothing else than a category of comodules over a certain dual quasi-Hopf algebra.

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139

Proposition 3.51 Let G be a group, φ ∈ H 3 (G, k∗ ) a normalized 3-cocycle and kφ [G] the dual quasi-Hopf algebra described in Example 3.50. Then the categories kφ [G] are monoidally isomorphic. VectG φ and M Proof We first identify VectG and M kφ [G] as usual categories. If X is a right kφ [G]-comodule with structure map ρ : X → X ⊗ kφ [G] then for any x ∈ X we write ρ (x) = ∑g∈G xg ⊗ g, for some elements xg ∈ X, g ∈ G, and this is possible because the elements of G define a basis for kφ [G]. From the coassociativity of ρ we obtain that

∑

(xg )h ⊗ h ⊗ g =

g,h∈G

∑ xg ⊗ g ⊗ g,

g∈G

and so we must have (xg )h = δg,h xg , for all h, g ∈ G. If for any g ∈ G we define Xg = {xg | x ∈ X}, then the latter equation shows that the sum ∑g∈G Xg is direct. But, on the other hand, since (IdX ⊗ ε )ρ = IdX we get that x = ∑g∈G xg , for all x ∈ X, and  therefore X = g∈G Xg . Thus X is a G-graded vector space. Conversely, if X is a G-graded vector space we define ρ X : X → X ⊗ kφ [G] by X ρ (x) = ∑g∈G xg ⊗ g, where x = ∑g∈G xg is the decomposition of x in homogeneous components. It can be easily seen that with this structure X becomes a right kφ [G]comodule, and the two correspondences described above provide inverse bijections. Now, recall that for the monoidal category of G-graded vector spaces VectG φ , the associativity constraint is defined by aX,Y,Z ((x ⊗ y) ⊗ z) = φ (|x|, |y|, |z|)x ⊗ (y ⊗ z), for all X,Y, Z ∈ VectG φ and homogeneous elements x ∈ X, y ∈ Y , z ∈ Z. If we regard X as a right kφ [G]-comodule then ρ X (x) = x ⊗ |x|, thus aX,Y,Z can be rewritten as aX,Y,Z ((x ⊗ y) ⊗ z) = φ (x(1) , y(1) , z(1) )x(0) ⊗ (y(0) ⊗ z(0) ). But this family of isomorphisms defines the associativity constraint of M kφ [G] , and kφ [G] identify as monoidal categories, as claimed. therefore VectG φ and M The category of corepresentations over a dual quasi-Hopf algebra H is left and/or right rigid, depending on the bijectivity of the antipode S of H. Proposition 3.52 Let H be a dual quasi-Hopf algebra with antipode S and distinguished elements α , β ∈ H ∗ , and denote by MfdH and H Mfd the category of right and left H-comodules, respectively, that are finite dimensional. Then MfdH is left rigid and H Mfd is right rigid. Furthermore, if S is bijective then MfdH and H Mfd are both rigid monoidal categories. Proof By duality reasons we sketch the proof by indicating only the left/right dual object associated to a finite-dimensional H-comodule.

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The left dual of V ∈ MfdH is V ∗ = Homk (V, k) with the right H-comodule structure

ρV ∗ (v∗ ) = v∗ , i v(0) i v ⊗ S(i v(1) ), for all v∗ ∈ V ∗ , where {i v}i is a basis in V with dual basis {i v}i in V ∗ , and ρV (v) = v(0) ⊗ v(1) ∈ V ⊗ H is the right H-comodule structure of V . The evaluation and coevaluation maps are defined by evV : V ∗ ⊗V → k,

evV (v∗ ⊗ v) = α (v(1) )v∗ (v(0) ),

coevV : k → V ⊗V ∗ ,

coevV (1) = β (i v(1) )i v(0) ⊗ i v,

for all v∗ ∈ V ∗ and v ∈ V . If S is bijective with inverse S−1 then ∗V = Homk (V, k) is a right H-comodule via the structure

ρ∗V (∗ v) = ∗ v, i v(0) i v ⊗ S−1 (i v(1) ), for all ∗ v ∈ ∗V , and together with the maps evV : V ⊗ ∗V → k,

evV (v ⊗ ∗ v) = α (S−1 (v(1) ))v∗ (v(0) ),

coevV : k → ∗V ⊗V,

coevV (1) = β (S−1 (i v(1) ))i v ⊗ i v(0)

define a right dual for V in MfdH . Consider now the category H Mfd . It is always right rigid since any object V of it admits as a right dual the object ∗V := Homk (V, k), considered a left H-comodule via the structure morphism

λ∗V : ∗V → H ⊗ ∗V,

λV (∗ v) = ∗ v, i v(0) S(i v(−1) ) ⊗ i v,

where λV (v) = v(−1) ⊗ v(0) is the left H-comodule structure of V . The evaluation and coevaluation maps are defined by evV : V ⊗ ∗V → k, coevV : k → ∗V ⊗V,

evV (v ⊗ ∗ v) = β (v(−1) )∗ v(v(0) ), coevV (1) = ∑ α (i v(−1) )i v ⊗ i v(0) . i

HM

H Assuming S is bijective we get that fd is rigid monoidal. For V ∈ Mfd we have ∗ that V = Homk (V, k) is a left H-comodule via the structure morphism given by

λV ∗ (v∗ ) = ∗ v, i v(0) S−1 (i v(−1) ) ⊗ i v, for all v∗ ∈ V ∗ , and a left dual of V in H Mfd if it is considered together with the maps evV : V ∗ ⊗V → k, coevV : k → V ⊗V ∗ ,

evV (v∗ ⊗ v) = β (S−1 (v(−1) ))v∗ (v(0) ), coevV (1) = ∑ α (S−1 (i v(−1) ))i v(0) ⊗ i v. i

So the proof is finished. Corollary 3.53 Let G be a group, k a field and φ ∈ H 3 (G, k∗ ) a normalized 3cocycle. Then vectG φ , the category of finite-dimensional vector spaces equipped with the monoidal structure of VectG φ , is rigid monoidal.

3.8 Further Examples of (Dual) Quasi-Hopf Algebras

141

Proof This follows immediately from Proposition 3.51 and Proposition 3.52. The explicit structure of the dual objects is exactly the one stated in Example 1.65, we leave the details to the reader. The dual version of Proposition 3.35 says that the isomorphism λX,Y : (X ⊗Y )∗ → in MfdH , where H is a dual quasi-Hopf algebra, is given by

Y ∗ ⊗ X∗

λX,Y (μ ) = f −1 (i x(1) , j y(1) )μ (i x(0) ⊗ j y(0) ) j y ⊗ i x, where f −1 is the inverse of the Drinfeld twist defined in Proposition 3.49, and {i x, i x}i and { j y, j y} j are dual bases in X and X ∗ , and in Y and Y ∗ , respectively. Its inverse is given by −1 ∗ λX,Y (y ⊗ x∗ )(x ⊗ y) = f (x(1) , y(1) )x∗ (x(0) )y∗ (y(0) ),

(3.7.8)

for all x∗ ∈ X ∗ , y∗ ∈ Y ∗ , x ∈ X and y ∈ Y , where f is the convolution inverse of f −1 . We leave it to the reader to compute the right-handed version λ  of λ , and λ  −1 .

3.8 Further Examples of (Dual) Quasi-Hopf Algebras By Example 3.50 we can construct examples of dual quasi-Hopf algebras starting with a pair (G, φ ) consisting of a group G and a normalized 3-cocycle φ on it. When G is finite, by duality, we also get examples of quasi-Hopf algebras; see the dual version of Proposition 3.48. The goal of this section is to provide concrete examples of (dual) quasi-Hopf algebras of the type mentioned above. In other words, we will present concrete examples of pairs (G, φ ) as above. Proposition 3.54 Let Cn = σ  be the cyclic group of order n and k a field that contains an nth root of unity q. Then φ (σ a , σ b , σ c ) := qabc is a normalized 3-cocycle on Cn . It is a coboundary if and only if q

n(n−1) 2

= 1.

Proof The fact that q is an nth root of unity in k implies that φ is well defined. Indeed, if a = a + n, b = b + n and c = c + n then abc = n3 + (a + b + c )n2 +       (a b + a c + b c )n + a b c , and so φ (σ a , σ b , σ c ) = qabc = qa b c = φ (σ a , σ b , σ c ). Now, the 3-cocycle condition follows because of the equality bcd + a(b + c)d + abc = ab(c + d) + (a + b)cd in Z. It is immediate that φ is normalized. Suppose now that φ is a coboundary; take g ∈ K 2 (Cn , k∗ ) such that g(σ b , σ c )g(σ a+b , σ c )−1 g(σ a , σ b+c )g(σ a , σ b )−1 = qabc , ∀ a, b, c ∈ Z,

(3.8.1)

and denote β := g(σ a , 1) = g(1, σ b ) and αc := g(σ , σ c ), for a, b, c ∈ {1, . . . , n − 1}. By taking a = 1, b = k and c = n − 1 in (3.8.1) we obtain g(σ k+1 , σ n−1 ) = qk g(σ k , σ n−1 )g(σ , σ k−1 )g(σ , σ k )−1 , ∀ k ∈ Z.

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By mathematical induction it follows that g(σ k , σ n−1 ) = q 2 ≤ k ≤ n − 1. We then have

k(k−1) 2

−1 αn−1 β αk−1 , for all

β = g(1, σ n−1 ) = g(σ (n−1)+1 , σ n−1 ) = qn−1 g(σ n−1 , σ n−1 )g(σ , σ n−2 )g(σ , σ n−1 )−1 = qn−1 q

(n−1)(n−2) 2

−1 −1 αn−1 β αn−2 αn−2 αn−1 =q

from which we conclude that q

n(n−1) 2

n(n−1) 2

β,

= 1, as stated.

n(n−1) 2

= 1 we consider the functions f : Z × Z → Z given by Conversely, if q , for all x, y ∈ Z, and g : Cn ×Cn → k defined by g(σ a , σ b ) = q f (a,b) , f (x, y) = − (x−1)xy 2 for all a, b ∈ {0, 1, . . . , n − 1}. If a = a + n and b = b + n then     n2 (n − 1) n(n − 1)  a (a − 1) + a b n − − b. f (a, b) − f (a , b ) = −a n2 − 2 2 2 n(n−1)

This relation together with qn = 1 and q 2 = 1 implies that g is well defined. Finally, a straightforward computation ensures that f (y, z) − f (x + y, z) + f (x, y + z) − f (x, y) = xyz, ∀ x, y, z ∈ Z, and this shows that Δ2 (g) = φ , that is, that φ is a coboundary. We move now to the situation when G is not a cyclic group. Proposition 3.55 If G = Z2 × Z2 × Z2 then φ (x,y,z) := (−1)(x×y)·z , for all x,y,z ∈ G, is a normalized coboundary 3-cocycle on G. Here (x ×y) ·z is the mixed double product of x,y,y ∈ G, where we used a vector notation for the elements of G. Proof Let A a be a square 3 × 3-matrix whose rows are given by the components of three vectors in G. Then the determinant of A coincides with the mixed double product of those three vectors. Hence, (x,y,z) := (x ×y) ·z is Z-linear in each argument. From here we can easily see that (y,z,w) + (x,y +z,w) + (x,y,z) = (x,y,z + w) + (x +y,z,w), for all x,y,z,w ∈ G, and this proves that φ is a 3-cocycle on G. Since the mixed double product is zero when a vector is zero it follows that φ is normalized. To show that φ is a coboundary, define f : G × G → Z by f (x,y) =

∑

xi y j + y1 x2 x3 + x1 y2 x3 + x1 x2 y3 , ∀ x,y ∈ G,

1≤i≤ j≤3

where we also made use of the multiplication of Z2 . Taking into account that −x = x in Z2 , by a direct computation one can easily see that f (y,z) − f (x +y,z) + f (x,y +z) − f (x,y) = −z1 (x2 y3 + y2 x3 ) − z2 (x1 y3 + y1 x3 ) − z3 (x1 y2 + y1 x2 ) = z1 (x2 y3 − y2 x3 ) − z2 (x1 y3 − y1 x3 ) + z3 (x1 y2 − y1 x2 ) = (x ×y) ·z. Thus g ∈ K 2 (G, k∗ ), given by g(x,y) = (−1) f (x,y) , for allx,y ∈ G, satisfies Δ2 (g) = φ , and from here we conclude that φ is a coboundary.

3.8 Further Examples of (Dual) Quasi-Hopf Algebras

143

Remark 3.56 If G is a direct product of a finite number of copies of Zn and q is an nth root of unity in k then any Z-trilinear map (, , ) : G × G × G → Z defines a normalized 3-cocycle on G, namely φ (x,y,z) := q(x,y,z) , for all x,y,z ∈ G, where we used the same vector notation for the elements of G as above. As we mentioned at the beginning of this section, classes of quasi-Hopf algebras can be obtained from groups and normalized 3-cocycles φ on them, by considering the dual structure of the dual quasi-Hopf algebra kφ [G]. A convenient situation occurs when G is a finite abelian group. Let k[G] be the group algebra of G over k. As we mentioned before, it has a Hopf algebra structure given by the comultiplication Δ(g) = g ⊗ g and counit ε (g) = 1, for all g ∈ G, extended by linearity and as algebra maps. The antipode S maps g ∈ G to its inverse g−1 . Obviously k[G] is a cocommutative k-coalgebra. Proposition 3.57 Let k be a field that contains a primitive nth root of unity. Then for any finite abelian group C of order n we have k[C] ∼ = k[C]∗ as Hopf algebras. Proof Consider first the situation when C = Cn , the cyclic group of order n. Since k contains a primitive nth root of unity, say ξ , we deduce that the characteristic of k does not divide n (this follows easily from n = (1 − ξ )(1 − ξ 2 ) · · · (1 − ξ n−1 )). Suppose C = c, written multiplicatively, and let {P1 , Pc , . . . , Pcn−1 } be the basis of k[Cn ]∗ dual to the basis {1, c, . . . , cn−1 } of k[Cn ]. Define f ∈ k[Cn ]∗ by f (ci ) = ξ i , for all 0 ≤ i ≤ n−1. Then f is a well-defined algebra map and f j (cs ) = ξ js , for all 0 ≤ s ≤ n − 1. We now claim that Ψ : k[Cn ] c j → f j ∈ k[Cn ]∗ , extended by linearity, is a Hopf algebra isomorphism. js s To see this we observe first that f j = ∑n−1 s=0 ξ Pc , for all 0 ≤ j ≤ n − 1, and then we compute Δk[Cn ]∗ (Ψ(c j )) = Δk[Cn ]∗ ( f j ) =

n−1

∑

s,t=0

=

n−1

n−1

s=0

t=0

f j (cs+t )Pcs ⊗ Pct =

n−1

∑

ξ (s+t) j Pcs ⊗ Pct

s,t=0

∑ ξ s j Pcs ⊗ ∑ ξ t j Pct = f j ⊗ f j = (Ψ ⊗ Ψ)Δk[Cn ] (c j )

and εk[Cn ]∗ (Ψ(c j )) = εk[Cn ]∗ ( f j ) = f j (1) = 1 = εk[Cn ] (c j ), for all 0 ≤ j ≤ n − 1. Thus, we have proved that Ψ is a coalgebra morphism. It can be easily checked that Ψ is an algebra morphism as well, and so it is a Hopf algebra morphism. To show that Ψ is an isomorphism we prove that {ε , f , . . . , f n−1 } is a basis of k[Cn ]∗ . Counting the dimensions it suffices to verify that {ε , f , . . . , f n−1 } is an indej pendent linear system in k[Cn ]∗ . Indeed, if α0 , . . . , αn−1 ∈ k are such that ∑n−1 j=0 α j f = js 0 then ∑n−1 j=0 α j ξ = 0, for all 0 ≤ s ≤ n − 1. In this way we get an n × n homogeneous linear system with indeterminates α0 , · · · , αn−1 whose discriminant is the Vandermonde discriminant of 1, ξ , . . . , ξ n−1 , usually denoted by V (1, ξ , . . . , ξ n−1 ).

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Since ξ is a primitive nth root of unity in k we deduce that V (1, ξ , . . . , ξ n−1 ) = 0, therefore α0 = · · · = αn−1 = 0, as needed. For further use note that the inverse of Ψ is given by Ψ−1 (Pc j ) =

1 n−1 (n−s) j s c , ∀ 0 ≤ j ≤ n − 1. ∑ξ n s=0

Indeed, for all 0 ≤ t ≤ n − 1, 1 n−1 (n−s) j s t f (c ) ∑ξ n s=0  1 1

1 + ξ t− j + (ξ t− j )2 + · · · + (ξ t− j )n−1 = = n n



n

if t = j

1−(ξ t− j )n 1−ξ t− j

if t = j

= δ j,t . So ΨΨ−1 (Pc j ) = Pc j , for all 0 ≤ j ≤ n − 1, thus Ψ and Ψ−1 are bijective inverses. Now take C an arbitrary finite abelian group of order n. If we write C as a direct product of finite cyclic groups then the Hopf algebra isomorphism k[C] ∼ = k[C]∗ follows from the first part of the proof and the fact that k[G × H] (g, h) → g ⊗ h ∈ k[G] ⊗ k[H], extended by linearity, is an isomorphism of Hopf algebras, for any groups G and H. Here k[G] ⊗ k[H] has the Hopf algebra structure given by the tensor product algebra and tensor product coalgebra structures. Actually, we have k[G × H]∗ ∼ = (k[G] ⊗ k[H])∗ ∼ = k[G]∗ ⊗ k[H]∗ ∼ = k[G] ⊗ k[H] ∼ = k[G × H] as Hopf algebras, and then the general case follows by mathematical induction. Note that the second Hopf algebra isomorphism used above is given by (k[G] ⊗ k[H])∗ μ →

∑

μ (g ⊗ h)Pg ⊗ Ph ∈ k[G]∗ ⊗ k[H]∗

g∈G,h∈H



and its inverse is Pg ⊗ Ph → g ⊗ h → δg,g δh,h . When G is abelian, k[G] is a commutative k-algebra. Thus in the abelian case a quasi-Hopf algebra structure on k[G] is completely determined by a reassociator, that is, an invertible element Φ ∈ k[G] ⊗ k[G] ⊗ k[G] satisfying (3.1.9) and (3.1.10). If G is, moreover, finite, as we have seen before, finding all the associators is equivalent to finding all the elements of H 3 (G, k∗ ). Next we point out how we can compute such elements Φ in some particular cases. Proposition 3.58 Let Cn = c be the cyclic group of order n written multiplicatively and k a field that contains a primitive nth root of unity, say ξ . Then for any k ∈ {0, 1, . . . , n − 1} the elements Φk ∈ k[Cn ] ⊗ k[Cn ] ⊗ k[Cn ] of the form Φk = 1 ⊗ 1 ⊗ 1 −

n−1 1 (1 − ck ) ⊗ ∑ (1 − nδi, j )(ξ j − nδ j,0 )ci ⊗ c j , 2 n i, j=0

are normalized 3-cocycles on k[G]. In particular, by considering n = 2 we get that

3.8 Further Examples of (Dual) Quasi-Hopf Algebras

145

k[C2 ] is a quasi-Hopf algebra with the non-trivial reassociator Φ1 = 1 ⊗ 1 ⊗ 1 − 2p− ⊗ p− ⊗ p− , where p− = 12 (1 − c). Proof Let Ψ be the isomorphism from k[Cn ] to k[Cn ]∗ defined in the proof of Proposition 3.57. Then we have   n−1 1 n−1 n−1 k j n−s j s j −1 ∑ q Ψ (Pc j ) = n ∑ ∑ ξ (ξ ) c j=0 s=0 j=0   n−1 n−1 1 1 n−1 = ∑ ∑ (ξ k−s ) j cs = ∑ nδk,s cs = ck . n s=0 j=0 n j=0 Thus, according to the above comments and the definition of φq given in Example 1.16, we have that all the elements Φ ∈ k[Cn ] ⊗ k[Cn ] ⊗ k[Cn ] of the form Φk : =

n−1

∑

ϕq (cu , cv , cs )Ψ−1 (Pcu ) ⊗ Ψ−1 (Pcv ) ⊗ Ψ−1 (Pcs )

u,v,s=0 n−1

=

∑ Ψ−1 (Pcu ) ⊗ ∑

u=0 n−1

+ ∑ qu Ψ−1 (Pcu ) ⊗ u=1

= 1⊗

∑

Ψ−1 (Pcv ) ⊗ Ψ−1 (Pcs )

v+s b)(a > b ) = aa0 x˜1ρ > (b · a1 x˜2ρ )(b · x˜3ρ ),

(5.2.2)

for all a, a ∈ A and b, b ∈ B. Then A > B is an associative algebra with unit 1A > 1B , called the generalized smash product of A and B. Proof If we regard B as an H-bimodule algebra with trivial left H-action, then obviously A > B coincides with A # B, which is a left H-module algebra with trivial left H-action, namely an associative algebra. The above constructions have left-handed versions (proofs are left to the reader). Proposition 5.17 Let H be a quasi-bialgebra, B a left H-comodule algebra and A an H-bimodule algebra. Define a multiplication on A ⊗ B by (ϕ # b)(ϕ  # b ) = (x˜1λ · ϕ )(x˜2λ b[−1] · ϕ  ) # x˜3λ b[0] b , ∀ ϕ , ϕ  ∈ A , b, b ∈ B, (5.2.3) where we write ϕ # b for ϕ ⊗ b, and denote this structure by A # B. Then A # B becomes a right H-module algebra with unit 1A # 1B and with right H-action (ϕ # b) · h = ϕ · h # b, ∀ ϕ ∈ A , h ∈ H, b ∈ B. We call A # B the quasi-smash product of A and B. Corollary 5.18 Let H be a quasi-bialgebra, A a left H-module algebra and B a left H-comodule algebra. Denote by A < B the k-vector space A ⊗ B with multiplication (a < b)(a < b ) = (x˜λ1 · a)(x˜λ2 b[−1] · a ) < x˜λ3 b[0] b ,

(5.2.4)

for all a, a ∈ A and b, b ∈ B. Then A < B is an associative algebra with unit 1A < 1B , called the generalized smash product of A and B. Remark 5.19 If H is a quasi-bialgebra, A a left H-module algebra and B a right H-module algebra, then A < H = A#H and H > B = H#B. The proof of the next result is similar to the proof of Proposition 5.9 and is left to the reader. Proposition 5.20 Let H be a quasi-bialgebra, A a left H-module algebra, B a right H-module algebra and A an H-bicomodule algebra. Then A < A becomes a right H-comodule algebra, with structure defined for all a ∈ A and u ∈ A by

ρ : A < A → (A < A) ⊗ H,

ρ (a < u) = (θ 1 · a < θ 2 u0 ) ⊗ θ 3 u1 ,

1 2 3 Φρ = (1A < X˜ ρ ) ⊗ X˜ ρ ⊗ X˜ ρ ∈ (A < A) ⊗ H ⊗ H,

5.2 Quasi-smash Products and Generalized Smash Products

187

and A > B becomes a left H-comodule algebra, with structure defined for all u ∈ A, b ∈ B by

λ : A > B → H ⊗ (A > B),

λ (u > b) = u[−1] θ 1 ⊗ (u[0] θ 2 > b · θ 3 ),

1 2 3 Φλ = X˜ λ ⊗ X˜ λ ⊗ (X˜ λ > 1B ) ∈ H ⊗ H ⊗ (A > B).

Proposition 5.21 Let H be a finite-dimensional quasi-Hopf algebra with bijective antipode. Define the linear map

μ : H # H ∗ → Endk (H), μ (h # ϕ )(h ) = ϕ (h2 p˜2 )hh1 p˜1 , for all h, h ∈ H and ϕ ∈ H ∗ , where pL = p˜1 ⊗ p˜2 is the element defined by (3.2.20). Then μ is a bijection, and therefore there exists a unique left H-module algebra structure on Endk (H) such that μ becomes an H-module algebra isomorphism. The multiplication, unit and left H-module structure of Endk (H) are given by (u◦v)(h) = u(v(hx3 X23 )S−1 (S(x1 X 2 )α x2 X13 ))S−1 (X 1 ), 1Endk (H) (h) =

hS−1 (β ),





(h · u)(h ) = u(h h2

)S−1 (h

1 ),

(5.2.5) (5.2.6)

for all u, v ∈ Endk (H) and h, h ∈ H. We denote by END(H) this left H-module algebra structure of Endk (H). Proof Let {ei }i=1,n be a basis of H and {ei }i=1,n the corresponding dual basis of H ∗ . We claim that the inverse of μ is the map μ −1 : Endk (H) → H # H ∗ given by n

μ −1 (u) = ∑ u(q˜2 (ei )2 )S−1 (q˜1 (ei )1 ) # ei , ∀ u ∈ Endk (H), i=1

where qL = q˜1 ⊗ q˜2 is defined by (3.2.20). Indeed, for h ∈ H, ϕ ∈ H ∗ we have: (μ −1 ◦ μ )(h # ϕ )

n

=

∑ μ (h # ϕ )(q˜2 (ei )2 )S−1 (q˜1 (ei )1 ) # ei

i=1 n

= (3.2.22)

=

(3.2.24)

=

∑ ϕ (q˜22 (ei )(2,2) p˜2 )hq˜21 (ei )(2,1) p˜1 S−1 (q˜1 (ei )1 ) # ei

i=1 n

∑ ϕ (q˜22 p˜2 ei )hq˜21 p˜1 S−1 (q˜1 ) # ei

i=1 n

∑ ϕ (ei )h # ei = h # ϕ .

i=1

Similarly, for u ∈ Endk (H) and h ∈ H we have (μ ◦ μ −1 )(u)(h) = u(h). By using the bijection μ , we transfer the H-module algebra structure from H # H ∗ to Endk (H). First we compute the transferred multiplication ◦: for all u, v ∈ Endk (H), we find u◦v = μ (μ −1 (u)μ −1 (v)) n

=

∑

μ ((u(q˜2 (ei )2 )S−1 (q˜1 (ei )1 ) # ei )(v(Q˜ 2 (e j )2 )S−1 (Q˜ 1 (e j )1 ) # e j ))

∑

 μ u(q˜2 (ei )2 )S−1 (q˜1 (ei )1 )[v(Q˜ 2 (e j )2 )S−1 (Q˜ 1 (e j )1 )]1 x1

i, j=1 n

=

i, j=1

188

Crossed Products

 # (ei  [v(Q˜ 2 (e j )2 )S−1 (Q˜ 1 (e j )1 )]2 x2 )(e j  x3 ) ,

where Q˜ 1 ⊗ Q˜ 2 is another copy of qL and  is the right H-action on H ∗ defined by (4.1.16). Note that (3.1.9) and (3.2.20) imply S(x1 )q˜1 x12 ⊗ q˜2 x22 ⊗ x3 = q˜1 X 1 ⊗ q˜21 X 2 ⊗ q˜22 X 3 .

(5.2.7)

By using the above arguments, a long but straightforward computation shows that (u◦v)(h) = u(v(hx3 X23 )S−1 (S(x1 X 2 )α x2 X13 ))S−1 (X 1 ), for all h ∈ H. Thus, we have obtained (5.2.5). Similar computations show that the transferred unit and the H-action on Endk (H) are given by (5.2.6).

5.3 Endomorphism H-module Algebras If H is a quasi-Hopf algebra with bijective antipode, M is a left H-module and A is a left H-module algebra, we show that setting a left A, H-module structure on M is equivalent to giving a morphism of left H-module algebras from A to a certain deformation of End(M). We begin with a lemma of independent interest. Lemma 5.22 Let H be a quasi-Hopf algebra, B,C associative algebras, η : B → C, j : H → B, v : H → C algebra maps such that η ◦ j = v. Then the map η : B j → Cv is a morphism of left H-module algebras (notation as in Proposition 4.3). Proof

This follows by a direct computation, using the formula (4.1.7).

Let H be a quasi-Hopf algebra and M a left H-module, with action denoted by h ⊗ m → h · m. Consider the (usual) associative algebra End(M) of k-linear endomorphisms of M (with composition) and define v : H → End(M), v(h)(m) = h · m, which is an algebra map, so we can consider the left H-module algebra End(M)v , whose multiplication, unit and H-action are given by (u ◦ u )(m) = X 1 · u(S(x1 X 2 )α x2 X13 · u (S(x3 X23 ) · m)),

(5.3.1)

1End(M)v (m) = v(β )(m) = β · m,

(5.3.2)

(h v u)(m) = h1 · u(S(h2 ) · m),

(5.3.3)

for all h ∈ H, u, u ∈ End(M)v , m ∈ M. Suppose we have also a left H-module algebra A and the antipode of H is bijective. Theorem 5.23 Setting a structure of a left A, H-module on M is equivalent to giving a morphism of left H-module algebras ϕ : A → End(M)v . The correspondence is given as follows: if M is a left A, H-module (with A-action denoted by a ⊗ m → a  m) then the map ϕ : A → End(M)v is given by

ϕ (a)(m) = (p1 · a)  (p2 · m), ∀ a ∈ A, m ∈ M,

(5.3.4)

5.3 Endomorphism H-module Algebras

189

where pR = p1 ⊗ p2 = x1 ⊗ x2 β S(x3 ). Conversely, if ϕ : A → End(M)v is a morphism of left H-module algebras, then M becomes a left A, H-module, with A-action a  m = q1 · ϕ (a)(S(q2 ) · m), ∀ a ∈ A, m ∈ M,

(5.3.5)

where qR = q1 ⊗ q2 = X 1 ⊗ S−1 (α X 3 )X 2 , and the H-action is the original H-module structure of M. Proof Suppose first that M is a left A, H-module, with A-action a ⊗ m → a  m. By Proposition 5.7, this is equivalent to M being a left A#H-module, with structure (a#h) · m = a  (h · m), ∀ a ∈ A, h ∈ H, m ∈ M. So, by considering the usual associative algebra End(M), we obtain an algebra map η : A#H → End(M), η (a#h)(m) = (a#h) · m. We also have the canonical algebra map j : H → A#H, j(h) = 1A #h; since we obviously have that η ◦ j = v, we can apply Lemma 5.22 and obtain that the map η : (A#H) j → End(M)v is a morphism of left H-module algebras. By Lemma 5.11, the map i0 : A → (A#H) j , i0 (a) = p1 · a#p2 , where pR = p1 ⊗ p2 = x1 ⊗ x2 β S(x3 ), is a morphism of left H-module algebras, so the composition ϕ = η ◦ i0 : A → End(M)v is also a morphism of left H-module algebras, and one can easily check that it is given by ϕ (a)(m) = (p1 · a)  (p2 · m), for all a ∈ A, m ∈ M. Conversely, let ϕ : A → End(M)v be a morphism of left H-module algebras; by applying the Universal Property of the smash product A#H (Theorem 5.12) for B = End(M), we obtain the algebra map ϕ #v : A#H → End(M), which (by using the formula (5.1.7)) can be expressed as follows: (ϕ #v)(a#h)(m) = q1 · ϕ (a)(S(q2 )h · m), ∀ a ∈ A, h ∈ H, m ∈ M. Hence, M becomes a left A#H-module (i.e. a left A, H-module) with action (a#h) · m = q1 · ϕ (a)(S(q2 )h · m), ∀ a ∈ A, h ∈ H, m ∈ M. In particular, the A-action is given by a  m = (a#1H ) · m = q1 · ϕ (a)(S(q2 ) · m), ∀ a ∈ A, m ∈ M, and, by using the fact that q1 β S(q2 ) = 1H (which follows from (3.2.2)), we obtain that the H-action is given by (1A #h) · m = q1 β S(q2 )h · m = h · m, ∀ h ∈ H, m ∈ M. The only thing left to prove is that the two correspondences are inverse to each other. If M is an A, H-module with A-action denoted by , ϕ is the associated map ϕ : A → End(M)v and  is the A-action associated to ϕ , we have a  m = q1 · ((p1 · a)  (p2 S(q2 ) · m)) (5.1.4)

(3.2.23)

= (q11 p1 · a)  (q12 p2 S(q2 ) · m) = a  m,

for all a ∈ A and m ∈ M. Conversely, if ϕ : A → End(M)v is a left H-module algebra

190

Crossed Products

map,  is the A-action obtained from ϕ and ϕ  is the map obtained from this A, Hmodule structure on M, we have (for all a ∈ A and m ∈ M):

ϕ  (a)(m) = (p1 · a)  (p2 · m) = q1 · ϕ (p1 · a)(S(q2 )p2 · m) = q1 · ((p1 v ϕ (a))(S(q2 )p2 · m)) (5.3.3) 1

= q · (p11 · ϕ (a)(S(p12 )S(q2 )p2 · m)) (3.2.23)

= q1 p11 · ϕ (a)(S(S−1 (p2 )q2 p12 ) · m) = ϕ (a)(m), finishing the proof. By taking A = End(M)v and ϕ = Id in Theorem 5.23, we obtain: Corollary 5.24 If H is a quasi-Hopf algebra with bijective antipode and M is a left H-module, then M becomes a left End(M)v , H-module (i.e. a left End(M)v #Hmodule), with End(M)v -action given, for all u ∈ End(M)v and m ∈ M, by u  m = q1 · u(S(q2 ) · m).

(5.3.6)

We now study the behavior of the construction End(M)v under twisting. Let H be a quasi-Hopf algebra, F ∈ H ⊗ H a gauge transformation and M a left H-module. Then M is also a left HF -module, with the same H-action. Denote by v : H → End(M) and vF : HF → End(M) the corresponding algebra maps, and consider the H-module algebra End(M)v and the HF -module algebra End(M)vF ; we also consider the HF module algebra End(M)vF −1 . We will prove that End(M)vF and End(M)vF −1 are isomorphic as left HF -module algebras. Actually, we will prove something more general. Let H be a quasi-Hopf algebra, F ∈ H ⊗ H a gauge transformation, B an algebra and v : H → B an algebra map, which will be denoted by vF when it is considered as a map from HF to B. Proposition 5.25 The map ψ : BvF −1 → BvF , ψ (b) = v(F 1 )bv(S(F 2 )), for all b ∈ B, is an isomorphism of left HF -module algebras. Proof The map ψ is obviously bijective, with inverse ψ −1 (b) = v(G1 )bv(S(G2 )), for all b ∈ B, where F −1 = G1 ⊗ G2 . Then one checks by a direct computation that ψ is a morphism of left HF -module algebras, by using the formulas for ΔF , ΦF , αF , βF and for the multiplications, units and actions in BvF and BvF −1 . By taking B = End(M), where M is a left H-module, we obtain: Corollary 5.26

End(M)vF ∼ = End(M)vF −1 as left HF -module algebras.

By taking B = H, v = IdH in Proposition 5.25, we obtain: Corollary 5.27 (H0 )F −1 ∼ = (HF )0 as left HF -module algebras, with an isomorphism given by ψ : (H0 )F −1 → (HF )0 , ψ (h) = F 1 hS(F 2 ) for all h ∈ H, where F = F 1 ⊗ F 2 is a gauge transformation on H and H0 is the left H-module algebra that appears in Definition 4.4.

5.4 Two-sided Smash and Crossed Products

191

Proposition 5.28 Let H be a finite-dimensional quasi-Hopf algebra with bijective antipode. Denote by M the vector space H viewed as a left H-module with action h · m = mS−1 (h) for all m, h ∈ H, and consider the left H-module algebra End(M)v constructed at the beginning of this section and the left H-module algebra END(H) as in Proposition 5.21. Then END(H) = End(M)v as left H-module algebras. Proof This is a straightforward verification, using the formulas (5.2.5) and (5.2.6).

5.4 Two-sided Smash and Crossed Products We introduce some new types of crossed products, which will turn out to be iterated (quasi-)smash products. Proposition 5.29 Let H be a quasi-bialgebra, A a right H-comodule algebra, B a left H-comodule algebra and A an H-bimodule algebra. On A ⊗ A ⊗ B define a multiplication by (a > ϕ < b)(a > ϕ  < b ) = aa0 x˜1ρ > (x˜1λ · ϕ · a1 x˜2ρ )(x˜2λ b[−1] · ϕ  · x˜3ρ ) < x˜3λ b[0] b ,

(5.4.1)

for all a, a ∈ A, b, b ∈ B and ϕ , ϕ  ∈ A , where we write a > ϕ < b for a ⊗ ϕ ⊗ b. Then this multiplication yields an associative algebra with unit 1A > 1A < 1B , denoted by A > A < B and called the two-sided crossed product. Proof

We check the associativity of the multiplication:

[(a > ϕ < b)(a > ϕ  < b )](a > ϕ  < b ) =

(aa0 x˜1ρ > (x˜1λ · ϕ · a1 x˜2ρ )(x˜2λ b[−1] · ϕ  · x˜3ρ ) < x˜3λ b[0] b )(a > ϕ  < b )

=

aa0 x˜1ρ a0 y˜1ρ > [((y˜1λ )1 x˜λ1 · ϕ · a1 x˜2ρ a11 (y˜2ρ )1 )((y˜1λ )2 x˜λ2 b[−1] · ϕ  ·x˜3ρ a12 (y˜2ρ )2 )](y˜2λ (x˜λ3 )[−1] b[0,−1] b[−1] · ϕ  · y˜3ρ ) < y˜3λ (x˜λ3 )[0] b[0,0] b[0] b

(4.3.1)

=

aa0 a0,0 x˜1ρ y˜1ρ > [((y˜1λ )1 x˜λ1 · ϕ · a1 a0,1 x˜2ρ (y˜2ρ )1 )((y˜1λ )2 x˜λ2 b[−1] · ϕ  ·a1 x˜3ρ (y˜2ρ )2 )](y˜2λ (x˜λ3 )[−1] b[0,−1] b[−1] · ϕ  · y˜3ρ ) < y˜3λ (x˜λ3 )[0] b[0,0] b[0] b

(4.1.13)

=

aa0 a0,0 x˜1ρ y˜1ρ > (X 1 (y˜1λ )1 x˜λ1 · ϕ · a1 a0,1 x˜2ρ (y˜2ρ )1 x1 ) [(X 2 (y˜1λ )2 x˜λ2 b[−1] · ϕ  · a1 x˜3ρ (y˜2ρ )2 x2 )(X 3 y˜2λ (x˜λ3 )[−1] b[0,−1] b[−1] · ϕ  · y˜3ρ x3 )] < y˜3λ (x˜λ3 )[0] b[0,0] b[0] b

(4.3.6)

=

aa0 a0,0 x˜1ρ y˜1ρ > (x˜λ1 · ϕ · a1 a0,1 x˜2ρ (y˜2ρ )1 x1 )[((x˜λ2 )1 y˜1λ b[−1] · ϕ  ·a1 x˜3ρ (y˜2ρ )2 x2 )((x˜λ2 )2 y˜2λ b[0,−1] b[−1] · ϕ  · y˜3ρ x3 )] < x˜λ3 y˜3λ b[0,0] b[0] b

(4.3.5)

=

aa0 a0,0 x˜1ρ y˜1ρ > (x˜λ1 · ϕ · a1 a0,1 x˜2ρ (y˜2ρ )1 x1 )[((x˜λ2 )1 b[−1]1 y˜1λ · ϕ  ·a1 x˜3ρ (y˜2ρ )2 x2 )((x˜λ2 )2 b[−1]2 y˜2λ b[−1] · ϕ  · y˜3ρ x3 )] < x˜λ3 b[0] y˜3λ b[0] b

(4.3.2)

=

aa0 a0,0 (y˜1ρ )0 x˜1ρ > (x˜λ1 · ϕ · a1 a0,1 (y˜1ρ )1 x˜2ρ )[((x˜λ2 )1 b[−1]1 y˜1λ · ϕ 

192

Crossed Products ·a1 y˜2ρ (x˜3ρ )1 )((x˜λ2 )2 b[−1]2 y˜2λ b[−1] · ϕ  · y˜3ρ (x˜3ρ )2 )] < x˜λ3 b[0] y˜3λ b[0] b

=

(a > ϕ < b)[a a0 x˜1ρ > (x˜1λ · ϕ  · a1 x˜2ρ )(x˜2λ b[−1] · ϕ  · x˜3ρ ) < x˜3λ b[0] b ]

=

(a > ϕ < b)[(a > ϕ  < b )(a > ϕ  < b )].

The fact that 1A > 1A < 1B is the unit is easy to check and left to the reader. Proposition 5.30 Let H be a quasi-bialgebra, A a left H-module algebra, B a right H-module algebra and A an H-bicomodule algebra. Define on A ⊗ A ⊗ B a multiplication by (a < u > b)(a < u > b ) = (x˜1λ · a)(x˜2λ u[−1] θ 1 · a ) < x˜3λ u[0] θ 2 u 0 x˜1ρ > (b · θ 3 u 1 x˜2ρ )(b · x˜3ρ ), (5.4.2) for all a, a ∈ A, u, u ∈ A, b, b ∈ B (where we write a < u > b for a ⊗ u ⊗ b), and denote this structure on A ⊗ A ⊗ B by A < A > B. Then A < A > B is an associative algebra with unit 1A < 1A > 1B , called the two-sided generalized smash product. Proof

For all a, a , a ∈ A, u, u , u ∈ A and b, b , b ∈ B we compute:

[(a < u > b)(a < u > b )](a < u > b ) (5.4.2)

=

2 {y˜1λ · [(x˜1λ · a)(x˜2λ u[−1] θ 1 · a )]}[y˜2λ (x˜3λ )[−1] u[0,−1] θ[−1] 1

2  u0[−1] (x˜1ρ )[−1] θ · a ] < y˜3λ (x˜3λ )[0] u[0,0] θ[0] u0[0] (x˜1ρ )[0] θ

2

3

u0 y˜1ρ > {[(b · θ 3 u1 x˜2ρ )(b · x˜3ρ )] · θ u1 y˜2ρ }(b · y˜3ρ ) (4.1.1)

=

(4.1.4)

[(X 1 (y˜1λ )1 x˜1λ · a]{[X 2 (y˜1λ )2 x˜2λ u[−1] θ 1 · a ] 1

2 [X 3 y˜2λ (x˜3λ )[−1] u[0,−1] θ[−1] u0[−1] (x˜1ρ )[−1] θ · a ]} 2

2  < y˜3λ (x˜3λ )[0] u[0,0] θ[0] u0[0] (x˜1ρ )[0] θ u0 y˜1ρ 3

3

> [b · θ 3 u1 x˜2ρ θ 1 u11 (y˜2ρ )1 x1 ]{[(b · x˜3ρ θ 2 u12 (y˜2ρ )2 x2 ](b · y˜3ρ x3 )} (4.3.6)

=

2 (y˜1λ · a){[(y˜2λ )1 x˜1λ u[−1] θ 1 · a ][(y˜2λ )2 x˜2λ u[0,−1] θ[−1] u0[−1] 1

2

2  (x˜1ρ )[−1] θ · a ]} < y˜3λ x˜3λ u[0,0] θ[0] u0[0] (x˜1ρ )[0] θ u0 y˜1ρ 3

3

> [b · θ 3 u1 x˜2ρ θ 1 u11 (y˜2ρ )1 x1 ]{[b · x˜3ρ θ 2 u12 (y˜2ρ )2 x2 ](b · y˜3ρ x3 )} (4.3.5)

=

(4.4.2)

1

(y˜1λ · a){y˜2λ u[−1] θ 1 · [(x˜1λ · a )(x˜2λ Θ1 u0[−1] (x˜1ρ )[−1] θ · a )]} 2

< y˜3λ u[0] θ 2 (x˜3λ )0 Θ2 u0[0] (x˜1ρ )[0] θ u0 y˜1ρ > [b · θ 3 (x˜3λ )1 3

3

Θ3 u1 x˜2ρ θ 1 u11 (y˜2ρ ))1 x1 ]{[b · x˜3ρ θ 2 u12 (y˜2ρ )2 x2 ](b · y˜3ρ x3 )} (4.4.1),(4.4.3)

=

(4.3.1)

1

(y˜1λ · a){y˜2λ u[−1] θ 1 · [((x˜1λ · a )(x˜2λ u[−1] θ · a )]} < y˜3λ u[0] θ 2 2

2

(x˜3λ )0 u[0]0 θ 0 u0,0 x˜1ρ y˜1ρ > [b · θ 3 (x˜3λ )1 u[0]1 θ 1 3

u0,1 x˜2ρ (y˜2ρ )1 x1 ]{[b · θ u1 x˜3ρ (y˜2ρ )2 x2 ](b · y˜3ρ x3 )}

5.4 Two-sided Smash and Crossed Products (4.3.2)

=

(4.1.5)

193

1

(y˜1λ · a){y˜2λ u[−1] θ 1 · [(x˜1λ · a )(x˜2λ u[−1] θ · a )]} < y˜3λ u[0] θ 2 2

2

(x˜3λ u[0] θ u0 y˜1ρ )0 x˜1ρ > [b · θ 3 (x˜3λ u[0] θ u0 y˜1ρ )1 x˜2ρ ] 3

{[(b · θ u1 y˜2ρ )(b · y˜3ρ )] · x˜3ρ } (5.4.2)

=

1

2

(a < u > b)[(x˜1λ · a )(x˜2λ u[−1] θ · a ) < x˜3λ u[0] θ u0 y˜1ρ 3

(5.4.2)

=

> (b · θ u1 y˜2ρ )(b · y˜3ρ )]

(a < u > b)[(a < u > b )(a < u > b )].

Finally, by (4.3.3), (4.3.4), (4.3.7), (4.3.8) and (4.4.4) it follows that 1A < 1A > 1B is the unit of A < A > B. Remarks 5.31 (i) The two-sided crossed product A > A < B cannot be particularized for A = k or B = k because, in general, k is not a right or left H-comodule algebra. For the algebra A < A > B, we can take A = k or B = k. In these cases we obtain the right or left generalized smash products A > B and A < A, respectively. (ii) Let A = H. In this particular case we will denote the algebra A < H > B by A#H#B (the elements will be written a#h#b, a ∈ A, h ∈ H, b ∈ B) and will call it the two-sided smash product. This terminology is based on the fact that when we take A = k or B = k the resulting algebra is the right or left version of the smash product algebra. Note that the multiplication of A#H#B is defined, for all a, a ∈ A, h, h ∈ H, b, b ∈ B, by (a#h#b)(a #h #b ) = (x1 · a)(x2 h1 y1 · a )#x3 h2 y2 h1 z1 #(b · y3 h2 z2 )(b · z3 ). It follows that the canonical maps i : A#H → A#H#B and j : H#B → A#H#B, i(a#h) = a#h#1B and j(h#b) = 1A #h#b, are algebra morphisms. Suppose again that H is a quasi-bialgebra, A is a left H-module algebra and F = F 1 ⊗ F 2 ∈ H ⊗ H is a gauge transformation with inverse F −1 = G1 ⊗ G2 . Suppose now that we also have a right H-module algebra B. If we introduce on B another multiplication, by b  b = (b · F 1 )(b · F 2 ) for all b, b ∈ B, and denote this structure by F B, then F B becomes a right HF -module algebra with the same unit and right Haction as for B. We have the following type of invariance under twisting for two-sided smash products: Proposition 5.32

With notation as before, we have an algebra isomorphism

ϕ : A#H#B ∼ = AF −1 #HF #F B, ϕ (a#h#b) = F 1 · a#F 2 hG1 #b · G2 , ∀ a ∈ A, h ∈ H, b ∈ B. In particular, by taking A = k, we have an algebra isomorphism H#B ∼ = HF #F B. Proof

This follows by a direct computation, similar to the one in Proposition 5.10.

194

Crossed Products

We now prove that the two-sided generalized smash product can be written (in two ways) as an iterated generalized smash product. Proposition 5.33 Let H be a quasi-bialgebra, A a left H-module algebra, B a right H-module algebra and A an H-bicomodule algebra. Consider the right and left H-comodule algebras A < A and A > B as in Proposition 5.20. Then we have algebra isomorphisms A < A > B ≡ (A < A) > B,

A < A > B ≡ A < (A > B),

given by the trivial identifications. In particular, we have A#H#B ≡ (A#H) > B,

A#H#B ≡ A < (H#B).

Proof We will prove the first isomorphism, the second is similar. We compute the multiplication in (A < A) > B. For a, a ∈ A, b, b ∈ B and u, u ∈ A we have: ((a < u) > b)((a < u ) > b ) = (a < u)(a < u )0 (1A < x˜1ρ ) > (b · (a < u )1 x˜2ρ )(b · x˜3ρ ) = (a < u)(θ 1 · a < θ 2 u 0 x˜1ρ ) > (b · θ 3 u 1 x˜2ρ )(b · x˜3ρ ) = ((x˜1λ · a)(x˜2λ u[−1] θ 1 · a ) < x˜3λ u[0] θ 2 u 0 x˜1ρ ) > (b · θ 3 u 1 x˜2ρ )(b · x˜3ρ ). Via the trivial identification, this is exactly the multiplication of A < A > B. Proposition 5.34 Let H be a quasi-bialgebra, A a right H-comodule algebra, B a left H-comodule algebra and A an H-bimodule algebra. Consider the left and right H-module algebras A # A and A # B as in Propositions 5.15 and 5.17. Then we have algebra isomorphisms A > A < B ≡ (A # A ) < B,

A > A < B ≡ A > (A # B),

obtained from the trivial identifications. Proof

This follows by direct computations.

Theorem 5.35 Let H be a quasi-bialgebra, A an H-bimodule algebra, A a right H-comodule algebra, B an H-bicomodule algebra and C a left H-comodule algebra. Then: (i) A > A < B admits a right H-comodule algebra structure; (ii) B > A < C admits a left H-comodule algebra structure; (iii) there is an algebra isomorphism (given by the trivial identification) (A > A < B) > A < C ≡ A > A < (B > A < C). Proof By writing A > A < B as (A # A ) < B, we obtain that this is a right H-comodule algebra (being a generalized smash product between a left H-module algebra and an H-bicomodule algebra), and we can explicitly write its structure:

ρ : A > A < B ≡ (A # A ) < B → ((A # A ) < B) ⊗ H ≡ (A > A < B) ⊗ H,

5.4 Two-sided Smash and Crossed Products

195

ρ (a > ϕ < b) = (a > θ 1 · ϕ < θ 2 b0 ) ⊗ θ 3 b1 , ∀ a ∈ A, ϕ ∈ A , b ∈ B, 1 2 3 Φρ = (1A > 1A < X˜ ρ ) ⊗ X˜ ρ ⊗ X˜ ρ ∈ (A > A < B) ⊗ H ⊗ H.

Similarly, by writing B > A < C as B > (A # C), we obtain that this is a left H-comodule algebra, with structure:

λ : B > A < C ≡ B > (A # C) → H ⊗ (B > (A # C)) ≡ H ⊗ (B > A < C), λ (b > ϕ < c) = b[−1] θ 1 ⊗ (b[0] θ 2 > ϕ · θ 3 < c), ∀ b ∈ B, ϕ ∈ A , c ∈ C, 1

2

3

Φλ = X˜ λ ⊗ X˜ λ ⊗ (X˜ λ > 1A < 1C ) ∈ H ⊗ H ⊗ (B > A < C). To prove (iii), we will use the identifications appearing in Propositions 5.33 and 5.34: (A > A < B) > A < C ≡ ((A # A ) < B) > A < C ≡ ((A # A ) < B) > (A # C) ≡ (A # A ) < B > (A # C), and A > A < (B > A < C) ≡ A > A < (B > (A # C)) ≡ (A # A ) < (B > (A # C)) ≡ (A # A ) < B > (A # C). So, we have proved that the two iterated two-sided crossed products that appear in (iii) are both isomorphic as algebras (via the trivial identifications) to the two-sided generalized smash product (A # A ) < B > (A # C). By using the same results, we can obtain another relation between the two-sided crossed product and the two-sided generalized smash product. Namely, let H be a quasi-bialgebra, A an H-bimodule algebra, A a left H-module algebra, B a right Hmodule algebra and A and B two H-bicomodule algebras. As we have seen before, A < A (resp. B > B) becomes a right (resp. left) H-comodule algebra, so we can consider the two-sided crossed product (A < A) > A < (B > B). On the other hand, by Theorem 5.35, A > A < B becomes a right H-comodule algebra and a left H-comodule algebra, but actually, by using the explicit formulas for its structures that we gave, one can prove that it is even an H-bicomodule algebra, with Φλ ,ρ = 1H ⊗ (1A > 1A < 1B ) ⊗ 1H , so we can consider the two-sided generalized smash product A < (A > A < B) > B. Proposition 5.36

We have an algebra isomorphism

(A < A) > A < (B > B) ≡ A < (A > A < B) > B obtained from the trivial identification. In particular, we have (A#H) > H ∗ < (H#B) ≡ A < (H > H ∗ < H) > B. Proof This can be proved by computing explicitly the multiplication rules in the two algebras and noting that they coincide. Alternatively, we provide a conceptual proof, by a sequence of identifications using the above results. We compute: A < (A > A < B) > B ≡ A < ((A > A < B) > B)

196

Crossed Products ≡ A < (((A # A ) < B) > B) ≡ A < ((A # A ) < (B > B)) ≡ A < (A > A < (B > B)) ≡ A < (A > (A # (B > B))) ≡ (A < A) > (A # (B > B)) ≡ (A < A) > A < (B > B),

where the fourth and the fifth identities hold since the left H-comodule algebra structures on (A > A < B) > B, A > A < (B > B) and A > (A # (B > B)) coincide (via the trivial identifications).

5.5 H ∗ -Hopf Bimodules ∗

∗

H ∗ The aim of this section is to prove that a suitably defined category H H ∗ MH ∗ of H Hopf bimodules, where H is a finite-dimensional quasi-Hopf algebra with bijective antipode, is isomorphic to a certain category of modules. Let H be a finite-dimensional quasi-bialgebra and A a left H-module algebra. We ∗ define the category MAH , whose objects are vector spaces M such that M is a right H ∗ -comodule (i.e. M is a left H-module, with action denoted by h ⊗ m → h  m) and A acts on M to the right (denote this action by m ⊗ a → m · a) such that m · 1A = m for all m ∈ M, and the following relations hold, for all a, a ∈ A, m ∈ M, h ∈ H:

(m · a) · a = (X 1  m) · [(X 2 · a)(X 3 · a )],

(5.5.1)

h  (m · a) = (h1  m) · (h2 · a).

(5.5.2)

H∗

Similarly, the category A M consists of vector spaces M such that M is a right H ∗ -comodule (i.e. a left H-module, with action denoted also by ) and A acts on M to the left (denote this action by a ⊗ m → a · m) such that 1A · m = m for all m ∈ M, and the following relations hold: a · (a · m) = [(x1 · a)(x2 · a )] · (x3  m),

(5.5.3)

h  (a · m) = (h1 · a) · (h2  m),

(5.5.4) H∗

for all a, a ∈ A, m ∈ M, h ∈ H. Clearly, the category A M coincides with the cate∗ gory A,H M from Definition 5.5; by Proposition 5.7, we have that A M H ∼ =A#H M . ∗ We need a description of MAH as a category of left modules over a right-handed smash product. Proposition 5.37 Let H be a quasi-Hopf algebra and A a left H-module algebra. Define on A a new multiplication, by putting a  a = (g1 · a )(g2 · a), ∀ a, a ∈ A,

(5.5.5)

where f −1 = g1 ⊗ g2 is given by (3.2.16), and denote this new structure by A. Then A becomes a right H-module algebra, with the same unit as A and right H-action given by a · h = S(h) · a, for all a ∈ A, h ∈ H. Proof

This is a straightforward computation, using (3.2.13) and (3.2.17).

5.5 H ∗ -Hopf Bimodules

197

Definition 5.38 Let H be a quasi-bialgebra and B a right H-module algebra. We say that M, a k-linear space, is a left H, B-module if (i) M is a left H-module with action denoted by h ⊗ m → h  m; (ii) B acts weakly on M from the left, that is, there exists a k-linear map B ⊗ M → M, denoted by b ⊗ m → b · m, such that 1B · m = m for all m ∈ M; (iii) the following compatibility conditions hold: b · (b · m) = x1  ([(b · x2 )(b · x3 )] · m),

(5.5.6)

b · (h  m) = h1  [(b · h2 ) · m],

(5.5.7)

for all b, b ∈ B, h ∈ H, m ∈ M. The category of all left H, B-modules, with morphisms being the H-linear maps that preserve the B-action, will be denoted by H,B M . Proposition 5.39 If H, B are as above, the categories H,B M and H#B M are isomorphic. The isomorphism is given as follows. If M ∈ H#B M , define h  m = (h#1B ) · m and b · m = (1H #b) · m. Conversely, if M ∈ H,B M , define (h#b) · m = h  (b · m). Proof

This is a straightforward computation.

Proposition 5.40 If H is a finite-dimensional quasi-Hopf algebra with bijective an∗ tipode and A is a left H-module algebra, then MAH is isomorphic to H#A M , where A is the right H-module algebra constructed in Proposition 5.37. The correspondences are given as follows (we fix {ei }i a basis in H with {ei }i the dual basis in H ∗ ): ∗

• If M ∈ H#A M , then M becomes an object in MAH with the following structures (we denote by h⊗m → hm the left H-module structure of M and by a⊗m → am the weak left A-action on M arising from Proposition 5.39): M → M ⊗ H ∗,

n

m → ∑ ei  m ⊗ ei , ∀ m ∈ M, i=1

M ⊗ A → M,

m ⊗ a → m · a = q1  ((S(q2 ) · a)  m),

where qR = q1 ⊗ q2 = X 1 ⊗ S−1 (α X 3 )X 2 ∈ H ⊗ H. ∗ • Conversely, if M ∈ MAH , if we denote the H ∗ -comodule structure of M by M → M ⊗ H ∗ , m → m(0) ⊗ m(1) , and the weak right A-action on M by m ⊗ a → ma, then M becomes an object in H#A M with the following structures (again via Proposition 5.39): M is a left H-module with action h  m = m(1) (h)m(0) , and the weak left A-action on M is given by a → m = (p1  m)(p2 · a), ∀ a ∈ A, m ∈ M, where pR = p1 ⊗ p2 = x1 ⊗ x2 β S(x3 ) ∈ H ⊗ H. Proof

Assume first that M ∈ H#A M ; then we have, by Propositions 5.39 and 5.37: a  (a  m) = x1  ([(g1 S(x3 ) · a )(g2 S(x2 ) · a)]  m),

(5.5.8)

a  (h  m) = h1  [(S(h2 ) · a)  m],

(5.5.9)

198

Crossed Products ∗

for all a, a ∈ A, h ∈ H, m ∈ M. We have to prove that M ∈ MAH . To prove (5.5.1), we compute (denoting by Q1 ⊗ Q2 another copy of qR ): (m · a) · a

Q1  [(S(Q2 ) · a )  (q1  [(S(q2 ) · a)  m])]

= (5.5.9)

Q1 q11  [(S(q12 )S(Q2 ) · a )  ((S(q2 ) · a)  m)]

(5.5.8)

Q1 q11 x1  [((g1 S(x3 )S(q2 ) · a)(g2 S(x2 )S(Q2 q12 ) · a ))  m]

= =

(3.2.26) 1

1 q X11  [((g1 S(X(2,2) )S(q22 ) f 1 X 2 · a)

=

1 (g2 S(X(2,1) )S(q21 ) f 2 X 3 · a ))  m] (3.2.13) 1

=

q X11  [((S(q2 X21 )1 X 2 · a)(S(q2 X21 )2 X 3 · a ))  m]

=

q1 X11  [(S(q2 X21 ) · ((X 2 · a)(X 3 · a )))  m]

=

q1  [X11  [(S(X21 ) · (S(q2 ) · ((X 2 · a)(X 3 · a ))))  m]]

(5.5.9)

=

q1  [(S(q2 ) · ((X 2 · a)(X 3 · a )))  (X 1  m)]

=

(X 1  m) · ((X 2 · a)(X 3 · a )). QED

To prove (5.5.2), we compute: (h1  m) · (h2 · a)

=

q1  ((S(q2 )h2 · a)  (h1  m))

(5.5.9)

q1 h(1,1)  ((S(h(1,2) )S(q2 )h2 · a)  m)

(3.2.21)

hq1  ((S(q2 ) · a)  m) = h  (m · a).

= =

∗

Obviously m · 1A = m, for all m ∈ M, hence indeed M ∈ MAH . ∗ Conversely, assume that M ∈ MAH , that is, (ma)a = (X 1  m)[(X 2 · a)(X 3 · a )], h  (ma) = (h1  m)(h2 · a),

(5.5.10) (5.5.11)

for all m ∈ M, a, a ∈ A, h ∈ H, and we have to prove that a → (a → m) = x1  ([(g1 S(x3 ) · a )(g2 S(x2 ) · a)] → m), a → (h  m) = h1  [(S(h2 ) · a) → m],

(5.5.12) (5.5.13)

for all a, a ∈ A, h ∈ H, m ∈ M. To prove (5.5.12), we compute (denoting by P1 ⊗ P2 another copy of pR ): a → (a → m)

=

(p1  [(P1  m)(P2 · a )])(p2 · a)

(5.5.11)

[(p11 P1  m)(p12 P2 · a )](p2 · a)

(5.5.10)

(X 1 p11 P1  m)[(X 2 p12 P2 · a )(X 3 p2 · a)]

(3.2.25)

1 1 (x11 p1  m)[(x(2,1) p21 g1 S(x3 ) · a )(x(2,2) p22 g2 S(x2 ) · a)]

= = =

(5.5.11) 1

=

x  [(p1  m)[(p21 g1 S(x3 ) · a )(p22 g2 S(x2 ) · a)]]

=

x1  [((g1 S(x3 ) · a )(g2 S(x2 ) · a)) → m]. QED

To prove (5.5.13), we compute: h1  [(S(h2 ) · a) → m]

=

h1  [(p1  m)(p2 S(h2 ) · a)]

5.5 H ∗ -Hopf Bimodules

199

(5.5.11)

=

(h(1,1) p1  m)(h(1,2) p2 S(h2 ) · a)

(3.2.21)

(p1 h  m)(p2 · a) = a → (h  m).

=

Obviously 1A → m = m, for all m ∈ M, hence indeed M ∈ H#A M . ∗ To show that MAH ∼ = H#A M , the only things left to prove are the following: (1) If M ∈ H#A M , then a → m = a  m, for all a ∈ A, m ∈ M. ∗ (2) If M ∈ MAH , then m · a = ma, for all a ∈ A, m ∈ M. To prove (1), we compute: a → m = (p1  m) · (p2 · a) = q1  [(S(q2 )p2 · a)  (p1  m)] (5.5.9) 1 1 (3.2.23) = q p1  [(S(p12 )S(q2 )p2 · a)  m] = a  m.

To prove (2), we compute: m·a

=

q1  [(S(q2 ) · a) → m]

=

q1  [(p1  m)(p2 S(q2 ) · a)]

(5.5.11)

=

(3.2.23)

(q11 p1  m)(q12 p2 S(q2 ) · a) = ma,

and the proof is finished. We also need the description of left modules over a two-sided smash product. Definition 5.41 Let H be a quasi-bialgebra, A a left H-module algebra and B a right H-module algebra. Define the category A,H,B M as follows: an object in this category is a left H-module M, with action denoted by h ⊗ m → h  m, and we have left weak actions of A and B on M, denoted by a ⊗ m → a · m and b ⊗ m → b · m, such that: (i) M ∈ A#H M , that is, the relations (5.5.3) and (5.5.4) hold; (ii) M ∈ H#B M , that is, the relations (5.5.6) and (5.5.7) hold; (iii) the following compatibility condition holds: b · (a · m) = (y1 · a) · [y2  ((b · y3 ) · m)],

(5.5.14)

for all a ∈ A, b ∈ B, m ∈ M. The morphisms in this category are the H-linear maps compatible with the two weak actions. Proposition 5.42 If H, A, B are as above, then phism being given as follows:

A#H#B M

∼ = A,H,B M , the isomor-

• If M ∈ A#H#B M , define a · m = (a#1H #1B ) · m, h  m = (1A #h#1B ) · m, b · m = (1A #1H #b) · m; • Conversely, if M ∈ A,H,B M , define (a#h#b) · m = a · (h  (b · m)). Proof Straightforward computation, using the formula for the multiplication in A#H#B. Let us point out how the condition (5.5.14) occurs: b · (a · m) = (1A #1H #b) · ((a#1H #1B ) · m)

200

Crossed Products = [(1A #1H #b)(a#1H #1B )] · m = (y1 · a#y2 #b · y3 ) · m = (y1 · a) · (y2  ((b · y3 ) · m)),

which is exactly (5.5.14). Let H be a finite-dimensional quasi-bialgebra and A, D two left H-module alge∗ bras. It is obvious that A M H coincides with the category of left A-modules within ∗ the monoidal category H M , and similarly MDH coincides with the category of right D-modules within H M . Hence, we can introduce the following new category: ∗

Definition 5.43 If H, A, D are as above, define A MDH as the category of A–D∗ bimodules within the monoidal category H M , that is, M ∈ A MDH if and only if ∗ ∗ M ∈ A M H , M ∈ MDH and the following relation holds, for all a ∈ A, m ∈ M, d ∈ D: (a · m) · d = (X 1 · a) · [(X 2  m) · (X 3 · d)],

(5.5.15)

where a ⊗ m → a · m and m ⊗ d → m · d are the weak actions. Proposition 5.44 Let H be a finite-dimensional quasi-Hopf algebra with bijective antipode and A, D two left H-module algebras. Then we have an isomorphism of ∗ categories A MDH ∼ = A#H#D M , where the algebra D ∈ MH is as in Proposition 5.37. ∗ ∗ Proof Since A M H ∼ = A#H M and MDH ∼ = H#D M , the only thing left to prove is that the compatibility (5.5.14) in A,H,D M is equivalent to the compatibility (5.5.15) ∗ in A MDH . Let us first note the following easy consequences of (3.1.9), (3.2.1):

X 1 p11 ⊗ X 2 p12 ⊗ X 3 p2 = y1 ⊗ y21 p1 ⊗ y22 p2 S(y3 ),

(5.5.16)

q11 y1 ⊗ q12 y2 ⊗ S(q2 y3 )

(5.5.17)

= X ⊗q 1

1

X12 ⊗ S(q2 X22 )X 3 ,

where pR = p1 ⊗ p2 and qR = q1 ⊗ q2 are again the standard elements in H ⊗ H. ∗ Now let M ∈ A MDH , with right D-action on M denoted by m ⊗ d → m · d. Then, by Proposition 5.40, the weak left D-action on M is given by d → m = (p1  m) · (p2 · d). We check (5.5.14); we compute: d → (a · m)

(p1  (a · m)) · (p2 · d)

= (5.5.4)

[(p11 · a) · (p12  m)] · (p2 · d)

(5.5.15)

=

(X 1 p11 · a) · [(X 2 p12  m) · (X 3 p2 · d)]

(5.5.16)

(y1 · a) · [(y21 p1  m) · (y22 p2 S(y3 ) · d)]

(5.5.2)

=

(y1 · a) · [y2  ((p1  m) · (p2 S(y3 ) · d))]

=

(y1 · a) · [y2  ((S(y3 ) · d) → m)]

=

(y1 · a) · [y2  ((d · y3 ) → m)].

= =

Conversely, assume that M ∈ A#H#D M , and denote the actions of A, H, D on M by a · m, h  m, d · m, respectively. Then, by Proposition 5.40, the right D-action on M is given by m · d = q1  ((S(q2 ) · d) · m). To check (5.5.15), we compute: (a · m) · d

=

q1  [(S(q2 ) · d) · (a · m)]

5.6 Diagonal Crossed Products

201

(5.5.14) 1

=

q  [(y1 · a) · (y2  ((S(q2 ) · d · y3 ) · m))]

=

q1  [(y1 · a) · (y2  ((S(q2 y3 ) · d) · m))]

(5.5.4)

(q11 y1 · a) · [q12 y2  ((S(q2 y3 ) · d) · m)]

(5.5.17)

=

(X 1 · a) · [q1 X12  ((S(q2 X22 )X 3 · d) · m)]

=

(X 1 · a) · [q1 X12  ((S(q2 )X 3 · d · X22 ) · m)]

(5.5.7)

=

(X 1 · a) · [q1  ((S(q2 )X 3 · d) · (X 2  m))]

=

(X 1 · a) · [(X 2  m) · (X 3 · d)],

=

and the proof is finished. Let H be a finite-dimensional quasi-bialgebra and A , D two H-bimodule alge∗ H∗ bras. Define the category H A MD as the category of A –D-bimodules within the monoidal category H MH . By regarding A and D as left module algebras over ∗ (H⊗H op )∗ H∗ ∼ H ⊗ H op , it is easy to see that H . Hence, as a consequence A MD = A MD of Proposition 5.44, we finally obtain: Theorem 5.45 If H is a finite-dimensional quasi-Hopf algebra with bijective antipode and A , D are two H-bimodule algebras, we have an isomorphism of ∗ H∗ ∼ H∗ H∗ ∼ categories H A MD = A #(H⊗H op )#D M . In particular, H ∗ MH ∗ = H ∗ #(H⊗H op )#H ∗ M .

5.6 Diagonal Crossed Products Let H be a quasi-Hopf algebra with bijective antipode, A an H-bimodule algebra and (δ , Ψ) a two-sided coaction of H on an algebra A. Define the following elements in H ⊗2 ⊗ A ⊗ H ⊗2 : 1

2

3

4

5

Ωδ = Ω1δ ⊗ · · · ⊗ Ω5δ = Ψ ⊗ Ψ ⊗ Ψ ⊗ S−1 ( f 1 Ψ ) ⊗ S−1 ( f 2 Ψ ), Ωδ

=

5 Ω1 δ ⊗ · · · ⊗ Ωδ

= S−1 (Ψ1 g1 ) ⊗ S−1 (Ψ2 g2 ) ⊗ Ψ3 ⊗ Ψ4 ⊗ Ψ5 .

(5.6.1) (5.6.2)

Here f = f 1 ⊗ f 2 is the twist defined in (3.2.15) and f −1 = g1 ⊗ g2 is its inverse. We denote by A  δ A and A  δ A the k-vector spaces A ⊗ A and A ⊗ A , respectively, furnished with the multiplications given by: (ϕ  δ u)(ϕ   δ u ) = (Ω1δ · ϕ · Ω5δ )(Ω2δ u(−1) · ϕ  · S−1 (u(1) )Ω4δ )  δ Ω3δ u(0) u , 

(5.6.3)



(u  δ ϕ )(u  δ ϕ ) 2 −1   4 1  5 = uu(0) Ω3 δ  δ (Ωδ S (u(−1) ) · ϕ · u(1) Ωδ )(Ωδ · ϕ · Ωδ ),

(5.6.4)

respectively, for all u, u ∈ A and ϕ , ϕ  ∈ A , where we write ϕ  δ u and u  δ ϕ in place of ϕ ⊗ u and u ⊗ ϕ , respectively, to distinguish the new algebraic structures, 5 and where Ωδ = Ω1δ ⊗ · · · ⊗ Ω5δ and Ωδ = Ω1 δ ⊗ · · · ⊗ Ωδ are the elements defined by (5.6.1) and (5.6.2), respectively. We call A  δ A and A  δ A the left and right generalized diagonal crossed products, respectively, between A and A.

202

Crossed Products

The following (technical) lemma, expressing some relations fulfilled by the elements Ωδ and Ωδ , will be essential in what follows. It will help to prove that the generalized diagonal crossed products defined above are associative algebras, and moreover it will allow us to regard an H-bicomodule algebra A, in two ways, as a left H ⊗ H op -comodule algebra. Lemma 5.46 Let H be a quasi-Hopf algebra with bijective antipode, A an algebra and (δ , Ψ) a two-sided coaction of H on A. 1 5 (a) Let Ωδ = Ω1δ ⊗ · · · ⊗ Ω5δ = Ωδ ⊗ · · · ⊗ Ωδ be the element defined by (5.6.1). Then for all u ∈ A the following relations hold: Ω1δ u(−1) ⊗ Ω2δ u(0,−1) ⊗ Ω3δ u(0,0) ⊗ S−1 (u(0,1) )Ω4δ ⊗ S−1 (u(1) )Ω5δ = u(−1)1 Ω1δ ⊗ u(−1)2 Ω2δ ⊗ u(0) Ω3δ ⊗ Ω4δ S−1 (u(1) )2 ⊗ Ω5δ S−1 (u(1) )1 , (5.6.5) 1

1

2

3

4

X 1 (Ωδ )1 Ω1δ ⊗ X 2 (Ωδ )2 Ω2δ ⊗ X 3 Ωδ (Ω3δ )(−1) ⊗ Ωδ Ω3(0) ⊗ S−1 ((Ω3δ )(1) )Ωδ x3 5

5

1

2

2

⊗ Ω4δ (Ωδ )2 x2 ⊗ Ω5δ (Ωδ )1 x1 = Ωδ ⊗ (Ωδ )1 Ω1δ ⊗ (Ωδ )2 Ω2δ 3

4

4

5

⊗ Ωδ Ω3δ ⊗ Ω4δ (Ωδ )2 ⊗ Ω5δ (Ωδ )1 ⊗ Ωδ . (5.6.6) 1

5

5 (b) Let Ωδ = Ω1 δ ⊗ · · · ⊗ Ωδ = Ωδ ⊗ · · · ⊗ Ωδ be the element defined by (5.6.2). Then for all u ∈ A the following relations hold: 2 −1 3 4 5 −1 Ω1 δ S (u(−1) ) ⊗ Ωδ S (u(0,−1) ) ⊗ u(0,0) Ωδ ⊗ u(0,1) Ωδ ⊗ u(1) Ωδ 2 3 4 5 −1 = S−1 (u(−1) )2 Ω1 δ ⊗ S (u(−1) )1 Ωδ ⊗ Ωδ u(0) ⊗ Ωδ u(1)1 ⊗ Ωδ u(1)2 , (5.6.7) 1

2

2

3

4

1 2 3 4 1 X 3 Ωδ ⊗ X 2 (Ωδ )2 Ω1 δ ⊗ X (Ωδ )1 Ωδ ⊗ Ωδ Ωδ ⊗ Ωδ (Ωδ )1 x 4

5

1

1

2

2 3 1 2 3 −1 ⊗ Ω5 δ (Ωδ )2 x ⊗ Ωδ x = (Ωδ )1 Ωδ ⊗ (Ωδ )2 Ωδ ⊗ Ωδ S ((Ωδ )(−1) ) 3

4

5

5

3 4 5 ⊗ (Ω3 δ )(0) Ωδ ⊗ (Ωδ )(1) Ωδ ⊗ Ωδ (Ωδ )1 ⊗ Ωδ (Ωδ )2 . (5.6.8)

Proof We will prove only (a), (b) being similar. The relation (5.6.5) follows easily by applying (5.6.1), (4.4.5) and (3.2.13); the details are left to the reader. We now prove (5.6.6). We compute: 1

1

2

3

X 1 (Ωδ )1 Ω1δ ⊗ X 2 (Ωδ )2 Ω2δ ⊗ X 3 Ωδ (Ω3δ )(−1) ⊗ Ωδ Ω3(0) 4

5

5

⊗ S−1 ((Ω3δ )(1) )Ωδ x3 ⊗ Ω4δ (Ωδ )2 x2 ⊗ Ω5δ (Ωδ )1 x1 (5.6.1)

=

(3.2.13)

1 1

1 2

2 3

3 3

4 3

X 1 Ψ1 ϒ ⊗ X 2 Ψ2 ϒ ⊗ X 3 Ψ ϒ(−1) ⊗ Ψ ϒ(0) ⊗ S−1 ( f 1 Ψ ϒ(1) )x3 5 4

5 5

⊗ S−1 (F 1 f12 Ψ1 ϒ )x2 ⊗ S−1 (F 2 f22 Ψ2 ϒ )x1 (4.4.6)

=

1

2

1

2

2

3

3

4

4

ϒ ⊗ ϒ1 Ψ ⊗ ϒ2 Ψ ⊗ ϒ Ψ ⊗ S−1 (S(x3 ) f 1 X 1 ϒ1 Ψ ) 4

5

5

⊗ S−1 (S(x2 )F 1 f12 X 2 ϒ2 Ψ ) ⊗ S−1 (S(x1 )F 2 f22 X 3 ϒ ) (3.1.5),(3.2.17)

=

(3.2.13) (5.6.1)

=

1

2

1

2

2

3

3

4

4

ϒ ⊗ ϒ1 Ψ ⊗ ϒ2 Ψ ⊗ ϒ Ψ ⊗ S−1 ( f 1 Ψ )S−1 (F 1 ϒ )2 5

4

5

⊗ S−1 ( f 2 Ψ )S−1 (F 1 ϒ )1 ⊗ S−1 (F 2 ϒ ) 1

2

2

3

4

4

5

Ωδ ⊗ (Ωδ )1 Ω1δ ⊗ (Ωδ )2 Ω2δ ⊗ Ωδ Ω3δ ⊗ Ω4δ (Ωδ )2 ⊗ Ω5δ (Ωδ )1 ⊗ Ωδ ,

5.6 Diagonal Crossed Products 1

203

5

as claimed. We denoted by ϒ ⊗· · ·⊗ϒ another copy of Ψ−1 and by F 1 ⊗F 2 another copy of the Drinfeld twist f defined in (3.2.15). Let A be an H-bicomodule algebra and let (δ , Ψ) = (δl/r , Ψl/r ) be the two-sided coactions defined by (4.4.24) and (4.4.25), respectively. For simplicity we denote Ω = Ωδl , ω = Ωδr , Ω = Ωδ and ω  = Ωδr . Concretely, the elements Ω, ω ∈ H ⊗2 ⊗ l A ⊗ H ⊗2 become 1

1

2 Ω = (X˜ ρ )[−1]1 x˜1λ θ 1 ⊗ (X˜ ρ )[−1]2 x˜2λ θ[−1] 2 ⊗ (X˜ ρ )[0] x˜3λ θ[0] ⊗ S−1 ( f 1 X˜ ρ θ 3 ) ⊗ S−1 ( f 2 X˜ ρ ), 1

2

3

(5.6.9)

1

ω = x˜1λ ⊗ x˜2λ Θ1 ⊗ (x˜3λ )0 X˜ ρ Θ20 2 3 ⊗ S−1 ( f 1 (x˜3λ )11 X˜ ρ Θ21 ) ⊗ S−1 ( f 2 (x˜3λ )12 X˜ ρ Θ3 ),

(5.6.10)

1 2 3 −1 1 2 3 1 2 3 where Φρ = X˜ ρ ⊗ X˜ ρ ⊗ X˜ ρ , Φ−1 λ = x˜λ ⊗ x˜λ ⊗ x˜λ , Φλ ,ρ = Θ ⊗ Θ ⊗ Θ , Φλ ,ρ = θ 1 ⊗ θ 2 ⊗ θ 3 and f = f 1 ⊗ f 2 is the twist defined in (3.2.15). For further use we record the fact that the formulas in Lemma 5.46 (a) specialize to (δl/r , Ψl/r ) as follows (for all u ∈ A):

Ω1 u0[−1] ⊗ Ω2 u0[0]

0[−1]

⊗ Ω3 u0[0]

= u0[−1] Ω ⊗ u0[−1] Ω ⊗ u0[0] Ω 1

2

1

1

2

X 1 Ω1 Ω1 ⊗ X 2 Ω2 Ω2 ⊗ X 3 Ω Ω30 5

1

2

1

3

2

1

⊗ S−1 (u0[0] )Ω4 ⊗ S−1 (u1 )Ω5

0[0]

⊗ Ω4 S−1 (u

5 −1 1 )2 ⊗ Ω S (u1 )1 ,

3

[−1]

4

(5.6.11)

5

⊗ Ω Ω30 ⊗ S−1 (Ω31 )Ω x3 ⊗ Ω4 Ω2 x2

2

[0]

3

4

4

5

⊗Ω5 Ω1 x1 = Ω ⊗ Ω1 Ω1 ⊗ Ω2 Ω2 ⊗ Ω Ω3 ⊗ Ω4 Ω2 ⊗ Ω5 Ω1 ⊗ Ω ,

(5.6.12)

and

ω 1 u[−1] ⊗ ω 2 u[0]0

[−1]

⊗ ω 3 u[0]0

[0]0

⊗ S−1 (u[0]0

[0]1

)ω 4 ⊗ S−1 (u[0]1 )ω 5

= u[−1]1 ω 1 ⊗ u[−1]2 ω 2 ⊗ u[0]0 ω 3 ⊗ ω 4 S−1 (u[0]1 )2 ⊗ ω 5 S−1 (u[0]1 )1 , (5.6.13) 3 3 3 ω 11 ω 1 ⊗ ω 12 ω 2 ⊗ ω 2 ω[−1] ⊗ ω 3 ω[0] ⊗ S−1 (ω[0] )ω 4 ⊗ ω 4 ω 52 ⊗ ω 5 ω 51 0 1

= x1 ω 1 ⊗ x2 ω 21 ω 1 ⊗ x3 ω 22 ω 2 ⊗ ω 3 ω 3 ⊗ ω 4 ω 42 X 3 ⊗ ω 5 ω 41 X 2 ⊗ ω 5 X 1 , (5.6.14) 1

5

respectively, where we denoted by Ω = Ω1 ⊗ · · · ⊗ Ω5 = Ω ⊗ · · · ⊗ Ω the element defined in (5.6.9) and by ω = ω 1 ⊗ · · · ⊗ ω 5 = ω 1 ⊗ · · · ⊗ ω 5 the element defined in (5.6.10). If (A, λ , ρ , Φλ , Φρ , Φλ ,ρ ) is an H-bicomodule algebra then, as we mentioned be321 321 op,cop -bicomodule fore, Aop,cop := (Aop , τA,H ◦ ρ , τH,A ◦ λ , Φ321 ρ , Φλ , Φλ ,ρ ) is an H op,cop we have that the Drinfeld twist (defined for an arbialgebra. Moreover, in H −1 = g2 ⊗ g1 , where f is trary quasi-Hopf algebra in (3.2.15)) is given by fop,cop = f21 the Drinfeld twist of H. Now, if we denote by Ωop,cop and ωop,cop the elements Ωδl/r corresponding to the H op,cop -bicomodule algebra Aop,cop , then one checks that Ω = (ωop,cop )54321 and ω  = (Ωop,cop )54321 ,

204

Crossed Products

so we restrict to the study of the elements Ω, ω and their associated constructions. If H is a quasi-Hopf algebra with bijective antipode, A an H-bimodule algebra and (A, λ , ρ , Φλ , Φρ , Φλ ,ρ ) an H-bicomodule algebra, we will denote A  δl A = A  A, A  δr A = A  A, A  δl A = A  A and A  δr A = A  A , where δl and δr are the two-sided coactions given by (4.4.24) and (4.4.25). We call the first two constructions left diagonal crossed products and the last two right diagonal crossed products. For example, the multiplications in A  A and A  A are given by (ϕ  u)(ϕ   u ) = (Ω1 · ϕ · Ω5 )(Ω2 u0[−1] · ϕ  · S−1 (u1 )Ω4 )  Ω3 u0[0] u , 

(5.6.15)



(ϕ  u)(ϕ  u ) = (ω 1 · ϕ · ω 5 )(ω 2 u[−1] · ϕ  · S−1 (u[0]1 )ω 4 )  ω 3 u[0]0 u ,

(5.6.16)

respectively, for all ϕ , ϕ  ∈ A and u, u ∈ A, where we write ϕ  u and ϕ  u instead of ϕ ⊗ u to distinguish the new algebraic structures. We are now ready to show that the generalized diagonal crossed products are unital associative algebras. Proposition 5.47 Let H be a quasi-Hopf algebra with bijective antipode, A an algebra and (δ , Ψ) a two-sided coaction of H on A. Consider A  δ A and A  δ A , the vector spaces A ⊗ A and A ⊗ A , respectively, with the multiplications defined in (5.6.3) and (5.6.4), respectively. Then these products define on A  δ A and A  δ A two associative algebra structures with unit 1A  δ 1A (resp. 1A  δ 1A ), containing A ≡ 1A  δ A (resp. A ≡ A  δ 1A ) as unital subalgebra. Consequently, if A is an H-bicomodule algebra and A is an H-bimodule algebra then A  A, A  A, A  A and A  A are associative unital algebras containing A as unital subalgebra. Proof We will give the proof only for A  δ A, the one for A  δ A being similar (it uses the relations satisfied by Ωδ , instead of the ones satisfied by Ωδ ). For ϕ , ϕ  , ϕ  ∈ A and u, u , u ∈ A we compute: (ϕ  δ u)[(ϕ   δ u )(ϕ   δ u )] (5.6.3)

(ϕ  δ u)[(Ω1δ · ϕ  · Ω5δ )(Ω2δ u(−1) · ϕ  · S−1 (u(−1) )Ω4δ )  δ Ω3δ u(0) u ]

(5.6.3)

(Ωδ · ϕ · Ωδ )[((Ωδ )1 u(−1)1 Ω1δ · ϕ  · Ω5δ S−1 (u(−1) )1 (Ωδ )1 )((Ωδ )2 u(−1)2

= =

(4.1.14)

(5.6.5)

=

1

5

2

4

2

Ω2δ u(−1) · ϕ  · S−1 (u(1) )Ω4δ S−1 (u(1) )2 (Ωδ )2 )]  δ Ωδ u(0) Ω3δ u(0) u 4

3

(Ωδ · ϕ · Ωδ )[(Ωδ )1 Ω1δ u(−1) · ϕ  · S−1 (u(1) )Ω5δ (Ωδ )1 ) 1

5

2

4

((Ωδ )2 Ω2δ u(0,−1) u(−1) · ϕ  · S−1 (u(0,1) u (1) )Ω4δ (Ωδ )2 )] 2

4

 δ Ωδ Ω3δ u(0,0) u(0) u 3

(5.6.6),(4.1.13)

=

[((Ωδ )1 Ω1δ · ϕ · Ω5δ (Ωδ )1 )((Ωδ )2 Ω2δ u(−1) · ϕ  · S−1 (u(1) )Ω4δ (Ωδ )2 )] 1

5

1

5

5.6 Diagonal Crossed Products

205

(Ωδ (Ω3δ )(−1) u(0,−1) u(−1) · ϕ  · S−1 ((Ω3δ )(1) u(0,1) u(1) )Ωδ ) 2

4

 δ Ωδ (Ω3δ )(0) u(0,0) u(0) u 3

(4.1.14), (5.6.3)

=

(5.6.3)

[(Ω1δ · ϕ · Ω5δ )(Ω2δ u(−1) · ϕ  · S−1 (u(1) )Ω4δ )  δ Ω3δ u(0) u ](ϕ   δ u ) [(ϕ  δ u)(ϕ   δ u )](ϕ   δ u ).

=

The fact that 1A  δ 1A is the unit follows easily from the (co)unit axioms. Remark 5.48 In the algebras A  δ A and A  δ A we have, for all ϕ ∈ A and u ∈ A, (ϕ  δ 1A )(1A  δ u) = ϕ  δ u and (u  δ 1A )(1A  δ ϕ ) = u  δ ϕ . Examples 5.49 (i) We know that, if H is a quasi-Hopf algebra with bijective antipode, then H ∗ is an H-bimodule algebra, hence it makes sense to consider the algebras H ∗  δ A and A  δ H ∗ , where A is an algebra and (δ , Ψ) is a two-sided coaction of H on A. (ii) Let A be a left H-module algebra. Then A becomes an H-bimodule algebra, with right H-action given via ε . In this particular case A  H and A  H both coincide to the smash product algebra A#H. Moreover, if we replace the quasi-Hopf algebra H by an arbitrary H-bicomodule algebra A, then A  A and A  A coincide with the generalized smash product algebra A < A. Therefore, the diagonal crossed products may be viewed as a generalization of the (generalized) smash product. (iii) As we know, H itself is an H-bicomodule algebra. So, in this case, the multiplications of the diagonal crossed products A  H and A  H specialize to (ϕ  h)(ϕ   h ) = (Ω1 · ϕ · Ω5 )(Ω2 h(1,1) · ϕ  · S−1 (h2 )Ω4 )  Ω3 h(1,2) h , 

(5.6.17)



(ϕ  h)(ϕ  h ) = (ω 1 · ϕ · ω 5 )(ω 2 h1 · ϕ  · S−1 (h(2,2) )ω 4 )  ω 3 h(2,1) h ,

(5.6.18)

for all ϕ , ϕ  ∈ A and h, h ∈ H, where Ω = Ω1 ⊗· · ·⊗Ω5 ∈ H ⊗5 , ω = ω 1 ⊗...⊗ ω 5 ∈ H ⊗5 are now given by 1 1 Ω = X(1,1) x1 y1 ⊗ X(1,2) x2 y21 ⊗ X21 x3 y22 ⊗ S−1 ( f 1 X 2 y3 ) ⊗ S−1 ( f 2 X 3 ),

ω = x ⊗x Y 1

2 1

3 3 ⊗ x13 X 1Y12 ⊗ S−1 ( f 1 x(2,1) X 2Y22 ) ⊗ S−1 ( f 2 x(2,2) X 3Y 3 ),

(5.6.19) (5.6.20)

and f = f 1 ⊗ f 2 is the twist defined in (3.2.15). Let H be a quasi-Hopf algebra with bijective antipode. For an H-bicomodule algebra A and an H-bimodule algebra A the multiplications of the right diagonal crossed products A  A and A  A are the following: if Ω = Ω1 ⊗ · · · ⊗ Ω5 and ω  = ω 1 ⊗ · · · ⊗ ω 5 we have (u  ϕ )(u  ϕ  ) = uu0[0] Ω3  (Ω2 S−1 (u0[−1] ) · ϕ · u1 Ω4 )(Ω1 · ϕ  · Ω5 ), (5.6.21)

206

Crossed Products (u  ϕ )(u  ϕ  ) = uu[0]0 ω 3  (ω 2 S−1 (u[−1] ) · ϕ · u[0]1 ω 4 )(ω 1 · ϕ  · ω 5 ), (5.6.22)

for all u, u ∈ A and ϕ , ϕ  ∈ A . We know from Proposition 5.47 that A  A and A  A are associative algebras with unit 1A  1A and 1A  1A , respectively, containing A as unital subalgebra. In fact, under the trivial permutation of tensor factors we have that A  A ≡ (A op  Aop,cop )op , A  A ≡ (A op  Aop,cop )op ,

(5.6.23)

where the left diagonal crossed products are made over H op,cop . Note that A op becomes an H op,cop -bimodule algebra via the actions h ·op ϕ ·op h = h · ϕ · h, for all h, h ∈ H and ϕ ∈ A . Lemma 5.50 Let H be a quasi-Hopf algebra with bijective antipode, A an Hbimodule algebra and A an H-bicomodule algebra. Then, for all ϕ ∈ A , we have

ϕ  1A = (1A  q˜1ρ )(( p˜1ρ )[−1] · ϕ · q˜2ρ S−1 ( p˜2ρ )  ( p˜1ρ )[0] ) in A  A, where p˜ρ and q˜ρ are given by (4.3.9). Proof

We compute:

(1A  q˜1ρ )(( p˜1ρ )[−1] · ϕ · q˜2ρ S−1 ( p˜2ρ )  ( p˜1ρ )[0] ) (5.6.15)

=

(4.3.12)

=

(q˜1ρ )0[−1] ( p˜1ρ )[−1] · ϕ · q˜2ρ S−1 ( p˜2ρ )S−1 ((q˜1ρ )1 )  (q˜1ρ )0[0] ( p˜1ρ )[0]

ϕ  1A ,

finishing the proof. Proposition 5.51 Let H be a quasi-Hopf algebra with bijective antipode, A an Hbimodule algebra and A an H-bicomodule algebra. Define the map Γ : A → A  A, Γ(ϕ ) = ( p˜1ρ )[−1] · ϕ · S−1 ( p˜2ρ )  ( p˜1ρ )[0] ,

(5.6.24)

for all ϕ ∈ A . Then A  A is generated as an algebra by A and Γ(A ). Proof By the previous lemma it follows that ϕ  1A = (1A  q˜1ρ )Γ(ϕ · q˜2ρ ), for all ϕ ∈ A , so for ϕ ∈ A and u ∈ A we can write ϕ  u = (1A  q˜1ρ )Γ(ϕ · q˜2ρ )(1A  u), finishing the proof. We prove now a sort of associativity property of diagonal crossed products with respect to tensoring by an arbitrary algebra. Proposition 5.52 Let H be a quasi-Hopf algebra with bijective antipode, A an H-bimodule algebra, A an H-bicomodule algebra and C an algebra. On A ⊗ C we have a (canonical) H-bicomodule algebra structure, yielding algebra isomorphisms A  (A ⊗C) ≡ (A  A) ⊗C, defined by the trivial identifications.

A  (A ⊗C) ≡ (A  A) ⊗C,

5.6 Diagonal Crossed Products

207

Proof The H-bicomodule algebra structure on A ⊗C is given such that everything happening on C is trivial, for instance the right H-comodule algebra structure is:

ρA⊗C : A ⊗C u ⊗ c → (u0 ⊗ c) ⊗ u1 ∈ (A ⊗C) ⊗ H, (Φρ )A⊗C = (X˜ρ1 ⊗ 1C ) ⊗ X˜ρ2 ⊗ X˜ρ3 ∈ (A ⊗C) ⊗ H ⊗ H, and one can easily check that A ⊗C indeed becomes an H-bicomodule algebra. Also, it is easy to see that the elements Ω and ω for A ⊗C are given by ΩA⊗C = Ω1 ⊗ Ω2 ⊗ (Ω3 ⊗ 1C ) ⊗ Ω4 ⊗ Ω5 ,

ωA⊗C = ω 1 ⊗ ω 2 ⊗ (ω 3 ⊗ 1C ) ⊗ ω 4 ⊗ ω 5 , where Ω = Ω1 ⊗ · · · ⊗ Ω5 and ω = ω 1 ⊗ · · · ⊗ ω 5 are the elements for A. By using these facts, one obtains that the multiplications in A  (A ⊗C) and A  (A ⊗C) coincide with those in (A  A) ⊗C and (A  A) ⊗C, respectively, via the trivial identifications. Let H be a quasi-Hopf algebra with bijective antipode and A an H-bicomodule algebra. We define two left H ⊗ H op -coactions on A, as follows:

λ1 , λ2 : A → (H ⊗ H op ) ⊗ A, λ1 (u) = (u0[−1] ⊗ S−1 (u1 )) ⊗ u0[0] := u(−1) ⊗ u(0) , ∀ u ∈ A, λ2 (u) = (u[−1] ⊗ S−1 (u[0]1 )) ⊗ u[0]0 := u(−1) ⊗ u(0) , ∀ u ∈ A. If we look at the element Ω ∈ H ⊗2 ⊗ A ⊗ H ⊗2 given by (5.6.9) and consider the element (Ω1 ⊗Ω5 )⊗(Ω2 ⊗Ω4 )⊗Ω3 , then one can check that this element is invertible in (H ⊗ H op ) ⊗ (H ⊗ H op ) ⊗ A, its inverse being given by (Θ1 X˜λ1 (x˜ρ1 )[−1]1 ⊗ S−1 (x˜ρ3 g2 )) ⊗ (Θ2[−1] X˜λ2 (x˜ρ1 )[−1]2 ⊗ S−1 (Θ3 x˜ρ2 g1 )) ⊗ Θ2[0] X˜λ3 (x˜ρ1 )[0] , where f −1 = g1 ⊗ g2 is the element given by (3.2.16). We will denote this inverse by Φλ1 ∈ (H ⊗ H op ) ⊗ (H ⊗ H op ) ⊗ A. Similarly, if we look at the element ω given by (5.6.10) and consider the element (ω 1 ⊗ ω 5 ) ⊗ (ω 2 ⊗ ω 4 ) ⊗ ω 3 , then one can check that this element is invertible in (H ⊗ H op ) ⊗ (H ⊗ H op ) ⊗ A, with inverse defined by 2 2 ˜3 2 1 ˜3 y˜ρ (Yλ )11 g1 )) ⊗ θ0 y˜ρ (Yλ )0 . (Y˜λ1 ⊗ S−1 (θ 3 y˜3ρ (Y˜λ3 )12 g2 )) ⊗ (θ 1Y˜λ2 ⊗ S−1 (θ1

We will denote this inverse by Φλ2 ∈ (H ⊗ H op ) ⊗ (H ⊗ H op ) ⊗ A. Proposition 5.53 With notation as above, (A, λ1 , Φλ1 ) and (A, λ2 , Φλ2 ) are left H ⊗ H op -comodule algebras, denoted by A1 and A2 , respectively. Proof It is easy to see that λ1 and λ2 are algebra maps, and also that the conditions (4.3.7) and (4.3.8) in the definition of a left comodule algebra are satisfied. Then the conditions (4.3.5) and (4.3.6) for (A, λ1 , Φλ1 ) (resp. for (A, λ2 , Φλ2 )) to be a left H ⊗ H op -comodule algebra are equivalent to the relations (5.6.11) and (5.6.12) fulfilled by Ω (resp. to the relations (5.6.13) and (5.6.14) fulfilled by ω ).

208

Crossed Products

We are now able to express the diagonal crossed products over H as some generalized smash products over H ⊗ H op . Proposition 5.54 Let H be a quasi-Hopf algebra with bijective antipode, A an H-bimodule algebra and A an H-bicomodule algebra. View A as a left H ⊗ H op module algebra with action (h ⊗ h ) · ϕ = h · ϕ · h for all h, h ∈ H and ϕ ∈ A , and consider the two left H ⊗ H op -comodule algebras A1 and A2 obtained from A as above. Then we have algebra isomorphisms A  A ≡ A < A1 ,

A  A ≡ A < A2 ,

defined by the trivial identifications. Proof We prove only the first isomorphism, the second being similar. The multiplication in A < A1 looks as follows (for all ϕ , ϕ  ∈ A and u, u ∈ A): (ϕ < u)(ϕ  < u ) = ((x˜λ1 )A1 · ϕ )((x˜λ2 )A1 u(−1) · ϕ  ) < (x˜λ3 )A1 u(0) u = ((Ω1 ⊗ Ω5 ) · ϕ )((Ω2 ⊗ Ω4 )(u0[−1] ⊗ S−1 (u1 )) · ϕ  ) < Ω3 u0[0] u = (Ω1 · ϕ · Ω5 )(Ω2 u0[−1] · ϕ  · S−1 (u1 )Ω4 ) < Ω3 u0[0] u , and via the trivial identification this is exactly the multiplication of A  A. Let us also record the fact that the two left H ⊗ H op -comodule algebra structures on H are defined as follows:

λ1 , λ2 : H → (H ⊗ H op ) ⊗ H, λ1 (h) = (h(1,1) ⊗ S−1 (h2 )) ⊗ h(1,2) , ∀ h ∈ H, λ2 (h) = (h1 ⊗ S−1 (h(2,2) )) ⊗ h(2,1) , ∀ h ∈ H, Φλ1 , Φλ2 ∈ (H ⊗ H op ) ⊗ (H ⊗ H op ) ⊗ H,

1 1 ⊗ S−1 (x3 g2 )) ⊗ (Y12 X 2 x(1,2) ⊗ S−1 (Y 3 x2 g1 )) ⊗Y22 X 3 x21 , Φλ1 = (Y 1 X 1 x(1,1)

3 3 Φλ2 = (Y 1 ⊗ S−1 (x3 y3Y(2,2) g2 )) ⊗ (x1Y 2 ⊗ S−1 (x22 y2Y(2,1) g1 )) ⊗ x12 y1Y13 ,

where f −1 = g1 ⊗ g2 is the element given by (3.2.16). Again let H be a quasi-Hopf algebra with bijective antipode, A an H-bimodule algebra and A an H-bicomodule algebra. We intend to prove that the two left diagonal crossed products A  A and A  A are isomorphic as algebras, using their description as generalized smash products. First we need a result on generalized smash products. Namely, let H be a quasibialgebra, A a left H-module algebra, B a left H-comodule algebra and U ∈ H ⊗ B an invertible element such that (ε ⊗ IdB )(U) = 1B . Recall from Section 4.3 that if we define a map λ  : B → H ⊗ B, λ  (b) = U λ (b)U −1 , for all b ∈ B, then this is a new left H-comodule algebra structure on B, with Φλ  = (1H ⊗U)(IdH ⊗ λ )(U)Φλ (Δ ⊗ IdB )(U −1 ),

5.6 Diagonal Crossed Products

209

which is denoted by B (and we say that B and B are twist equivalent). So, we can consider the generalized smash products A < B and A < B . Proposition 5.55

The linear map

f : A < B → A < B , f (a < b) = U · (a < b) = U 1 · a < U 2 b, is an algebra isomorphism, and moreover f (1A < b) = 1A < b, for all b ∈ B (so A < B and A < B are equivalent extensions of B). Proof

This follows by a direct computation.

In view of this proposition, it suffices to prove that if A is an H-bicomodule algebra, then the two left H ⊗ H op -comodule algebras A1 and A2 constructed earlier are twist equivalent. To prove this, we first need a technical lemma. Lemma 5.56 Let H be a quasi-Hopf algebra with bijective antipode and A an Hbicomodule algebra. Consider the elements Ω and ω given by (5.6.9) and (5.6.10). Then the following relations hold: Θ11 Ω1 ⊗ Θ12 Ω2 ⊗ Θ2 Ω3 ⊗ Ω5 S−1 (Θ3 )1 ⊗ Ω4 S−1 (Θ3 )2 = Θ11 x˜λ1 ⊗ Θ12 x˜λ2 Θ

1

2 3 ⊗ X˜ρ1 Θ20 (x˜λ3 )0 Θ ⊗ S−1 ( f 2 X˜ρ3 Θ3 ) ⊗ S−1 ( f 1 X˜ρ2 Θ21 (x˜λ3 )1 Θ ), 2 Θ11 Ω1 θ 1 ⊗ S−1 (θ 3 )Ω5 S−1 (Θ3 )1 ⊗ Θ12 Ω2 θ0 [−1]

⊗S

−1

(5.6.25)

2 (θ1 )Ω4 S−1 (Θ3 )2

2 ⊗ Θ2 Ω3 θ0 = ω 1 ⊗ ω 5 ⊗ ω 2 Θ1 ⊗ S−1 (Θ3 )ω 4 ⊗ ω 3 Θ2 . [0]

(5.6.26)

Proof The relation (5.6.25) follows by applying (3.2.13), (4.4.3) and (4.4.2), we leave the details to the reader. We prove now (5.6.26). We compute: 2 Θ11 Ω1 θ 1 ⊗ S−1 (θ 3 )Ω5 S−1 (Θ3 )1 ⊗ Θ12 Ω2 θ0 [−1] 2 2 ⊗ S−1 (θ1 )Ω4 S−1 (Θ3 )2 ⊗ Θ2 Ω3 θ0 [0] (5.6.25)

=

2 Θ11 x˜λ1 θ 1 ⊗ S−1 ( f 2 X˜ρ3 Θ3 θ 3 ) ⊗ Θ12 x˜λ2 Θ θ0 [−1] 1

2 2 ⊗ S−1 ( f 1 X˜ρ2 Θ21 (x˜λ3 )1 Θ θ1 ) ⊗ X˜ρ1 Θ20 (x˜λ3 )0 Θ θ0 [0] 3

(4.4.2)

=

2

˜ 1 θ 1 ⊗ S−1 ( f 2 X˜ρ3 (x˜3 )1 Θ3 Θ ˜ 3 θ 3 ) ⊗ x˜2 Θ1 Θ ˜ 2 Θ θ2 x˜λ1 Θ 0[−1] λ [−1] λ 1

˜ 2 Θ3 θ 2 ) ⊗ X˜ρ1 (x˜3 )0 Θ2 Θ ˜ 2 Θ2 θ 2 ⊗ S−1 ( f 1 X˜ρ2 (x˜λ3 )01 Θ21 Θ 1 0[0] [0]1 λ 0 0 [0]0 (4.4.1)

=

x˜λ1 ⊗ S−1 ( f 2 X˜ρ3 (x˜λ3 )1 Θ3 ) ⊗ x˜λ2 Θ1 Θ

1

3 2 ⊗ S−1 ( f 1 X˜ρ2 (x˜λ3 )01 Θ21 Θ ) ⊗ X˜ρ1 (x˜λ3 )00 Θ20 Θ (4.3.1)

=

1 x˜λ1 ⊗ S−1 ( f 2 (x˜λ3 )12 X˜ρ3 Θ3 ) ⊗ x˜λ2 Θ1 Θ 3 2 ⊗ S−1 ( f 1 (x˜λ3 )11 X˜ρ2 Θ21 Θ ) ⊗ (x˜λ3 )0 X˜ρ1 Θ20 Θ

(5.6.10)

=

ω 1 ⊗ ω 5 ⊗ ω 2 Θ ⊗ S−1 (Θ )ω 4 ⊗ ω 3 Θ ,

as required.

1

3

2

210

Crossed Products

Proposition 5.57 Let H be a quasi-Hopf algebra with bijective antipode and A an H-bicomodule algebra. Then the left H ⊗ H op -comodule algebras A1 and A2 are twist equivalent. More precisely, for the element U ∈ (H ⊗ H op ) ⊗ A given by U = (Θ1 ⊗ S−1 (Θ3 )) ⊗ Θ2 , we have

λ2 (u) = U λ1 (u)U −1 , ∀ u ∈ A, Φλ2 = (1 ⊗U)(Id ⊗ λ1 )(U)Φλ1 (Δ ⊗ Id)(U −1 ). Proof The first relation follows immediately from (4.4.1), and the second is equivalent to the relation (5.6.26) proved in the previous lemma. As a consequence of these results and (5.6.23), we obtain: Corollary 5.58 Let H be a quasi-Hopf algebra with bijective antipode, A an Hbimodule algebra and A an H-bicomodule algebra. Then the two left (resp. right) diagonal crossed products A  A and A  A (resp. A  A and A  A ) are isomorphic as algebras, and moreover they are equivalent extensions of A. Remark 5.59 Let H be a quasi-Hopf algebra with bijective antipode, A an Hbimodule algebra and A an H-bicomodule algebra with Φλ ,ρ = 1H ⊗ 1A ⊗ 1H . Then, by (4.4.1), it follows that (λ ⊗IdH )◦ ρ = (IdH ⊗ ρ )◦ λ , and by (5.6.26) it follows that Ω = ω . So, in this case we have that A  A and A  A are not only isomorphic, but actually coincide, and A1 and A2 also coincide. Our aim now is to show that the left generalized diagonal crossed products are isomorphic, as algebras, to the right generalized diagonal crossed products. Proposition 5.60 Let H be a quasi-Hopf algebra with bijective antipode, (δ , Ψ) a two-sided coaction of H on an algebra A, and A an H-bimodule algebra. Then the linear map ϑ : A  δ A → A  δ A defined for all ϕ ∈ A and u ∈ A by

ϑ (ϕ  δ u) = q2δ u(0)  S−1 (q1δ u(−1) ) · ϕ · q3δ u(1) is an algebra isomorphism, where qδ = q1δ ⊗q2δ ⊗q3δ is the element defined in (4.4.16). In particular, if A is an H-bicomodule algebra then we get that all four diagonal crossed products A  A, A  A , A  A and A  A are isomorphic as associative unital algebras. Proof We show that ϑ is multiplicative. For any ϕ , ϕ  ∈ A and u, u ∈ A we have (we denote by Q1δ ⊗ Q2δ ⊗ Q3δ another copy of qδ and by F 1 ⊗ F 2 another copy of f ):

ϑ ((ϕ  δ u)(ϕ   δ u ))   (5.6.3),(5.6.1) 1 5 2 4 3 = ϑ (Ψ · ϕ · S−1 ( f 2 Ψ ))(Ψ u(−1) · ϕ  · S−1 ( f 1 Ψ u(1) ))  δ Ψ u(0) u  (4.1.14) 3 3 1 = q2δ Ψ(0) u(0,0) u(0)  δ S−1 (F 2 (q1δ )2 Ψ(−1)2 u(0,−1)2 u(−1)2 g2 )Ψ (3.2.13)  5 3 · ϕ · S−1 ( f 2 Ψ )(q3δ )1 Ψ(1)1 u(0,1)1 u(1)1  3 S−1 (F 1 (q1δ )1 Ψ(−1)1 u(0,−1)1 u(−1)1 g1 )

5.6 Diagonal Crossed Products

211



Ψ u(−1) · ϕ  · S−1 ( f 1 Ψ u(1) )(q3δ )2 Ψ(1)2 u(0,1)2 u(1)2  (4.4.22) = q2δ (Q2δ )(0) Ψ3 u(0,0) u(0)  δ S−1 (q1δ (Q2δ )(−1) Ψ2 u(0,−1)2 u(−1)2 g2 ) · ϕ  · q3δ (Q2δ )(1) Ψ4 u(0,1)1 u(1)1 S−1 (Q1δ Ψ1 u(0,−1)1 u(−1)1 g1 )u(−1) · ϕ   · S−1 (u(1) )Q3δ Ψ5 u(0,1)2 u(1)2  (4.4.23) 3 2 = q2δ u(0) Ψ u(0)  δ S−1 (q1δ q2L u(−1)(2,2) Ψ2 u(−1)2 g2 ) (4.4.6)  4 2 1 · ϕ · q3δ q1R u(1)(1,1) Ψ1 u(1)1 S−1 (q1L u(−1)(2,1) Ψ1 u(−1)1 g1 )u(−1)1 Ψ · ϕ   5 4 · S−1 (u(1)2 Ψ )q2R u(1)(1,2) Ψ2 u(1)2  (3.2.22),(3.2.21) 2 = qδ u(0) (Q2δ )(0) Ψ3 u(0)  δ S−1 (q1δ u(−1) (Q2δ )(−1) Ψ2 u(−1)2 g2 ) · ϕ (4.4.23)   · q3δ u(1) (Q2δ )(1) Ψ4 u(1)1 S−1 (Q1δ Ψ1 u(−1)1 g1 ) · ϕ  · Q3δ Ψ5 u(1)2 2

(4.4.5)

4

3

q2δ u(0) (Q2δ )(0) u(0,0) Ω3  δ (Ω2 S−1 (q1δ u(−1) (Q2δ )(−1) u(0,−1) ) · ϕ

=

(5.6.2)

· q3δ u(1) (Q2δ )(1) u(0,1) Ω4 )(Ω1 S−1 (Q1δ u(−1) ) · ϕ  · Q3δ u(1) Ω5 )

(5.6.4)

(q2δ u(0)  δ S−1 (q1δ u(−1) ) · ϕ · q3δ u(1) )

=

(Q2δ u(0)  δ S−1 (Q1δ u(−1) ) · ϕ  · Q3δ u(1) )

ϑ (ϕ  δ u)ϑ (ϕ   δ u ). QED

=

One can see that the unit and counit properties imply ϑ (1A  δ 1A ) = 1A  δ 1A , so it remains to show that ϑ is bijective. For this, define ϑ −1 : A  δ A → A  δ A given for all u ∈ A and ϕ ∈ A by

ϑ −1 (u  δ ϕ ) = u(−1) p1δ · ϕ · S−1 (u(1) p3δ )  δ u(0) p2δ , where pδ = p1δ ⊗ p2δ ⊗ p3δ is the element defined in (4.4.15). We claim that ϑ and ϑ −1 are inverses. Indeed, ϑ ◦ ϑ −1 = IdA δ A because of (4.4.18) and (4.4.20), and ϑ ◦ ϑ −1 = IdA  δ A because of (4.4.17) and (4.4.19) (we leave the verification of the details to the reader). We now present a Universal Property of the diagonal crossed product. Proposition 5.61 Let H be a quasi-Hopf algebra with bijective antipode, A an Hbimodule algebra, A an H-bicomodule algebra, B an algebra, γ : A → B an algebra map and v : A → B a linear map such that the following conditions are satisfied:

γ (u0 )v(ϕ · u1 ) = v(u[−1] · ϕ )γ (u[0] ), v(ϕϕ  ) = γ (X˜ρ1 )v(θ 1 X˜λ1 · ϕ · X˜ρ2 )γ (θ 2 )v(X˜λ2 · ϕ  · X˜ρ3 θ 3 )γ (X˜ 3 ),

(5.6.27)

v(1A ) = 1B ,

(5.6.29)

λ

ϕ, ϕ

(5.6.28)

for all ∈ A and u ∈ A. Consider the algebra map j : A → A  A, j(u) = 1A  u, and the map Γ : A → A  A defined in Proposition 5.51. Then there exists a unique algebra map w : A  A → B such that w ◦ Γ = v and w ◦ j = γ . Moreover,

212

Crossed Products

w is given by the formula w(ϕ  u) = γ (q˜1ρ )v(ϕ · q˜2ρ )γ (u),

(5.6.30)

for all ϕ ∈ A and u ∈ A, where q˜ρ = q˜1ρ ⊗ q˜2ρ is given by formula (4.3.9). Proof We first prove the uniqueness of w. From the proof of Proposition 5.51, we know that ϕ  u = (1A  q˜1ρ )Γ(ϕ · q˜2ρ )(1A  u), for all ϕ ∈ A and u ∈ A, hence we can write w(ϕ  u) = w( j(q˜1ρ )Γ(ϕ · q˜2ρ ) j(u)) = w( j(q˜1ρ ))w(Γ(ϕ · q˜2ρ ))w( j(u)) = γ (q˜1ρ )v(ϕ · q˜2ρ )γ (u), showing that w is unique. We prove the existence part. Define w by formula (5.6.30); it is obvious that w is unital and satisfies w ◦ j = γ . We check that w ◦ Γ = v: (w ◦ Γ)(ϕ )

=

w(( p˜1ρ )[−1] · ϕ · S−1 ( p˜2ρ )  ( p˜1ρ )[0] )

=

γ (q˜1ρ )v(( p˜1ρ )[−1] · ϕ · S−1 ( p˜2ρ )q˜2ρ )γ (( p˜1ρ )[0] )

(5.6.27)

=

(4.3.13)

γ (q˜1ρ )γ (( p˜1ρ )0 )v(ϕ · S−1 ( p˜2ρ )q˜2ρ ( p˜1ρ )1 ) = v(ϕ ).

Thus, the only thing left to prove is that w is multiplicative. We denote by Q˜ 1ρ ⊗ Q˜ 2ρ another copy of the element q˜ρ , and we record the obvious relation Q˜ 1ρ x˜ρ1 ⊗ S−1 (x˜ρ3 )Q˜ 2ρ x˜ρ2 = 1H ⊗ S−1 (α ). Now we compute: w((ϕ  u)(ϕ   u )) =

γ (q˜1ρ )v([Ω1 · ϕ · Ω5 (q˜2ρ )1 ][Ω2 u0[−1] · ϕ  · S−1 (u1 )Ω4 (q˜2ρ )2 ]) γ (Ω3 u0[0] u )

(5.6.28)

=

γ (q˜1ρ )γ (X˜ρ1 )v(θ 1 X˜λ1 Ω1 · ϕ · Ω5 (q˜2ρ )1 X˜ρ2 )γ (θ 2 ) v(X˜λ2 Ω2 u0[−1] · ϕ  · S−1 (u1 )Ω4 (q˜2ρ )2 X˜ρ3 θ 3 )γ (X˜λ3 )γ (Ω3 u0[0] u )

(5.6.9)

1 γ (q˜1ρ X˜ρ1 )v(θ 1 (Y˜ρ1 )[−1] θ · ϕ · S−1 ( f 2Y˜ρ3 )(q˜2ρ )1 X˜ρ2 )γ (θ 2 )   2 3 v (Y˜ρ1 )[0][−1] θ [−1] u0[−1] · ϕ  · S−1 ( f 1Y˜ρ2 θ u1 )(q˜2ρ )2 X˜ρ3 θ 3   2 γ (Y˜ρ1 )[0][0] θ [0] u0[0] γ (u )  (5.6.27) 1 = γ (q˜1ρ X˜ρ1 )v θ 1 (Y˜ρ1 )[−1] θ · ϕ  2 · S−1 ( f 2Y˜ρ3 )(q˜2ρ )1 X˜ρ2 )γ (θ 2 (Y˜ρ1 )[0]0 θ 0 u00   3 2 v ϕ  · S−1 ( f 1Y˜ρ2 θ u1 )(q˜2ρ )2 X˜ρ3 θ 3 (Y˜ρ1 )[0]1 θ 1 u00 γ (u )  (4.4.1) 1 = γ (q˜1ρ X˜ρ1 )v (Y˜ρ1 )0[−1] θ 1 θ · ϕ

=

(5.6.31)

5.6 Diagonal Crossed Products

(4.4.3)

=

213



2 · S−1 ( f 2Y˜ρ3 )(q˜2ρ )1 X˜ρ2 )γ ((Y˜ρ1 )0[0] θ 2 θ 0 u00   3 2 v ϕ  · S−1 ( f 1Y˜ρ2 θ u1 )(q˜2ρ )2 X˜ρ3 (Y˜ρ1 )1 θ 3 θ 1 u01 γ (u )   γ (q˜1ρ X˜ρ1 )v (Y˜ρ1 )0[−1] (y˜1ρ )[−1] θ 1 · ϕ · S−1 ( f 2Y˜ρ3 )(q˜2ρ )1 X˜ρ2   γ (Y˜ρ1 )0[0] (y˜1ρ )[0] θ 2 Z˜ ρ1 u00   v ϕ  · S−1 ( f 1Y˜ρ2 y˜3ρ θ23 Z˜ ρ3 u1 )(q˜2ρ )2 X˜ρ3 (Y˜ρ1 )1 y˜2ρ θ13 Z˜ ρ2 u01 γ (u )

(5.6.27)

γ (q˜1ρ X˜ρ1 (Y˜ρ1 )00 (y˜1ρ )0 )   v θ 1 · ϕ · S−1 ( f 2Y˜ρ3 )(q˜2ρ )1 X˜ρ2 (Y˜ρ1 )01 (y˜1ρ )1 γ (θ 2 Z˜ ρ1 u00 )   v ϕ  · S−1 ( f 1Y˜ρ2 y˜3ρ θ23 Z˜ ρ3 u1 )(q˜2ρ )2 X˜ρ3 (Y˜ρ1 )1 y˜2ρ θ13 Z˜ ρ2 u01 γ (u )   (4.3.1) = γ (q˜1ρ (Y˜ρ1 )0 y˜1ρ X˜ρ1 )v θ 1 · ϕ · S−1 ( f 2Y˜ρ3 )(q˜2ρ )1 (Y˜ρ1 )11 (y˜2ρ )1 x1 X˜ρ2 (4.3.2)  γ (θ 2 Z˜ ρ1 u00 )v ϕ  · S−1 ( f 1Y˜ρ2 y˜3ρ x3 (X˜ρ3 )2 θ23 Z˜ ρ3 u1 )  (q˜2ρ )2 (Y˜ρ1 )12 (y˜2ρ )2 x2 (X˜ρ3 )1 θ13 Z˜ ρ2 u01 γ (u )   (4.3.1) = γ (q˜1ρ (Q˜ 1ρ )0 x˜ρ1 y˜1ρ X˜ρ1 )v θ 1 · ϕ · q˜2ρ (Q˜ 1ρ )1 x˜ρ2 (y˜2ρ )1 x1 X˜ρ2 γ (θ 2 u0 Z˜ ρ1 ) (4.3.15)   v ϕ  · S−1 (y˜3ρ x3 (X˜ρ3 )2 θ23 u12 Z˜ ρ3 )Q˜ 2ρ x˜ρ3 (y˜2ρ )2 x2 (X˜ρ3 )1 θ13 u11 Z˜ ρ2 γ (u )   (4.3.2) = γ (q˜1ρ (Q˜ 1ρ )0 (x˜ρ1 )0 )v θ 1 · ϕ · q˜2ρ (Q˜ 1ρ )1 (x˜ρ1 )1 γ (θ 2 u0 Z˜ ρ1 )   v ϕ  · S−1 (x˜ρ3 θ23 u12 Z˜ ρ3 )Q˜ 2ρ x˜ρ2 θ13 u11 Z˜ ρ2 γ (u ) =

(5.6.31)

γ (q˜1ρ )v(ϕ · q˜2ρ )γ (uZ˜ ρ1 )v(ϕ  · S−1 (Z˜ ρ3 )S−1 (α )Z˜ ρ2 )γ (u ) = γ (q˜1ρ )v(ϕ · q˜2ρ )γ (u)γ (Q˜ 1ρ )v(ϕ  · Q˜ 2ρ )γ (u ) =

=

w(ϕ  u)w(ϕ   u ),

finishing the proof. As a consequence of Proposition 5.61, we immediately obtain a new kind of Universal Property for the quasi-Hopf smash product: Proposition 5.62 Let H be a quasi-Hopf algebra with bijective antipode and A a left H-module algebra. Denote by i : A → A#H, i(a) = a#1H and j : H → A#H, j(h) = 1A #h. Let B be an algebra, γ : H → B an algebra map and v : A → B a linear map satisfying the following conditions, for all a, a ∈ A and h ∈ H:

γ (h)v(a) = v(h1 · a)γ (h2 ), 



(5.6.32)

v(aa ) = v(X · a)v(X · a )γ (X ),

(5.6.33)

v(1A ) = 1B .

(5.6.34)

1

2

3

214

Crossed Products

Then there exists a unique algebra map w : A#H → B such that w◦i = v and w◦ j = γ . Moreover, w is given by the formula w(a#h) = v(a)γ (h), for a ∈ A, h ∈ H. Proposition 5.62 may be easily extended to a Universal Property of the two-sided smash product (the proof is left to the reader): Proposition 5.63 Let H be a quasi-Hopf algebra with bijective antipode, A a left H-module algebra and B a right H-module algebra. Denote by iA , iB , j the standard inclusions of A, B and H, respectively, into A#H#B. Let X be an algebra, γ : H → X an algebra map and vA : A → X, vB : B → X two linear maps satisfying the conditions:

γ (h)vA (a) = vA (h1 · a)γ (h2 ), vA (aa ) = vA (X 1 · a)vA (X 2 · a )γ (X 3 ), vB (b)γ (h) = γ (h1 )vB (b · h2 ), vB (bb ) = γ (X 1 )vB (b · X 2 )vB (b · X 3 ), vA (1A ) = 1X = vB (1B ), vB (b)vA (a) = vA (x1 · a)γ (x2 )vB (b · x3 ), for all a, a ∈ A, b, b ∈ B and h ∈ H. Then there exists a unique algebra map w : A#H#B → X such that w ◦ iA = vA , w ◦ iB = vB and w ◦ j = γ . Moreover, w is given by the formula w(a#h#b) = vA (a)γ (h)vB (b), for all a ∈ A, h ∈ H, b ∈ B.

5.7 L–R-smash Products We introduce a new type of crossed product associated to a quasi-bialgebra, which in the case of a quasi-Hopf algebra with bijective antipode will turn out to be isomorphic to a diagonal crossed product. Proposition 5.64 Let H be a quasi-bialgebra, A an H-bimodule algebra and A an H-bicomodule algebra. Define on A ⊗ A the product (ϕ u)(ψ u ) = (x˜λ1 · ϕ · θ 3 u1 x˜ρ2 )(x˜λ2 u[−1] θ 1 · ψ · x˜ρ3 ) x˜λ3 u[0] θ 2 u0 x˜ρ1 ,

(5.7.1)

−1 −1 1 2 3 1 2 3 for ϕ , ψ ∈ A and u, u ∈ A, where Φ−1 ρ = x˜ρ ⊗ x˜ρ ⊗ x˜ρ , Φλ = x˜λ ⊗ x˜λ ⊗ x˜λ , Φλ ,ρ = 1 2 3 θ ⊗ θ ⊗ θ , and we write ϕ u instead of ϕ ⊗ u to distinguish the new algebraic structure. Then this product defines on A ⊗ A a structure of an associative algebra with unit 1A 1A , denoted by A A and called the L–R-smash product.

Proof

For ϕ , ψ , ξ ∈ A and u, u , u ∈ A we compute:

[(ϕ u)(ψ u )](ξ u ) (5.7.1)

[(x˜λ1 · ϕ · θ 3 u1 x˜ρ2 )(x˜λ2 u[−1] θ 1 · ψ · x˜ρ3 ) x˜λ3 u[0] θ 2 u0 x˜ρ1 ](ξ u )

(5.7.1)

[((y˜1λ )1 x˜λ1 · ϕ · θ 3 u1 x˜ρ2 θ 1 u11 (y˜2ρ )1 )((y˜1λ )2 x˜λ2 u[−1] θ 1 · ψ

= =

3

3

1

2 · x˜ρ3 θ 2 u12 (y˜2ρ )2 )](y˜2λ (x˜λ3 )[−1] u[0,−1] θ[−1] u0[−1] (x˜ρ1 )[−1] θ · ξ · y˜3ρ ) 2

2 

y˜3λ (x˜λ3 )[0] u[0,0] θ[0] u0[0] (x˜ρ1 )[0] θ u0 y˜1ρ (4.3.6)

=

(4.4.3)

2

[(t 1 y˜1λ · ϕ · θ 3 u1 θ˜ 3 (θ )1 x˜ρ2 u11 (y˜2ρ )1 )(t 2 (y˜2λ )1 x˜λ1 u[−1] θ 1 · ψ

5.7 L–R-smash Products

215

3

1

2 u0[−1] θ˜ 1 θ · ξ · y˜3ρ ) · θ x˜ρ3 u12 (y˜2ρ )2 )](t 3 (y˜2λ )2 x˜λ2 u[0,−1] θ[−1] 2

2 

y˜3λ x˜λ3 u[0,0] θ[0] u0[0] θ˜ 2 (θ )0 x˜ρ1 u0 y˜1ρ (4.1.13),(4.3.1)

=

(4.3.5)

2

(y˜1λ · ϕ · θ 3 u1 θ˜ 3 (θ )1 u0,1 x˜ρ2 (y˜2ρ )1t 1 )[((y˜2λ )1 u[−1]1 x˜λ1 θ 1 · ψ 3

1

2 · θ u1 x˜ρ3 (y˜2ρ )2t 2 )((y˜2λ )2 u[−1]2 x˜λ2 θ[−1] u0[−1] θ˜ 1 θ · ξ · y˜3ρ t 3 )] 2

2 

y˜3λ u[0] x˜λ3 θ[0] u0[0] θ˜ 2 (θ )0 u0,0 x˜ρ1 y˜1ρ (4.4.1)

=

2

(y˜1λ · ϕ · θ 3 θ˜ 3 u[0]1 (θ )1 u0,1 x˜ρ2 (y˜2ρ )1t 1 )[((y˜2λ )1 u[−1]1 x˜λ1 θ 1 · ψ 3

1

2 ·θ u1 x˜ρ3 (y˜2ρ )2t 2 )((y˜2λ )2 u[−1]2 x˜λ2 θ[−1] θ˜ 1 u[−1] θ · ξ · y˜3ρ t 3 )] 2

2 ˜2  θ u[0]0 (θ )0 u0,0 x˜ρ1 y˜1ρ

y˜3λ u[0] x˜λ3 θ[0] (4.3.2)

=

(4.4.2)

2

(y˜1λ · ϕ · θ 3 (x˜λ3 )1 u[0]1 (θ )1 u0,1 (x˜ρ1 )1 y˜2ρ )[((y˜2λ )1 u[−1]1 θ11 x˜λ1 · ψ 3

1

·θ u1 x˜ρ2 (y˜3ρ )1 )((y˜2λ )2 u[−1]2 θ21 x˜λ2 u[−1] θ · ξ · x˜ρ3 (y˜3ρ )2 )] 2

y˜3λ u[0] θ 2 (x˜λ3 )0 u[0]0 (θ )0 u0,0 (x˜ρ1 )0 y˜1ρ (5.7.1)

(ϕ u)[(x˜λ1 · ψ · θ u1 x˜ρ2 )(x˜λ2 u[−1] θ · ξ · x˜ρ3 ) x˜λ3 u[0] θ u0 x˜ρ1 ]

(5.7.1)

(ϕ u)[(ψ u )(ξ u )],

= =

3

1

2

hence the multiplication is associative. It is easy to check that 1A 1A is the unit. Remark 5.65 It is easy to see that, in A A, we have (1A u)(1A u ) = 1A uu for all u, u ∈ A, hence the map A → A A, u → 1A u, is an algebra map, and (ϕ 1A )(1A u) = ϕ · u1 u0 , for all ϕ ∈ A and u ∈ A. The examples below justify the name of this construction. Examples 5.66 (1) Let A be a left H-module algebra. Then A becomes an Hbimodule algebra, with right H-action given via ε . In this case the multiplication of A A becomes (a u)(a u ) = (x˜λ1 · a)(x˜λ2 u[−1] · a ) x˜λ3 u[0] u , for all a, a ∈ A and u, u ∈ A, hence in this case A A coincides with the generalized smash product A < A. (2) As we know, H itself is an H-bicomodule algebra. So, in this case, the multiplication of A H specializes to (ϕ h)(ψ h ) = (x1 · ϕ · t 3 h2 y2 )(x2 h1t 1 · ψ · y3 ) x3 h2t 2 h1 y1 ,

(5.7.2)

for all ϕ , ψ ∈ A and h, h ∈ H. If the right H-module structure of A is trivial, then A H coincides with the smash product A #H. Next, we show that either a two-sided smash product or a two-sided crossed product can be identified, up to isomorphism, to certain L–R-smash products.

216

Crossed Products

Proposition 5.67 Let H be a quasi-bialgebra, A a left H-module algebra, B a right H-module algebra and A an H-bicomodule algebra. If we consider A ⊗ B as an H-bimodule algebra, with H-actions h · (a ⊗ b) · h = h · a ⊗ b · h , ∀ a ∈ A, h, h ∈ H, b ∈ B,

(5.7.3)

then we have an algebra isomorphism

φ : (A ⊗ B) A ∼ = A < A > B, φ ((a ⊗ b) u) = a < u > b, ∀ a ∈ A, b ∈ B, u ∈ A. Proof

We compute:

φ ([(a ⊗ b) u)][(a ⊗ b ) u ]) = φ ((x˜λ1 · (a ⊗ b) · θ 3 u1 x˜ρ2 )(x˜λ2 u[−1] θ 1 · (a ⊗ b ) · x˜ρ3 ) x˜λ3 u[0] θ 2 u0 x˜ρ1 ) = φ ((x˜λ1 · a ⊗ b · θ 3 u1 x˜ρ2 )(x˜λ2 u[−1] θ 1 · a ⊗ b · x˜ρ3 ) x˜λ3 u[0] θ 2 u0 x˜ρ1 ) = φ (((x˜λ1 · a)(x˜λ2 u[−1] θ 1 · a ) ⊗ (b · θ 3 u1 x˜ρ2 )(b · x˜ρ3 )) x˜λ3 u[0] θ 2 u0 x˜ρ1 ) = (x˜λ1 · a)(x˜λ2 u[−1] θ 1 · a ) < x˜λ3 u[0] θ 2 u0 x˜ρ1 > (b · θ 3 u1 x˜ρ2 )(b · x˜ρ3 )

= (a < u > b)(a < u > b ) = φ ((a ⊗ b) u)φ ((a ⊗ b ) u ), finishing the proof.

Proposition 5.68 Let H be a quasi-bialgebra, A a right H-comodule algebra, B a left H-comodule algebra and A an H-bimodule algebra. Consider A ⊗ B as an H-bicomodule algebra, with the following structure: ρ (a ⊗ b) = (a0 ⊗ b) ⊗ a1 , λ (a⊗b) = b[−1] ⊗(a⊗b[0] ), Φρ = (X˜ρ1 ⊗1B )⊗ X˜ρ2 ⊗ X˜ρ3 , Φλ = X˜λ1 ⊗ X˜λ2 ⊗(1A ⊗ X˜λ3 ), Φλ ,ρ = 1H ⊗ (1A ⊗ 1B ) ⊗ 1H , for all a ∈ A and b ∈ B. Then we have an algebra isomorphism

τ : A (A ⊗ B) ∼ = A > A < B, τ (ϕ (a ⊗ b)) = a > ϕ < b, ∀ ϕ ∈ A , a ∈ A, b ∈ B. Proof

We compute:

τ ((ϕ (a ⊗ b))(ϕ  (a ⊗ b ))) = τ ((x˜λ1 · ϕ · (a ⊗ b )1 x˜ρ2 )(x˜λ2 (a ⊗ b)[−1] · ϕ  · x˜ρ3 )

(1A ⊗ x˜λ3 )(a ⊗ b)[0] (a ⊗ b )0 (x˜ρ1 ⊗ 1B )) = τ ((x˜λ1 · ϕ · a1 x˜ρ2 )(x˜λ2 b[−1] · ϕ  · x˜ρ3 ) (1A ⊗ x˜λ3 )(a ⊗ b[0] )(a0 ⊗ b )(x˜ρ1 ⊗ 1B )) = τ ((x˜λ1 · ϕ · a1 x˜ρ2 )(x˜λ2 b[−1] · ϕ  · x˜ρ3 ) (aa0 x˜ρ1 ⊗ x˜λ3 b[0] b )) = aa0 x˜ρ1 > (x˜λ1 · ϕ · a1 x˜ρ2 )(x˜λ2 b[−1] · ϕ  · x˜ρ3 ) < x˜λ3 b[0] b

= (a > ϕ < b)(a > ϕ  < b ) = τ (ϕ (a ⊗ b))τ (ϕ  (a ⊗ b )), and the proof is finished.

5.7 L–R-smash Products

217

Lemma 5.69 Let H be a quasi-Hopf algebra with bijective antipode and A an Hbicomodule algebra. Consider the element Ω ∈ H ⊗2 ⊗ A ⊗ H ⊗2 given by (5.6.9). If we denote by Q˜ 1ρ ⊗ Q˜ 2ρ another copy of the element q˜ρ given by formula (4.3.9) and by 1 ⊗ 2 ⊗ 3 another copy of Φλ ,ρ , then we have: Θ11 Ω1 ⊗ Θ12 Ω2 ⊗ q˜1ρ (Θ2 Ω3 )0 ⊗ Ω5 S−1 (Θ3 )1 (q˜2ρ )1 (Θ2 Ω3 )11 ⊗Ω4 S−1 (Θ3 )2 (q˜2ρ )2 (Θ2 Ω3 )12 1 2 = x˜1λ Θ1 ⊗ x˜2λ 1 Θ2[−1] Θ ⊗ x˜3λ q˜1ρ (2 Θ2[0] Q˜ 1ρ Θ0 )0 x˜1ρ 2 3 2 ⊗ S−1 (3 Θ3 )q˜2ρ (2 Θ2[0] Q˜ 1ρ Θ0 )1 x˜2ρ ⊗ S−1 (Θ )Q˜ 2ρ Θ1 x˜3ρ , (5.7.4) 1 2 2 Θ u0[−1] ⊗ (Q˜ 1ρ Θ0 )0 x˜1ρ u0[0] u 0 ⊗ (Q˜ 1ρ Θ0 )1 x˜2ρ u0[0] 0

3 2 ⊗ S−1 (Θ u1 )Q˜ 2ρ Θ1 x˜3ρ u0[0]

12

11

u 11

1 2 u 12 = u[−1] Θ ⊗ (u[0] Q˜ 1ρ )0 (Θ u )0,0 x˜1ρ

⊗ (u[0] Q˜ 1ρ )1 (Θ u )0,1 x˜2ρ ⊗ S−1 (Θ )Q˜ 2ρ (Θ u )1 x˜3ρ , 2

3

2

(5.7.5)

Θ1 ⊗ q˜1ρ Θ20 ⊗ S−1 (Θ3 )q˜2ρ Θ21 = (q˜1ρ )[−1] θ 1 ⊗ (q˜1ρ )[0] θ 2 ⊗ q˜2ρ θ 3 .

(5.7.6)

Proof We only indicate the main steps and leave details to the reader. The relation (5.7.4) follows by using (5.6.25), (4.3.15), (4.4.2) and using (4.3.11) and (4.3.1) several times. By using (4.3.1), (4.4.1) and (4.3.11) one obtains (5.7.5). Finally, (5.7.6) follows by using (4.3.9), (4.4.3), (3.2.1) and (4.4.4). Theorem 5.70 Let H be a quasi-Hopf algebra with bijective antipode, A an Hbimodule algebra and A an H-bicomodule algebra. Then the linear map

ν : A  A → A A, ν (ϕ  u) = Θ1 · ϕ · S−1 (Θ3 )q˜2ρ Θ21 u1 q˜1ρ Θ20 u0 , (5.7.7) for all ϕ ∈ A and u ∈ A, is an algebra isomorphism, with inverse

ν −1 : A A → A  A, ν −1 (ϕ u) = θ 1 · ϕ · S−1 (θ 3 u1 p˜2ρ )  θ 2 u0 p˜1ρ . (5.7.8) Proof

First we establish that ν is an algebra map. We compute:

ν ((ϕ  u)(ψ  u )) (5.6.15)

=

(5.7.7)

(5.7.4)

=

(Θ11 Ω1 · ϕ · Ω5 S−1 (Θ3 )1 (q˜2ρ )1 (Θ2 Ω3 )11 u0[0] u 11 ) 11  1 2 4 −1 3 −1 2 Θ2 Ω u0[−1] · ψ · S (u1 )Ω S (Θ )2 (q˜ρ )2 (Θ2 Ω3 )12 u0[0]

12

2

x˜3λ q˜1ρ 20 Θ2[0]0 (Q˜ 1ρ Θ0 )0 x˜1ρ u0[0] u 0 0

(5.7.5)

=

12

q˜1ρ (Θ2 Ω3 )0 u0[0] u 0 0   2 1 1 3 3 2 2 −1 x˜λ Θ · ϕ · S ( Θ )q˜ρ 1 Θ2[0]1 (Q˜ 1ρ Θ0 )1 x˜2ρ u0[0] u 11 11   1 3 2 3 2 1 2 −1 2 ˜ x˜λ  Θ[−1] Θ u0[−1] · ψ · S (Θ u1 )Qρ Θ1 x˜ρ u0[0] u 12   2 x˜1λ Θ1 · ϕ · S−1 (3 Θ3 )q˜2ρ 21 Θ2[0]1 (u[0] Q˜ 1ρ )1 (Θ u )0,1 x˜2ρ

u 12



218

Crossed Products   1 3 2 x˜2λ 1 Θ2[−1] u[−1] Θ · ψ · S−1 (Θ )Q˜ 2ρ (Θ u )1 x˜3ρ

(5.7.6)

=

(4.4.1)

=

(5.7.1)

=

(5.7.7)

=

2

x˜3λ q˜1ρ 20 Θ2[0]0 (u[0] Q˜ 1ρ )0 (Θ u )0,0 x˜1ρ   2 x˜1λ Θ1 · ϕ · S−1 (Θ3 )q˜2ρ θ 3 Θ2[0]1 (u[0] Q˜ 1ρ )1 (Θ u )0,1 x˜2ρ   1 3 2 x˜2λ (q˜1ρ )[−1] θ 1 Θ2[−1] u[−1] Θ · ψ · S−1 (Θ )Q˜ 2ρ (Θ u )1 x˜3ρ 2

x˜3λ (q˜1ρ )[0] θ 2 Θ2[0]0 (u[0] Q˜ 1ρ )0 (Θ u )0,0 x˜1ρ   2 x˜1λ Θ1 · ϕ · S−1 (Θ3 )q˜2ρ Θ21 u1 θ 3 (Q˜ 1ρ )1 (Θ u )0,1 x˜2ρ   1 3 2 x˜2λ (q˜1ρ )[−1] Θ20[−1] u0[−1] θ 1 Θ · ψ · S−1 (Θ )Q˜ 2ρ (Θ u )1 x˜3ρ 2

x˜3λ (q˜1ρ )[0] Θ20[0] u0[0] θ 2 (Q˜ 1ρ )0 (Θ u )0,0 x˜1ρ   Θ1 · ϕ · S−1 (Θ3 )q˜2ρ Θ21 u1 q˜1ρ Θ20 u0   1 3 2 2 Θ · ψ · S−1 (Θ )Q˜ 2ρ Θ1 u1 Q˜ 1ρ Θ0 u0

ν (ϕ  u)ν (ψ  u ),

as needed. The fact that ν (1A  1A ) = 1A 1A is trivial. We prove now that ν and ν −1 are inverses. Indeed, we have: 2 νν −1 (ϕ u) = Θ1 θ 1 · ϕ · S−1 (θ 3 u1 p˜2ρ )S−1 (Θ3 )q˜2ρ Θ21 θ1 u0,1 ( p˜1ρ )1 (4.3.11)

2

q˜1ρ Θ20 θ0 u0,0 ( p˜1ρ )0 = ϕ u, (4.3.13)

and similarly

ν −1 ν (ϕ  u) θ 1 Θ1 · ϕ · S−1 (Θ3 )q˜2ρ Θ21 u1 S−1 (θ 3 (q˜1ρ )1 Θ20,1 u0,1 p˜2ρ )

=

 θ 2 (q˜1ρ )0 Θ20,0 u0,0 p˜1ρ (4.3.10),(4.3.12)

=

θ 1 Θ1 · ϕ · S−1 (θ 3 Θ3 )  θ 2 Θ2 u = ϕ  u,

so we are done. Examples 5.71 (1) If A is a left H-module algebra regarded as an H-bimodule algebra with trivial right H-action, then A  A and A A both coincide with A < A, and the isomorphism ν is just the identity. (2) If A = H then ν : A  H → A H and ν −1 : A H → A  H are given by

ν (ϕ  h) = X 1 · ϕ · S−1 (X 3 )q2 X22 h2 q1 X12 h1 , ν −1 (ϕ h) = x1 · ϕ · S−1 (x3 h2 p2 )  x2 h1 p1 , for all ϕ ∈ A and h ∈ H, where qR = q1 ⊗ q2 and pR = p1 ⊗ p2 are from (3.2.19). As consequences of Propositions 5.67, 5.68 and Theorem 5.70, we obtain:

5.7 L–R-smash Products

219

Corollary 5.72 With the hypotheses of Proposition 5.67 and assuming, moreover, that H is a quasi-Hopf algebra with bijective antipode, we have an algebra isomorphism

μ : (A ⊗ B)  A ∼ = A < A > B, μ ((a ⊗ b)  u) = Θ1 · a < q˜1ρ Θ20 u0 > b · S−1 (Θ3 )q˜2ρ Θ21 u1 , for all a ∈ A, b ∈ B and u ∈ A. Corollary 5.73 With the hypotheses of Proposition 5.68 and assuming, moreover, that H is a quasi-Hopf algebra with bijective antipode, we have an algebra isomorphism

η : A > A < B ∼ = A  (A ⊗ B), η (a > ϕ < b) = ϕ · S−1 (a1 p˜2ρ )  (a0 p˜1ρ ⊗ b), for all a ∈ A, b ∈ B and ϕ ∈ A . As a consequence of these two results, we obtain: Corollary 5.74 Let H be a quasi-Hopf algebra with bijective antipode, A a left H-module algebra, B a right H-module algebra, A a right H-comodule algebra and B a left H-comodule algebra. Then we have algebra isomorphisms A < (A ⊗ B) > B ∼ = A > (A ⊗ B) < B. = (A ⊗ B)  (A ⊗ B) ∼ We now study a kind of invariance under twisting of the L–R-smash product. Let H be a quasi-bialgebra, A an H-bimodule algebra and F ∈ H ⊗ H a gauge transformation. If we introduce on A another multiplication, ϕ ◦ ϕ  = (G1 · ϕ · F 1 ) (G2 · ϕ  · F 2 ) for all ϕ , ϕ  ∈ A , where F −1 = G1 ⊗ G2 , and denote this structure by F AF −1 , then one can check that F AF −1 is an HF -bimodule algebra, with the same unit and H-actions as for A . Suppose that B is a left H-comodule algebra; then on the algebra structure of B −1 one can introduce a left HF -comodule algebra structure (denoted by BF in what −1 −1 follows) by putting λ F = λ and ΦFλ = Φλ (F −1 ⊗ 1B ). Similarly, if A is a right H-comodule algebra, one can introduce on the algebra structure of A a right HF comodule algebra structure (denoted by F A in what follows) by putting F ρ = ρ and F Φ = (1 ⊗ F)Φ . One checks that if A is an H-bicomodule algebra, the left and ρ ρ A −1 right HF -comodule algebras AF and F A, respectively, actually define the structure −1 of an HF -bicomodule algebra on A, denoted by F AF , which has the same Φλ ,ρ as A. Proposition 5.75

With notation as above, we have an algebra isomorphism A A≡

given by the trivial identification.

F AF −1

−1

F AF ,

220

Crossed Products

Proof Let F 1 ⊗ F 2 and G 1 ⊗ G 2 be two more copies of F and F −1 , respectively. −1 We compute the multiplication in F AF −1 F AF : (ϕ u)(ψ u ) = (F 1 x˜λ1 · ϕ · θ 3 u1 x˜ρ2 G1 ) ◦ (F 2 x˜λ2 u[−1] θ 1 · ψ · x˜ρ3 G2 ) x˜λ3 u[0] θ 2 u0 x˜ρ1 = (G 1 F 1 x˜λ1 · ϕ · θ 3 u1 x˜ρ2 G1 F 1 )(G 2 F 2 x˜λ2 u[−1] θ 1 · ψ · x˜ρ3 G2 F 2 ) x˜λ3 u[0] θ 2 u0 x˜ρ1 = (x˜λ1 · ϕ · θ 3 u1 x˜ρ2 )(x˜λ2 u[−1] θ 1 · ψ · x˜ρ3 ) x˜λ3 u[0] θ 2 u0 x˜ρ1 , which is the multiplication of A A.

5.8 A Duality Theorem for Quasi-Hopf Algebras We present a situation when a certain quasi-Hopf smash product is isomorphic to a usual tensor product of associative algebras. As an application, we obtain a duality theorem for quasi-Hopf algebras. Proposition 5.76 Let H be a quasi-Hopf algebra, B an algebra and v : H → B an algebra map. Denote by η the algebra map η : H → B ⊗ H, η (h) = v(h1 ) ⊗ h2 . Define the map u : B → B ⊗ H, u(b) = v(x1 )bv(S(x23 X 3 ) f 1 ) ⊗ x2 X 1 β S(x13 X 2 ) f 2 , (5.8.1) where f = f 1 ⊗ f 2 is the Drinfeld twist given by (3.2.15). Then u is a morphism of left H-module algebras from Bv to (B ⊗ H)η . Proof The fact that u(v(β )) = η (β ) follows immediately from (3.2.14). We check now that u is a morphism of left H-modules: h η u(b) =

η (h1 )u(b)η (S(h2 ))

=

v(h(1,1) )v(x1 )bv(S(x23 X 3 ) f 1 )v(S(h2 )1 ) ⊗ h(1,2) x2 X 1 β S(x13 X 2 ) f 2 S(h2 )2

(3.2.13)

=

v(h(1,1) x1 )bv(S(h(2,2) x23 X 3 ) f 1 ) ⊗ h(1,2) x2 X 1 β S(h(2,1) x13 X 2 ) f 2

(3.1.7)

v(x1 h1 )bv(S(x23 h(2,2,2) X 3 ) f 1 ) ⊗ x2 h(2,1) X 1 β S(x13 h(2,2,1) X 2 ) f 2

=

(3.1.7), (3.2.1)

=

v(x1 h1 )bv(S(x23 X 3 h2 ) f 1 ) ⊗ x2 X 1 β S(x13 X 2 ) f 2 = u(h v b).

Now we check that u is multiplicative (F = F 1 ⊗ F 2 is another copy of f ): u(b) ◦ u(b ) =

η (Y 1 )u(b)η (S(y1Y 2 )α y2Y13 )u(b )η (S(y3Y23 ))

=

3 z1 )b v(Y11 x1 )bv(S(x23 X 3 ) f 1 S(y1Y 2 )1 α1 y21Y(1,1)

v(S(z32 Z 3 )F 1 S(y3Y23 )1 ) ⊗Y21 x2 X 1 β S(x13 X 2 ) f 2 S(y1Y 2 )2 α2 3 y22Y(1,2) z2 Z 1 β S(z31 Z 2 )F 2 S(y3Y23 )2

5.8 A Duality Theorem for Quasi-Hopf Algebras (3.2.13)

=

(3.2.14)

221

3 3 v(Y11 x1 )bv(S(T 2t21 y12Y22 x23 X 3 )α T 3t 2 y21Y(1,1) z1 )b v(S(y32Y(2,2) z32 Z 3 ) f 1 ) 3 3 z2 Z 1 β S(y31Y(2,1) z31 Z 2 ) f 2 ⊗Y21 x2 X 1 β S(T 1t11 y11Y12 x13 X 2 )α t 3 y22Y(1,2)

(3.1.7)

=

(3.2.1) (3.1.9),(3.1.7)

=

(3.2.1) (3.1.9)

=

(3.1.7)

v(Y11 x1 )bv(S(T 2t21 y12Y22 x23 X 3 )α T 3t 2 y21 z1Y13 )b v(S(y32 z32 Z 3Y23 ) f 1 ) ⊗Y21 x2 X 1 β S(T 1t11 y11Y12 x13 X 2 )α t 3 y22 z2 Z 1 β S(y31 z31 Z 2 ) f 2 v(Y11 x1 )bv(S(T 2t21Y22 x23 X 3 )α T 3t 2Y13 )b v(S(y32 Z 3t 3Y23 ) f 1 ) ⊗Y21 x2 X 1 β S(y1 T 1t11Y12 x13 X 2 )α y2 Z 1 β S(y31 Z 2 ) f 2 3 2 3 v(x1Y 1 )bv(S(T 2 x(1,1,2) t21W22Y(2,2) X 3 )α T 3 x(1,2) t 2W13Y13 )b

v(S(y32 Z 3 x23t 3W23Y23 ) f 1 ) ⊗ x2W 1Y12 X 1 β 3 2 S(y1 T 1 x(1,1,1) t11W12Y(2,1) X 2 )α y2 Z 1 β S(y31 Z 2 ) f 2 (3.1.7)

=

(3.2.1) (3.1.7)

=

(3.2.1) (3.1.9)

=

(3.2.1)

2 v(x1Y 1 )bv(S(T 2t21W22Y(2,2) X 3 )α T 3t 2W13Y13 )b v(S(y32 Z 3 x23t 3W23Y23 ) f 1 ) 2 ⊗ x2W 1Y12 X 1 β S(y1 x13 T 1t11W12Y(2,1) X 2 )α y2 Z 1 β S(y31 Z 2 ) f 2

v(x1Y 1 )bv(S(T 2t21W22 X 3Y 2 )α T 3t 2W13Y13 )b v(S(y32 Z 3 x23t 3W23Y23 ) f 1 ) ⊗ x2W 1 X 1 β S(y1 x13 T 1t11W12 X 2 )α y2 Z 1 β S(y31 Z 2 ) f 2 3 v(x1Y 1 )bv(S(T 2t21 z2 X13Y 2 )α T 3t 2 z31 X(2,1) Y13 )b 3 v(S(y32 Z 3 x23t 3 z32 X(2,2) Y23 ) f 1 )

⊗ x2 X 1 β S(y1 x13 T 1t11 z1 X 2 )α y2 Z 1 β S(y31 Z 2 ) f 2 (3.1.9)

=

(3.2.1)

3 v(x1Y 1 )bv(S(T 2t21 z2 X13Y 2 )α T 3t 2 z31 X(2,1) Y13 )b 3 v(S(x23t 3 z32 X(2,2) Y23 ) f 1 ) ⊗ x2 X 1 β S(x13 T 1t11 z1 X 2 )S(y1 )α y2 β S(y3 ) f 2

(3.2.2)

=

(3.1.9)

3 3 v(x1Y 1 )bv(S(y21t 1 X13Y 2 )α y22t 2 X(2,1) Y13 )b v(S(x23 y3t 3 X(2,2) Y23 ) f 1 )

⊗ x2 X 1 β S(x13 y1 X 2 ) f 2 (3.1.7),(3.2.1)

=

v(x1Y 1 )bv(S(t 1Y 2 )α t 2Y13 )b v(S(x23 X 3t 3Y23 ) f 1 ) ⊗ x2 X 1 β S(x13 X 2 ) f 2

=

u(v(Y 1 )bv(S(t 1Y 2 )α t 2Y13 )b v(S(t 3Y23 ))) = u(b ◦ b ),

finishing the proof. Corollary 5.77

The linear map ψ : Bv #H → B ⊗ H, given by

ψ (b#h) = v(X 1 x11 )bv(S(X 2 x21 )α X 3 x2 h1 ) ⊗ x3 h2 ,

(5.8.2)

for all b ∈ B and h ∈ H, is an algebra map. Proof The Universal Property of the smash product (Theorem 5.12), applied to the maps η and u from Proposition 5.76, provides an algebra map Bv #H → B ⊗ H, which is exactly the map ψ given by (5.8.2); we leave the details to the reader. Proposition 5.78 If H is a quasi-Hopf algebra, B an algebra, v : H → B an algebra map, then: (i) θ : B → Bv #H, θ (b) = v(z1 )bv(Z 1 β S(z2 Z 2 ))#z3 Z 3 , is an algebra map; (ii) μ : H → Bv #H, μ (h) = v(z1 Z 1 β S(z2 h1 Z 2 ))#z3 h2 Z 3 , is an algebra map;

222

Crossed Products

(iii) for all h ∈ H and b ∈ B, the following relation holds:

θ (b)μ (h) = μ (h)θ (b) = v(z1 )bv(Z 1 β S(z2 h1 Z 2 ))#z3 h2 Z 3 ; (iv) consequently, the linear map

ξ : B ⊗ H → Bv #H, ξ (b ⊗ h) = v(z1 )bv(Z 1 β S(z2 h1 Z 2 ))#z3 h2 Z 3 ,

(5.8.3)

is an algebra map. Proof We only prove (i) and leave the rest to the reader. Clearly θ (1B ) = v(β )#1H , so we only have to check that θ is multiplicative. We compute:

θ (b)θ (b ) [x1 v (v(z1 )bv(Z 1 β S(z2 Z 2 )))] ◦ [x2 z31 Z13 v (v(t 1 )b v(T 1 β S(t 2 T 2 )))]

=

#x3 z22 Z23t 3 T 3 3 v(X 1 x11 z1 )bv(Z 1 β S(y1 X 2 x21 z2 Z 2 )α y2 X13 x12 z3(1,1) Z(1,1) t 1)

=

3 t 2 T 2 ))#x3 z22 Z23t 3 T 3 b v(T 1 β S(y3 X23 x22 z3(1,2) Z(1,2) (3.1.9)

=

(3.2.1) (3.1.9)

=

(3.2.1) (3.1.7)

=

(3.2.1)

3 v(X 1 w11 z1 y1 )bv(Z 1 β S(X 2 w12 z2 y21 x1 Z 2 )α X 3 w2 z31 y2(2,1) x12 Z(1,1) t 1) 3 t 2 T 2 ))#y3 x3 Z23t 3 T 3 b v(T 1 β S(w3 z32 y2(2,2) x22 Z(1,2) 3 v(y1 )bv(Z 1 β S(z1 y21 x1 Z 2 )α z2 y2(2,1) x12 Z(1,1) t 1) 3 t 2 T 2 ))#y3 x3 Z23t 3 T 3 b v(T 1 β S(z3 y2(2,2) x22 Z(1,2) 3 3 v(y1 )bv(Z 1 β S(z1 x1 Z 2 )α z2 x12 Z(1,1) t 1 )b v(T 1 β S(y2 z3 x22 Z(1,2) t 2 T 2 ))

#y3 x3 Z23t 3 T 3 (3.1.9)

=

(3.2.1)

2 2 v(y1 )bv(Y 1 Z 1 β S(z1Y12 Z 2 )α z2Y(2,1) Z13t 1 )b v(T 1 β S(y2 z3Y(2,2) Z23t 2 T 2 ))

#y3Y 3t 3 T 3 (3.1.7)

=

(3.2.1) (3.1.9), (3.2.1)

=

(3.2.2)

=

v(y1 )bv(Y 1 Z 1 β S(z1 Z 2 )α z2 Z13t 1 )b v(T 1 β S(y2Y 2 z3 Z23t 2 T 2 )) #y3Y 3t 3 T 3 v(y1 )bv(Y 1 Z 1 β S(Z 2 )α Z 3t 1 )b v(T 1 β S(y2Y 2t 2 T 2 ))#y3Y 3t 3 T 3 v(y1 )bb v(T 1 β S(y2 T 2 ))#y3 T 3 = θ (bb ),

finishing the proof. Theorem 5.79 The maps ψ and ξ given by (5.8.2) and (5.8.3), respectively, are inverse to each other, providing thus an algebra isomorphism Bv #H ∼ = B ⊗ H. Proof

We compute:

ξ (ψ (b#h)) =

v(z1 X 1 x11 )bv(S(X 2 x21 )α X 3 x2 h1 Z 1 β S(z2 x13 h(2,1) Z 2 ))#z3 x23 h(2,2) Z 3

(3.1.7), (3.2.1)

v(z1 X 1 x11 )bv(S(X 2 x21 )α X 3 x2 Z 1 β S(z2 x13 Z 2 ))#z3 x23 Z 3 h

(3.1.9), (3.2.1)

1 1 v(z1 X 1Y(1,1) t11 )bv(S(X 2Y(1,2) t21 )α X 3Y21t 2 β S(z2Y 2t 3 ))#z3Y 3 h

= =

5.9 Notes (3.1.7), (3.2.1)

v(X 1t11 )bv(S(X 2t21 )α X 3t 2 β S(t 3 ))#h

(3.1.9), (3.2.1)

bv(S(t 1 )α t 2 β S(t 3 ))#h = b#h,

= =

223

(3.2.2)

and similarly

ψ (ξ (b ⊗ h)) =

v(X 1 x11 z1 )bv(Z 1 β S(X 2 x21 z2 h1 Z 2 )α X 3 x2 z31 h(2,1) Z13 ) ⊗ x3 z32 h(2,2) Z23

(3.1.9), (3.2.1)

bv(Z 1 β S(t 1 h1 Z 2 )α t 2 h(2,1) Z13 ) ⊗ t 3 h(2,2) Z23

(3.1.7), (3.2.1)

bv(Z 1 β S(t 1 Z 2 )α t 2 Z13 ) ⊗ ht 3 Z23

(3.1.9), (3.2.1)

bv(Z 1 β S(Z 2 )α Z 3 ) ⊗ h = b ⊗ h,

= = =

(3.2.2)

finishing the proof. Corollary 5.80 Let H be a quasi-Hopf algebra and consider the left H-module algebra H0 that appears in Definition 4.4. Then we have an algebra isomorphism Ψ : H0 #H → H ⊗ H, Ψ(h ⊗ h) = X 1 x11 h S(X 2 x21 )α X 3 x2 h1 ⊗ x3 h2 . (5.8.4) Proof

Take B = H and v = IdH in Theorem 5.79.

As an application of Theorem 5.79, we obtain a duality theorem for quasi-Hopf algebras: Theorem 5.81 If H is a finite-dimensional quasi-Hopf algebra with bijective antipode, then the two-sided crossed product H > H ∗ < H is isomorphic to End(H)⊗ H as associative unital algebras. Proof

By Proposition 5.34, we have the identification of algebras H > H ∗ < H ≡ (H#H ∗ )#H.

By Propositions 5.21 and 5.28, H#H ∗ is isomorphic, as left H-module algebras, to End(H)v , where End(H) is regarded as an associative algebra in the usual way and v : H → End(H) is a certain algebra map. We can thus apply Theorem 5.79 for B = End(H) and obtain the desired result.

5.9 Notes The content of Section 5.1 is taken from [59]. The content of Section 5.2 is taken from [46] and [60]. The results in Section 5.3 appear in [178]. The two-sided crossed and smash products in Section 5.4 appear in [107] and [60], respectively. Theorem 5.35 is from [107], but the proof is taken from [60]. The content of Section 5.5 is taken from [60]. The content of Section 5.6 is taken from [107] and [60], except for Propositions 5.61, 5.62 and 5.63, which appear in [8]. The content of Section 5.7 is taken from [179], while the duality theorem for quasi-Hopf algebras and the rest of Section 5.8 are taken from [8].

6 Quasi-Hopf Bimodule Categories

We present some structure theorems for quasi-Hopf bimodules. We also show that for a quasiHopf algebra H the category of quasi-Hopf H-bimodules is monoidally equivalent to the category of left H-representations. As an application, we prove a structure theorem for quasiHopf comodule algebras.

6.1 Quasi-Hopf Bimodules Let H be a quasi-bialgebra over a field k. We saw at the end of Section 4.2 that H has a natural structure of a coalgebra within the monoidal category H MH . So we can consider left and right H-comodules within H MH . Definition 6.1 Let H be a quasi-bialgebra. We call a right (resp. left) H-comodule within H MH a right (resp. left) quasi-Hopf H-bimodule. More precisely, a right quasi-Hopf H-bimodule M is an H-bimodule together with an H-bimodule map ρ : M → M ⊗ H such that the following relations hold: (IdM ⊗ ε ) ◦ ρ = IdM ,

(6.1.1)

Φ · (ρ ⊗ IdM )(ρ (m)) = (IdM ⊗ Δ)(ρ (m)) · Φ, ∀ m ∈ M.

(6.1.2)

A morphism between two right quasi-Hopf H-bimodules is an H-comodule morphism in H MH , that is, an H-bimodule map f : M → M  satisfying ρ  ◦ f = ( f ⊗ IdH ) ◦ ρ . We denote by H MHH the category of right quasi-Hopf H-bimodules and morphisms of right quasi-Hopf H-bimodules. Likewise, a left quasi-Hopf H-bimodule is an H-bimodule N together with an Hbimodule map λ : N → H ⊗ N such that the following relations hold: (ε ⊗ IdN ) ◦ λ = IdN ,

(6.1.3)

(IdH ⊗ λ )(λ (n)) · Φ = Φ · (Δ ⊗ IdN )(λ (n)), ∀ n ∈ N.

(6.1.4)

A morphism between two left quasi-Hopf H-bimodules is an H-bimodule map that is at the same time left H-colinear. By H H MH we denote the category of left quasi-Hopf H-bimodules and morphisms of left quasi-Hopf H-bimodules.

226

Quasi-Hopf Bimodule Categories op

H It can be easily checked that H H MH ≡ H op MH op via natural identification. Hence any result proved for right quasi-Hopf H-bimodules is valid for left quasi-Hopf Hbimodules as well, and vice versa. Definition 6.1 admits the following natural generalization.

Definition 6.2 Let H be a quasi-bialgebra, A a right H-comodule algebra and C an H-bimodule coalgebra. Then a right quasi-Hopf (H, A,C)-module is an (H, A)bimodule M together with a k-linear map ρ : M m → m(0) ⊗ m(1) ∈ M ⊗C such that the following relations hold: (IdM ⊗ ε ) ◦ ρ = IdM ,

(6.1.5)

ρ (h · m · a) = h1 · m(0) · a0 ⊗ h2 · m(1) · a1 ,

(6.1.6)

Φ · (ρ ⊗ IdH )(ρ (m)) = (IdM ⊗ Δ)(ρ (m)) · Φρ ,

(6.1.7)

for all h ∈ H, m ∈ M and a ∈ A. Denote by H MAC the category of quasi-Hopf (H, A,C)-modules and (H, A)-bimodule maps that are at the same time right Ccolinear maps. In a similar manner, for B a left comodule algebra and C a bimodule coalgebra over a quasi-bialgebra H, one can introduce the category of left quasi-Hopf (B, H,C)-modules, denoted in what follows by CB MH . For A = H (resp. B = H) we have that H MHC (resp. CH MH ) is the category of right (resp. left) corepresentations over C within H MH . Of course, when C = H, too, we reduce to the category of right (resp. left) quasi-Hopf H-bimodules. Examples of right quasi-Hopf (H, A,C)-modules can be obtained from the input datum (H, A,C). In what follows we call a right quasi-Hopf bimodule datum a triple (H, A,C) consisting of a quasi-bialgebra H, a right H-comodule algebra A and an H-bimodule coalgebra C. Proposition 6.3 Let (H, A,C) be a right quasi-Hopf bimodule datum. If M is an (H, A)-bimodule then M ⊗C becomes an object in H MAC via the structure given by h · (m ⊗ c) · a = h1 · m · a0 ⊗ h2 · c · a1 ,

ρ : M ⊗C m ⊗ c → x

1

1 · m · X˜ ρ

⊗x

2

2 · c1 · X˜ ρ

⊗x

3

3 · c2 · X˜ ρ

∈ M ⊗C ⊗C,

(6.1.8) (6.1.9)

for all h ∈ H, m ∈ M, c ∈ C and a ∈ A. Consequently, we have a functor F : H MA → C C H MA . F sends an (H, A)-bimodule M to M ⊗ C regarded as object in H MA via  (6.1.8) and (6.1.9), and an (H, A)-bimodule morphism f : M → M to f ⊗ IdC : M ⊗ C → M  ⊗C. Proof It can be easily proved that M ⊗C with the structure in (6.1.8) is an (H, A)bimodule. Now the coaction ρ in (6.1.9) is an (H, A)-bimodule map since

ρ (h · (m ⊗ c)) = ρ (h1 · m ⊗ h2 · c) 1 2 3 = x1 h1 · m · X˜ ρ ⊗ x2 h(2,1) · c1 · X˜ ρ ⊗ x3 h(2,2) · c2 · X˜ ρ (3.1.7)

1 2 3 = h(1,1) x1 · m · X˜ ρ ⊗ h(1,2) x2 · c1 · X˜ ρ ⊗ h2 x3 · c2 · X˜ ρ

6.1 Quasi-Hopf Bimodules

227

= h1 · (m ⊗ c)(0) ⊗ h2 · (m ⊗ c)(1) = h · ρ (m ⊗ c), and similarly

ρ ((m ⊗ c) · a) = ρ (m · a0 ⊗ c · a1 ) 1

2

3

= x1 · m · a0 X˜ ρ ⊗ x2 · c1 · a11 X˜ ρ ⊗ x3 · c2 · a12 X˜ ρ (4.3.1) 1

1

2

3

= x · m · X˜ ρ a0,0 ⊗ x2 · c1 · a0,1 X˜ ρ ⊗ x3 · c2 · X˜ ρ a1 = (m ⊗ c)(0) · a0 ⊗ (m ⊗ c)(1) · a1 = ρ (m ⊗ c) · a,

for all h ∈ H, m ∈ M, c ∈ C and a ∈ A. The map ρ is also right C-colinear since Φ · (ρ ⊗ IdC )(ρ (m ⊗ c)) 1 2 3 = Φ · (ρ (x1 · m · X˜ ρ ⊗ x2 · c1 · X˜ ρ ) ⊗ x3 · c2 · X˜ ρ ) 1

1

1

2

2

3

= X11 y1 x1 · m · X˜ ρ Y˜ ρ ⊗ X21 y2 x12 · c(1,1) · (X˜ ρ )1Y˜ ρ ⊗ X 2 y3 x22 · c(1,2) · (X˜ ρ )2Y˜ ρ 3 ⊗ X 3 x3 · c2 · X˜ ρ (3.1.7) 1 1 1 2 1 3 2 = x · m · X˜ ρ (Y˜ ρ )0 ⊗ x2 X 1 · c(1,1) · y1 X˜ ρ (Y˜ ρ )1 ⊗ x13 X 2 · c(1,2) · y2 (X˜ ρ )1Y˜ ρ (4.3.2) 3 3 ⊗ x23 X 3 · c2 · y3 (X˜ ρ )2Y˜ ρ (4.2.9) 1 1 1 2 1 3 2 = x · m · X˜ ρ (Y˜ ρ )0 ⊗ x2 · c1 · X˜ ρ (Y˜ ρ )1 ⊗ x13 · c(2,1) · (X˜ ρ )1Y˜ ρ 3 3 ⊗ x23 · c(2,2) · (X˜ ρ )2Y˜ ρ 1 2 3 = (m ⊗ c)(0) · Y˜ ρ ⊗ (m ⊗ c)(1)1 · Y˜ ρ ⊗ (m ⊗ c)(1)2 · Y˜ ρ

= (IdM⊗C ⊗ ΔC )(ρ (m ⊗ c)) · Φρ , as required. Finally, it can be easily checked that any (H, A)-bimodule morphism f : M → M  gives rise to a morphism f ⊗ IdC : M ⊗C → M  ⊗C in H MAC , and so we have a well-defined functor F : H MA → H MAC . This finishes the proof. We specialize Proposition 6.3 for various types of right quasi-Hopf bimodule data. Example 6.4 Let (H, A,C) be a right quasi-Hopf bimodule datum. Then for any right A-module M the k-vector space M ⊗C is an object in H MAH with the following structure: • M ⊗C is an (H, A)-bimodule via h · (m ⊗ c) · a = m · a0 ⊗ h · c · a1 ;

(6.1.10)

• the right coaction of H on M ⊗C is given by

ρ (m ⊗ c) = m · X˜ρ1 ⊗ c1 · X˜ρ2 ⊗ c2 · X˜ρ3 ,

(6.1.11)

for all m ∈ M, h ∈ H, c ∈ C and a ∈ A. Proof Consider M as a left H-module via the trivial action h · m = ε (h)m, for all h ∈ H and m ∈ M. Then M is an (H, A)-bimodule and so Proposition 6.3 applies. The resulting structures on M ⊗ C that make it an object in H MAC are exactly the ones stated in (6.1.10) and (6.1.11).

228

Quasi-Hopf Bimodule Categories

Examples 6.5 Let H be a quasi-Hopf algebra with bijective antipode, (A, ρ , Φρ ) a right H-comodule algebra and C an H-bimodule coalgebra. (1) V := A ⊗C ∈ H MAC . The structure maps are h  (a ⊗ c) = a ⊗ h · c, 1

(a ⊗ c) ≺ a = aa0 ⊗ c · a1 , 2

3

ρV (a ⊗ c) = aX˜ ⊗ c1 · X˜ ⊗ c2 · X˜ , for all h ∈ H, a, a ∈ A and c ∈ C. (2) U := C ⊗ A ∈ H MAC . Now the structure maps are given by the following formulas, for all h ∈ H, a, a ∈ A and c ∈ C: (c ⊗ a) ≺ a = c ⊗ aa , 3 1 3 2 ρU (c ⊗ a) = c1 · S−1 (q2L X˜ 2 g2 ) ⊗ X˜ a0 ⊗ c2 · S−1 (q1L X˜ 1 g1 )X˜ a1 .

h  (c ⊗ a) = h · c ⊗ a,

(6.1.12)

Here qL = q1L ⊗ q2L and f −1 = g1 ⊗ g2 are the elements defined by the formulas (3.2.20) and (3.2.16). Proof (1) This follows from Example 6.4 specialized for M = A, regarded as a right A-module via its multiplication. (2) Consider θ : V → U given by

θ (a ⊗ c) = c · S−1 (a1 p˜2ρ ) ⊗ a0 p˜1ρ , for all c ∈ C and a ∈ A, where, as before, we use the notation p˜ρ = p˜1ρ ⊗ p˜2ρ = x˜1 ⊗ x˜2 β S(x˜3 ) ∈ A ⊗ H. We claim that θ is bijective; its inverse θ −1 : U → V is defined as follows:

θ −1 (c ⊗ a) = q˜1ρ a0 ⊗ c · q˜2ρ a1 , 1 3 2 with notation q˜ρ = q˜1ρ ⊗ q˜2ρ = X˜ ⊗ S−1 (α X˜ )X˜ ∈ A ⊗ H. Furthermore, θ is a morphism of quasi-Hopf (H, A)-bimodules, and we conclude that U = C ⊗ A and A ⊗C = V are isomorphic in H MAC . Indeed, by using (4.3.10) and (4.3.13) one can show easily that θ and θ −1 are inverses, and that U is an (H, A)-bimodule via the actions  and ≺. One can finally compute the right Hcoaction on U carried from the coaction on V by using θ , and then see that it coincides with (6.1.12). For, observe that (4.3.9), (4.3.2) and (4.3.4) imply 1 2 1 3 X˜ 1 p˜2ρ S(X˜ ) ⊗ X˜ 0 p˜1ρ ⊗ X˜ = x˜2 S(x˜31 p1L ) ⊗ x˜1 ⊗ x˜32 p2L ,

(6.1.13)

where pL = p1L ⊗ p2L is the element defined in (3.2.20). We also mention that the computation uses the formula (4.3.15); the details are left to the reader. We now consider the other way around. Example 6.6 For any left H-module V the k-vector space V ⊗ H is a right quasiHopf H-bimodule via the structure h · (v ⊗ x) · h = h1 · v ⊗ h2 xh defined for all h, x, h ∈ H and v ∈ V .

and ρV ⊗H (v ⊗ h) = x1 · v ⊗ x2 h1 ⊗ x3 h2 ,

6.1 Quasi-Hopf Bimodules

229

Proof In Proposition 6.3 take A = C = H and regard a left H-module V as an Hbimodule with the trivial right H-action, that is, v·h = ε (h)v, for all h ∈ H, v ∈ V . For H a quasi-bialgebra denote by Γ(H) the set of algebra morphisms from H to k, which is a monoid with unit ε via the multiplication χ ∗ χ  (h) = χ (h1 )χ  (h2 ), for all χ , χ  ∈ Γ(H) and h ∈ H. If, moreover, H is a quasi-Hopf algebra then Γ(H) is a group since any χ ∈ Γ(H) admits χ −1 := χ ◦ S as inverse; see Remark 3.15. Note that if S is bijective then χ −1 = χ ◦ S−1 , too, and that these assertions hold because of (3.2.1) and (3.2.2). Example 6.7 Let χ ∈ Γ(H) and denote by Hχ the k-vector space H endowed with the left and right H-actions given by h · h¯ := χ (h1 )h2 h¯ and h¯ · h = h¯ h, for all h, h¯ ∈ H. Then Hχ is an H-bimodule and, moreover, a right quasi-Hopf H-bimodule with

ρχ : Hχ h¯ → χ (x1 )x2 h¯ 1 ⊗ x3 h¯ 2 ∈ Hχ ⊗ H. Proof Take V = k in Example 6.6 and consider it as a left H-module via the action defined by h¯ · κ := χ (¯h)κ , for all h¯ ∈ H and κ ∈ k. Then the stated quasi-Hopf Hbimodule structure on Hχ is obtained through the identification Hχ ≡ k ⊗ H. Examples of quasi-Hopf bimodules can be also obtained by tensoring two quasiHopf bimodules. Proposition 6.8 Let H be a quasi-bialgebra and M, N two objects in H MHH . Then M ⊗H N is a right quasi-Hopf H-bimodule with the structure given by (6.1.14) h · (m ⊗H n) · h = h · m ⊗H n · h , 

ρM⊗H N : M ⊗H N m ⊗H n → m(0) ⊗H n(0) ⊗ m(1) n(1) ∈ (M ⊗H N) ⊗ H, (6.1.15) for all m ∈ M, n ∈ N and h, h ∈ H. In this way the category H MHH becomes a strict monoidal category with unit object H = Hε , considered in H MHH as in Example 6.7. Proof It is immediate that the left and right actions in (6.1.14) are well defined, and that they endow M ⊗H N with an H-bimodule structure. The coaction ρM⊗H N is also well defined since, for all m ∈ M, h ∈ H and n ∈ N, we have

ρ (m · h ⊗H n) = m(0) h1 ⊗H n(0) ⊗ m(1) h2 n(1) = m(0) ⊗H h1 · n(0) ⊗ m(1) h2 n(1) = m(0) ⊗H (h · n)(0) ⊗ m(1) (h · n)(1) = ρM⊗H N (m ⊗H h · n). Furthermore, it is coassociative up to conjugation by Φ since Φ(ρM⊗H N ⊗ IdH )(ρM⊗H N (m ⊗H n)) = X 1 · m(0,0) ⊗H n(0,0) ⊗ X 2 m(0,1) n(0,1) ⊗ X 3 m(1) n(1) = m(0) · X 1 ⊗H n(0,0) ⊗ m(1)1 X 2 n(0,1) ⊗ m(1)2 X 3 n(1) = m(0) ⊗H X 1 · n(0,0) ⊗ m(1)1 X 2 n(0,1) ⊗ m(1)2 X 3 n(1)

230

Quasi-Hopf Bimodule Categories = m(0) ⊗H n(0) · X 1 ⊗ m(1)1 n(1)1 X 2 ⊗ m(1)2 n(1)2 X 3 = (IdM⊗H N ⊗ Δ)(ρM⊗H N (m ⊗H n))Φ,

for all m ∈ M and n ∈ N. Here, and everywhere else, we adopt the notation (ρ ⊗ IdH )(ρ (m)) = m(0,0) ⊗ m(0,1) ⊗ m(1) , etc. Now observe that for any two right quasi-Hopf H-bimodule morphisms f : M → M  and g : N → N  we have that f ⊗H g : M ⊗H N → M  ⊗H N  is a right quasi-Hopf bimodule morphism as well. Hence ⊗H : H MH × H MH → H MH induces a functor, still denoted by ⊗H , from H MHH × H MHH to H MHH . Furthermore, it endows H MHH with a strict monoidal structure since the isomorphisms M ⊗H H ∼ = M and H ⊗H M ∼ = H H M in H MH are in H MH , provided that M ∈ H MH .

6.2 The Dual of a Quasi-Hopf Bimodule The category of left or right quasi-Hopf bimodules does not have duality. However, owing to the general result in Proposition 2.43 we can associate to any finitedimensional left (resp. right) quasi-Hopf H-bimodule a right (resp. left) one. Actually we will apply this general result to the case when C = H MH , the category of H-bimodules. This category is monoidal since it can be identified with the category of left modules over the quasi-Hopf algebra H ⊗ H op . The monoidal structure on H MH obtained in this way was described explicitly in Lemma 4.13. Furthermore, if we restrict to the category of finite-dimensional H-bimodules then it is left and right rigid, provided that the antipode of S is bijective (this follows from Proposition 3.33 applied to the quasi-Hopf algebra H ⊗ H op ). In fact, we have the following result. Recall that the linear dual V ∗ of a right (resp. left) H-module V is a left (resp. right) H-module via (h  v∗ )(v) = v∗ (v · h) (resp. (v∗  h)(v) = v∗ (h · v)). Lemma 6.9 Let {vi }i be a basis of a finite-dimensional H-bimodule V , with dual basis {vi }i of its linear dual V ∗ . The left dual of V is V ∗ with H-bimodule structure h · v∗ · h = (h ⊗ h) · v∗ = v∗  (S−1 (h ) ⊗ S(h)) = S−1 (h )  v∗  S(h). (6.2.1) The evaluation morphism evV : V ∗ ⊗V → k and the coevaluation morphism coevV : k → V ⊗V ∗ are given by the formulas evV (v∗ ⊗ v) = v∗ ((S−1 (β ) ⊗ α ) · v) = v∗ (α · v · S−1 (β )), coevV (1) = ∑(S i

−1

(α ) ⊗ β ) · vi ⊗ v = ∑ β · vi · S i

−1

(6.2.2)

(α ) ⊗ v . i

(6.2.3)

i

The right dual of V is V ∗ , now with H-bimodule structure h · v∗ · h = (h ⊗ h) · v∗ = v∗  (S(h ) ⊗ S−1 (h)) = S(h )  v∗  S−1 (h).

6.2 The Dual of a Quasi-Hopf Bimodule

231

The evaluation evV : V ⊗V ∗ → k and coevaluation coevV : k → V ∗ ⊗V are given by evV (v ⊗ v∗ ) = v∗ ((β ⊗ S−1 (α )) · v) = v∗ (S−1 (α ) · v · β ), coevV (1) = ∑ vi ⊗ (α ⊗ S−1 (β )) · vi = ∑ vi ⊗ S−1 (β ) · vi · α . i

i

Proof The two assertions follow easily from the canonical monoidal identification H MH = H op ⊗H M and the rigid monoidal structure of the category of finitedimensional modules over a quasi-Hopf algebra. The details are left to the reader. With all these structures in mind one can prove the following result. Proposition 6.10 Let H be a quasi-Hopf algebra with bijective antipode, C an Hbimodule coalgebra and M a finite-dimensional H-bimodule with basis {mi }i , and let {mi }i be the corresponding dual basis of M ∗ . (i) If M is a right quasi-Hopf (H, H,C)-module then M ∗ is a left quasi-Hopf (H, H,C)-module with structure h · m∗ · h = S−1 (h )  m∗  S(h),

λM∗ (m∗ ) = ∑ m∗ (S( p˜2 ) f 1 · (mi )(0) · S−1 (q˜2 g2 ))S( p˜1 ) f 2 · (mi )(1) · S−1 (q˜1 g1 ) ⊗ mi . i

Here pL = p˜1 ⊗ p˜2 and qL = q˜1 ⊗ q˜2 are the elements defined in (3.2.20), f = f 1 ⊗ f 2 is the Drinfeld twist from (3.2.15) and f −1 = g1 ⊗ g2 is its inverse from (3.2.16). (ii) If M is a left quasi-Hopf (H, H,C)-module then ∗ M is a right quasi-Hopf (H, H,C)-module with structure h · ∗ m · h = S(h )  ∗ m  S−1 (h) ,

ρ∗ M (∗ m) = ∑ ∗ m(S−1 ( f 1 p1 ) · (mi )[0] · g2 S(q1 ))mi ⊗ S−1 ( f 2 p2 ) · (mi )[−1] · g1 S(q2 ), i

where pR = p1 ⊗ p2 and qR = q1 ⊗ q2 are the elements presented in (3.2.19). Proof We will prove (i), and leave (ii) to the reader. Actually, we will show that the structure on M ∗ as stated in (i) is precisely the structure that we obtain after applying part (ii) of Proposition 2.43 in the case when we have a coalgebra C in C = H MH and V = M. Consider H as a bimodule via multiplication. An object M ∈ H MHC is actually a right C-comodule in H MH . Since M is finite dimensional we get that M ∗ , the left dual of M in H MH , is a left H-comodule in C , and therefore an object of CH MH . By (6.2.1) we deduce that the H-bimodule structure of H ∗ is the one mentioned in part (i) of the statement. In order to find the left C-coaction on M ∗ we specialize Proposition 2.43 (ii) for the monoidal structure of H MH , and use (6.2.2) and (6.2.3) to compute λM∗ as the following composition: IdM ∗ ⊗coevM

m∗

∑ m∗ ⊗ (β · mi · S−1 (α ) ⊗ mi ) = ∑ m j (β · mi · S−1 (α ))m∗ ⊗ (m j ⊗ mi ) i

IdM ∗ ⊗(ρM ⊗IdM ∗ )

i, j

∑ m j (β · mi · S−1 (α ))m∗ ⊗ (((m j )(0) ⊗ (m j )(1) ) ⊗ mi ) i, j

232

Quasi-Hopf Bimodule Categories

IdM ∗ ⊗aM,C,M ∗

∑ m j (β S(X 3 ) · mi · S−1 (α x3 )) i, j

m∗ ⊗ (X 1 · (m j )(0) x1 ⊗ (X 2 · (m j )(1) x2 ⊗ mi )) a−1 M ∗ ,M,C⊗M ∗

∑ m j (β S(y32 X 3 ) · mi · S−1 (α x3Y23 )) i, j

evM ⊗IdC⊗M ∗

−1 1 S (Y )  m∗  S(y1 )

 ⊗ y2 X 1 · (m j )(0) · x1Y 2 ⊗ (y31 X 2 · (m j )(1) · x2Y13 ⊗ mi )

∑ m∗ (S(y1 )α y2 X 1 · (m j )(0) · x1Y 2 S−1 (Y 1 β )) j

y31 X 2 · (m j )(1) · x2Y13 ⊗ S−1 (α x3Y23 )  m j  β S(y32 X 3 ). We then have, for all m∗ ∈ M ∗ , that (3.2.20)

λM∗ (m∗ ) =

∑m∗ , q˜1 X 1 · (m j )(0) · x1 p˜1  j

q˜21 X 2 · (m j )(1) · x2 p˜21 ⊗ S−1 (α x3 p˜22 )  m j  β S(q˜22 X 3 ) =

∑m∗ , q˜1 X 1 β1 S(q˜22 X 3 )1 · (mi )(0) · S−1 (α x3 p˜22 )1 x1 p˜1  i

q˜21 X 2 β2 S(q˜22 X 3 )2 · (mi )(1) · S−1 (α x3 p˜22 )2 x2 p˜21 ⊗ mi (3.2.14)

=

(3.2.13) (3.2.5)

=

(3.2.6)

m∗ , q˜1 X 1 δ 1 S(q˜2(2,2) X23 ) f 1 · (mi )(0) · S−1 (γ 2 x23 p˜2(2,2) g2 )x1 p˜1  q˜21 X 2 δ 2 S(q˜2(2,1) X13 ) f 2 · (mi )(1) · S−1 (γ 1 x13 p˜2(2,1) g1 )x2 p˜21 ⊗ mi

∑m∗ , q˜1 β S(q˜2(2,2) X 3 ) f 1 · (mi )(0) · S−1 (α x3 p˜2(2,2) g2 ) p˜1  i

q˜21 X 1 β S(q˜2(2,1) X 2 ) f 2 · (mi )(1) · S−1 (α x2 p˜2(2,1) g1 )x1 p˜21 ⊗ mi (3.1.7)

=

(3.2.1)

∑m∗ , q˜1 β S(X 3 q˜2 ) f 1 · (mi )(0) · S−1 (α p˜2 x3 g2 ) p˜1  i

X 1 β S(X 2 ) f 2 · (mi )(1) · S−1 (α x2 g1 )x1 ⊗ mi (3.2.20)

=

(3.2.2)

∑m∗ , S(X 3 ) f 1 · (mi )(0) · S−1 (x3 g2 ) i

X 1 β S(X 2 ) f 2 · (mi )(1) · S−1 (α x2 g1 )x1 ⊗ mi (3.2.20)

=

∑m∗ , S( p˜2 ) f 1 · (mi )(0) · S−1 (q˜2 g2 ) i

S( p˜1 ) f 2 · (mi )(1) · S−1 (q˜1 g1 ) ⊗ mi , as stated. We now describe the morphisms of quasi-Hopf (H, H,C)-bimodules between C and one of its dual objects. We start with a more general result. Lemma 6.11 Let C be a coalgebra in a monoidal category C and V,W objects of C . Then the following assertions hold. (i) If V has a right dual object ∗V in C then HomC (W, ∗V ) ∼ = HomC (V ⊗ W, 1).

6.2 The Dual of a Quasi-Hopf Bimodule

233

Furthermore, if V is a left C-comodule and W a right C-comodule in C then giving a right C-colinear morphism from W to ∗V is equivalent to giving a morphism Σ : V ⊗W → 1 that satisfies V W

V W

P P

  =

Σ

C

V W

, where Σ =

Σ

,

Σ

(6.2.4)

1

C

W

V

P  is the left C-coaction on V . P is the right C-coaction on W and  W C

C V

∼ HomC (V ⊗ W, 1). (ii) If W has a left dual object W ∗ in C then HomC (V,W ∗ ) = If, moreover, V is a left C-comodule and W a right C-comodule in C then giving a left C-colinear morphism from V to W ∗ in C is equivalent to giving a morphism Σ : V ⊗W → 1 obeying (6.2.4). Proof The assertions in (ii) follow from those in (i) by replacing C with C . The first assertion in (i) is a particular case of Proposition 1.81. To be concise, the two correspondences below V W

HomC (W, ∗V ) f → Σ f :=

h ∈ HomC (V ⊗W, 1) ,

• f

1 W

 • HomC (V ⊗W, 1) Σ → fΣ :=

Σ

∈ HomC (W, ∗V )

∗V

are inverse to each other. Thus, if V is a left and W is a right comodule over C then from Proposition 2.43 we have that f : W → ∗V is right C-colinear if and only if W

 •

W

fh ⇔  

•

P P h =

f

∗V

C ∗V

C

V W

V W

P P  fh  fh = ⇔

•

• C

C

V W

V W

P P Σf

  Σf

= C

,

C

as stated. This finishes the proof of the lemma. Consider again the situation when C is a coalgebra in the category of bimodules over a quasi-Hopf algebra. Proposition 6.12 Let H be a quasi-Hopf algebra with bijective antipode, C an H-bimodule coalgebra, and take N ∈ CH MH and M ∈ H MHC . Then:

234

Quasi-Hopf Bimodule Categories

(i) If N is finite dimensional then the assignment f → Σ f , where Σ f (n ⊗ m) = f (m)(S−1 (α ) · n · β ), for all m ∈ M and n ∈ N, provides a one-to-one correspondence between H-bilinear maps f : M → ∗ N and H-bilinear forms Σ f : N ⊗ M → k. Its inverse sends Σ : N ⊗ M → k in H MH to fΣ : M m → Σ(p1L · ni · q1L ⊗ p2L · m · q2L )ni ∈ ∗ N, an H-bimodule map, where {ni , ni }i are dual bases in N and N ∗ . Consequently, giving a morphism f : M → ∗ N in H MHC is equivalent to giving an H-bilinear form Σ f : N ⊗ M → k satisfying, for all m ∈ M and n ∈ N, Σ f (x1 · n · X 1 ⊗ x2 · m(0) · X 2 )x3 · m(1) · X 3 = Σ f (X 2 · n[0] · x2 ⊗ X 3 · m · x3 )X 1 · n[−1] · x1 .

(6.2.5)

(ii) Likewise, if M is finite dimensional then the assignment g → Σg , where Σg (n ⊗ m) = g(n)(α · m · S−1 (β )), for all m ∈ M and n ∈ N, provides a one-to-one correspondence between H-bilinear morphisms g : N → M ∗ and H-bilinear forms Σg : N ⊗ M → k. Its inverse associates to an H-bilinear map Σ : N ⊗ M → k the H-bilinear morphism gΣ : N n → Σ(p1R · n · q1R ⊗ p2R · m j · q2R )m j ∈ M ∗ , where this time {m j , m j } j are dual bases in M and M ∗ . Consequently, giving a morphism g : N → M ∗ in H MHC is equivalent to giving an H-bilinear form Σg : N ⊗M → k obeying (6.2.5), for all m ∈ M and n ∈ N. Proof This follows by specializing Lemma 6.11 for C = H MH . The computations are similar to those in the proof of Proposition 6.10, and this is why we leave all the details to the reader. For instance to a morphism f : M → ∗ N in H MH we associate Σ f (n ⊗ m) = evN (n ⊗ f (m)) = f (m)(S−1 (α ) · n · β ), an H-bilinear form from N ⊗ M to k, as stated. Corollary 6.13 Let H be a quasi-Hopf algebra with bijective antipode, C a finitedimensional H-bimodule coalgebra. There are bijective correspondences between (i) morphisms f : C → ∗C in H MHC ; (ii) H-bimodule maps Σ f : C ⊗C → k satisfying, for all c, c ∈ C, Σ(x1 · c · X 1 ⊗ x2 · c1 · X 2 )x3 · c2 · X 3 = Σ f (X 2 · c2 · x2 ⊗ X 3 · c · x3 )X 1 · c1 · x1 ; (iii) morphisms g : C → C∗ in CH MH . Proof Consider M = N = C in Proposition 6.12 and view it alternatively as a left or right quasi-Hopf (H, H,C)-module.

6.3 Structure Theorems for Quasi-Hopf Bimodules

235

6.3 Structure Theorems for Quasi-Hopf Bimodules If H is a quasi-Hopf algebra and A is a right H-comodule algebra we will show that H H MA is equivalent to the category of right A-modules, MA . The essential role in the proof is played by the definition of the set of coinvariants of an object in H MAH , and then by a certain structure theorem for quasi-Hopf (H, A, H)-modules. Of course we can formulate and prove a left-handed version for the result stated above: if A is a right H-comodule algebra then the categories A MHH and A M are equivalent. As we shall see, this requires a different definition for the set of coinvariants of an object N ∈ A MHH . Thus when A = H we get two different concepts of coinvariants for a quasi-Hopf H-bimodule. For consistency, we will call them the set of coinvariants of the first type and the set of alternative coinvariants, respectively. More details will be presented in what follows. Definition 6.14 Let H be a quasi-Hopf algebra with bijective antipode, A a right H-comodule algebra and M ∈ H MAH . We call m ∈ M a coinvariant element of M of the first type if

ρM (m) = S−1 (q2 (X˜ρ3 )2 g2 ) · m · X˜ρ1 ⊗ S−1 (q1 (X˜ρ3 )1 g1 )X˜ρ2 ,

(6.3.1)

where qL := q1 ⊗ q2 is the element defined by (3.2.20) and f −1 = g1 ⊗ g2 is the element defined by (3.2.16). We will denote by M co(H) the set of coinvariants of M of the first type. Lemma 6.15 Let M ∈ H MAH and let M co(H) be its set of coinvariants of the first type. Then M co(H) becomes a right A-module via m ← a := S−1 (a1 ) · m · a0 ,

(6.3.2)

for all m ∈ M co(H) and a ∈ A. Consequently, we have a functor G : H MAH → MA . G sends M ∈ H MAH to M co(H) and a morphism to its restriction to M co(H) . Proof The difficult part is to show that ← is well defined. To this end we compute, for all m ∈ M co(H) and a ∈ A, that

ρ (m ← a) =

S−1 (a1 )1 · m(0) · a0,0 ⊗ S−1 (a1 )2 m(1) a0,1

(3.2.13) −1

=

(4.3.1)

=

S

3

1

3

2

S−1 (q2 a1(2,2) (X˜ ρ )2 g2 ) · m · a0 X˜ ρ ⊗ S−1 (q1 a1(2,1) (X˜ ρ )1 g1 )a11 X˜ ρ 3

(3.2.22) −1

=

(q2 (X˜ ρ )2 a12 g2 ) · m · X˜ ρ a0,0 ⊗ S−1 (q1 (X˜ ρ )1 a11 g1 )X˜ ρ a0,1

S

1

3

2

(q2 (X˜ ρ )2 g2 ) · (m ← a) · X˜ ρ ⊗ S−1 (q1 (X˜ ρ )1 g1 )X˜ ρ , 2

1

3

2

as required. The fact that ← defines on M co(H) a right A-module structure is immediate, thus G in the statement is a well-defined functor. We can now prove the desired equivalence of categories.

236

Quasi-Hopf Bimodule Categories

Theorem 6.16 If H is a quasi-Hopf algebra with bijective antipode and A is a right H-comodule algebra then the category of quasi-Hopf (H, A, H)-modules is equivalent to the category of right A-modules, MA . Proof From Example 6.4 we have a functor F = • ⊗ H : MA → H MAH . We recall that for any M ∈ MA we can endow M ⊗ H with a structure of an object in H MAH as follows (for all m ∈ M, h, h ∈ H and a ∈ A): h · (m ⊗ h ) · a = m · a0 ⊗ hh a1 , ρ (m ⊗ h) = m · X˜ρ1 ⊗ h1 X˜ρ2 ⊗ h2 X˜ρ3 .

(6.3.3) (6.3.4)

We claim that F and G define inverse equivalences, where G is the functor defined in Lemma 6.15. Indeed, let M ∈ MA and consider M ⊗ H as an object in H MAH with the structure defined above. We then have that (M ⊗ H)co(H) = {m · q˜1ρ ⊗ q˜2ρ | m ∈ M}, where q˜ρ = q˜1ρ ⊗ q˜2ρ is the element defined in (4.3.9). Actually, by using the above definitions we have that m ⊗ h ∈ (M ⊗ H)co(H) if and only if m · X˜ρ1 ⊗ h1 X˜ρ2 ⊗ h2 X˜ρ3 = m · (X˜ρ1 )0 ⊗ S−1 (q2 (X˜ρ3 )2 g2 )h(X˜ρ1 )1 ⊗ S−1 (q1 (X˜ρ3 )1 g1 )X˜ρ2 , (6.3.5) where, as usual, qL = q1 ⊗ q2 is the element defined in (3.2.20). Therefore, if m ⊗ h ∈ (M ⊗ H)co(H) then by applying IdM ⊗ ε ⊗ IdH to the equality (6.3.5) we obtain m ⊗ h = ε (h)m · X˜ρ1 ⊗ S−1 (α X˜ρ3 )X˜ρ2 = ε (h)m · q˜1ρ ⊗ q˜2ρ , as wanted. For the converse inclusion we need the formula 1 3 2 3 Q˜ 1ρ (x˜1ρ )0 ⊗ S−1 (x˜2ρ )Q˜ 2ρ (x˜1ρ )1 ⊗ x˜3ρ = X˜ ρ ⊗ S−1 (q1 (X˜ ρ )1 )X˜ ρ ⊗ q2 (X˜ ρ )2 , (6.3.6)

which follows easily from the definitions of q˜ρ = Q˜ 1ρ ⊗ Q˜ 2ρ and qL and the equation (4.3.1). Then we compute, for all m ∈ M:

ρ (m · q˜1ρ ⊗ q˜2ρ ) =

1

2

3

m · q˜1ρ X˜ ρ ⊗ (q˜2ρ )1 X˜ ρ ⊗ (q˜2ρ )2 X˜ ρ

(4.3.15)

m · q˜1ρ (Q˜ 1ρ (x˜1ρ )0 )0 ⊗ S−1 (x˜3ρ g2 )q˜2ρ (Q˜ 1ρ (x˜1ρ )0 )1 ⊗ S−1 (x˜2ρ g1 )Q˜ 2ρ (x˜1ρ )1

(6.3.6)

=

1 3 1 3 2 m · q˜1ρ (X˜ ρ )0 ⊗ S−1 (q2 (X˜ ρ )2 g2 )q˜2ρ (X˜ ρ )1 ⊗ S−1 (q1 (X˜ ρ )1 g1 )X˜ ρ

=

3 1 3 2 S−1 (q2 (X˜ ρ )2 g2 ) · (m · q˜1ρ ⊗ q˜2ρ ) · X˜ ρ ⊗ S−1 (q1 (X˜ ρ )1 g1 )X˜ ρ ,

=

as required. Now, it is not hard to see that

ξM : GF(M) = {m · q˜1ρ ⊗ q˜2ρ | m ∈ M} → M, ξM (m · q˜1ρ ⊗ q˜2ρ ) = m is a well-defined isomorphism in MA with ξM−1 (m) = m · q˜1ρ ⊗ q˜2ρ , for any m ∈ M.

6.3 Structure Theorems for Quasi-Hopf Bimodules

237

Conversely, take M ∈ H MAH and define E : M → M by E(m) = S−1 (α m(1) ) · m(0) , ∀ m ∈ M. We have Im(E) ⊆ M co(H) since

ρ (E(m))

=

S−1 (α m(1) )1 · m(0,0) ⊗ S−1 (α m(1) )2 · m(0,1)

(3.2.13),(3.2.14) −1

=

(3.2.5)

=

S

(γ 2 m(1)2 g2 ) · m(0,0) ⊗ S−1 (γ 1 m(1)1 g1 )m(0,1)

S−1 (α x3 X23 m(1)2 g2 )X 1 · m(0,0) ⊗ S−1 (α x2 X13 m(1)1 g1 )x1 X 2 m(0,0)

(6.1.5)

=

S−1 (α x3 m(1)(2,2) (X˜ ρ )2 g2 )m(0) · X˜ ρ 3

1

⊗ S−1 (α x2 m(1)(2,1) (X˜ ρ )1 g1 )x1 m(1)1 X˜ ρ 3

(3.1.7),(3.2.1)

=

(3.2.20)

=

2

S−1 (α m(1) x3 (X˜ ρ )2 g2 )m(0) · X˜ ρ ⊗ S−1 (α x2 (X˜ ρ )1 g1 )x1 X˜ ρ 3

1

3

2

3 1 3 S−1 (q2 (X˜ ρ )2 g2 ) · E(m) · X˜ ρ ⊗ S−1 (q1 (X˜ ρ )1 g1 ),

for all m ∈ M. Therefore, the map

ζM : M → M co(H) ⊗ H = F(G(M)),

ζM (m) = E(m(0) ) ⊗ m(1) , ∀ m ∈ M

is well defined. We show that ζM is an isomorphism in H MAH with inverse

ζM−1 (m ⊗ h) = hS−1 ( p˜2ρ ) · m · p˜1ρ , for all m ∈ M co(H) and h ∈ H. Here p˜ρ = p˜1ρ ⊗ p˜2ρ is the element defined in (4.3.9). Indeed, the fact that ζM is left H-linear is a consequence of the fact that E(h · m) = S−1 (h2 m(1) )h1 · m(0) = ε (h)E(m), ∀ h ∈ H, m ∈ M. Likewise, the fact that E(m · a) = S−1 (α m(1) a1 ) · m(0) · a0 = E(m) ← a, ∀ m ∈ M, a ∈ A implies that ζM is right A-linear, and so an (H, A)-bimodule morphism. It is right H-colinear as well since, for all m ∈ M, we have 1

2

3

1

2

3

E(m(0) ) ← X˜ ρ ⊗ m(1)1 X˜ ρ ⊗ m(1)2 X˜ ρ = E(m(0) · X˜ ρ ) ⊗ m(1)1 X˜ ρ ⊗ m(1)2 X˜ ρ (6.1.5)

= E(X 1 · m(0,0) ) ⊗ X 2 m(0,1) ⊗ X 3 m(1) = E(m(0,0) ) ⊗ m(0,1) ⊗ m(1) .

We now check that ζM and ζM−1 are inverses. On the one hand we have

ζM−1 ζM (m) = m(1) S−1 ( p˜2ρ ) · E(m(0) ) · p˜1ρ = m(1) S−1 (α m(0,1) p˜2ρ ) · m(0,0) · p˜1ρ = m(1) x˜3ρ S−1 (α m(0,1) x˜2ρ β ) · m(0,0) · x˜1ρ (6.1.5) 3

= x m(1)2 S−1 (α x2 m(1)1 β )x1 · m(0)

238

Quasi-Hopf Bimodule Categories (3.2.2)

= x3 S−1 (α x2 β )x1 · m = m, for all m ∈ M. On the other hand, for all m ∈ M co(H) and h ∈ H we have

ζM ζM−1 (m ⊗ h) = E(h1 S−1 ( p˜2ρ )1 · m(0) · ( p˜1ρ )0 ⊗ h2 S−1 ( p˜2ρ )2 m(1) ( p˜2ρ )1 = E(m(0) ) ← ( p˜1ρ )0 ⊗ hS−1 ( p˜2ρ )m(1) ( p˜1ρ )1 = E(S−1 (q2 (X˜ ρ )2 g2 ) · m · X˜ ρ ) ← ( p˜1ρ )0 ⊗ hS−1 (q1 (X˜ ρ )1 g1 p˜2ρ )X˜ ρ ( p˜1ρ )1 3

1

3

2

1 3 2 = E(m) ← X˜ ρ ( p˜1ρ )0 ⊗ S−1 (α X˜ ρ p˜2ρ )X˜ ρ ( p˜1ρ )1 (4.3.13)

= E(m) ← q˜1ρ ( p˜1ρ )0 ⊗ hS−1 ( p˜2ρ )q˜2ρ ( p˜1ρ )1 = m ⊗ h, where in the last equality we also use the fact that m ∈ M co(H) implies 3 3 2 1 E(m) = S−1 (α m(1) ) · m(0) = S−1 (q2 (X˜ ρ )2 g2 α S−1 (q1 (X˜ ρ )1 g1 )X˜ ρ ) · m · X˜ ρ = m.

Finally, it is straightforward to check that the maps ξM and ζM define natural transformations, so the proof is finished. Remark 6.17 By the above proposition it follows that the category H MHH is equivalent to the category of right H-modules. It is equivalent to the category of left Hmodules, too. To see this we use the equivalence between the categories A MHH (≡ H op H op MAop ) and A M (≡ MAop ). More precisely, the equivalence is produced by the functors AM

F  G

H A MH .

If M ∈ A M then F(M) = M ⊗ H regarded as an object in A MHH via a(m ⊗ h)h = a0 · m ⊗ a1 hh ,

ρ (m ⊗ h) = x˜ρ1 · m ⊗ x˜ρ2 h1 ⊗ x˜ρ3 h2 , for all a ∈ A, m ∈ M and h, h ∈ H. If M ∈ A MHH then G(M) = M co(H) , where M co(H) := {m ∈ M | ρ (m) = x˜ρ1 · m · S((x˜ρ3 )2 X 3 ) f 1 ⊗ x˜ρ2 X 1 β S((x˜ρ3 )1 X 2 ) f 2 }. Note that M co(H) = {E(m) | m ∈ M}, where E : M → M is given by E(m) := m(0) · β S(m(1) ), ∀ m ∈ M,

(6.3.7)

and that M co(H) can be regarded as a left A-module via the action a → m := a0 · m · S(a1 ), ∀ a ∈ A, m ∈ M.

(6.3.8)

Then M m → E(m(0) ) ⊗ m(1) ∈ M co(H) ⊗ H is an isomorphism in A MHH with inverse given by M co(H) ⊗ H m ⊗ h → q˜1ρ · m · S(q˜2ρ )h ∈ M. By taking A = H this gives a second structure theorem for right quasi-Hopf H-bimodules.

6.4 The Categories H MHH and H M

239

6.4 The Categories H MHH and H M For H a quasi-Hopf algebra, we have proved so far two structure theorems for quasiHopf bimodules. A third one will be presented in this section. As we shall see, it will provide a monoidal equivalence between the categories H MHH and H M . Once more this third structure theorem is possible due to the choice of the set of coinvariants for a right quasi-Hopf H-bimodule. Definition 6.18 Let H be a quasi-Hopf algebra with bijective antipode and M a right quasi-Hopf H-bimodule. We define E : M → M by E(m) = X 1 · m(0) · β S(X 2 m(1) )α X 3 = q1 · m(0) · β S(q2 m(1) ),

(6.4.1)

for all m ∈ M, where M m → ρM (m) := m(0) ⊗ m(1) ∈ M ⊗ H denotes the right coaction of H on M and qR = q1 ⊗ q2 is the element defined in (3.2.19). The space M co(H) = {n ∈ M | E(n) = n} is called the space of coinvariants of M of the second type. Let M ∈ H MHH . To avoid confusion, we call the elements of M co(H) (see Remark 6.17) alternative coinvariants for the right quasi-Hopf H-bimodule M. As we see next, M co(H) and M co(H) are isomorphic as left H-modules. Lemma 6.19 For h ∈ H and m ∈ M define h¬m = E(h · m). Then, for all m ∈ M and h, h ∈ H, the following relations hold: E(m · h) = ε (h)E(m),

E(h · E(m)) = E(h · m),

h · E(m) = E(h1 · m) · h2 = [h1 ¬m] · h2 ,

(6.4.2) (6.4.3)

E = E, E(m(0) ) · m(1) = m and E(E(m)(0) ) ⊗ E(m)(1) = E(m) ⊗ 1H , (6.4.4) 2

(hh )¬m = h¬(h ¬m). Proof

(6.4.5)

For all m ∈ M and h ∈ H we have (3.2.1)

E(m · h) = q1 · m(0) · h1 β S(q2 m(1) h2 ) = ε (h)E(m), and this implies that E(h · E(m)) = E(hq1 · m(0) · β S(q2 m(1) )) = E(h · m). Next we compute: [h1 ¬m] · h2

=

E(h1 · m) · h2

=

q1 h(1,1) · m(0) · β S(q2 h(1,2) m(1) )h2

(3.2.21)

=

h · E(m).

Also, by using the first relation in (6.4.2) we have E 2 (m) = E(q1 · m(0) · β S(q2 m(1) )) = ε (β S(q2 m(1) ))E(q1 · m(0) ) = E(m),

240

Quasi-Hopf Bimodule Categories

for all m ∈ M. Finally, for m ∈ M, we have E(m(0) ) · m(1) = X 1 · m(0,0) · β S(X 2 m(0,1) )α X 3 m(1) (6.1.2)

= m(0) · X 1 β S(m(1)1 X 2 )α m(1)2 X 3 (3.2.2)

(3.2.1)

= m · X 1 β S(X 2 )α X 3 = m,

and similarly E(E(m)(0) ) ⊗ E(m)(1) E(q11 · m(0,0) · β1 S(q2 m(1) )1 ) ⊗ q12 m(0,1) β2 S(q2 m(1) )2

= (6.4.2)

E(q11 · m(0,0) ) ⊗ q12 m(0,1) β S(q2 m(1) )

(6.1.2)

E(q11 x1 · m(0) · X 1 ) ⊗ q12 x2 m(1)1 X 2 β S(q2 x3 m(1)2 X 3 )

= =

(6.4.2),(3.2.1)

=

(3.2.19),(3.2.23)

=

E(q11 x1 · m) ⊗ q12 x2 β S(q2 x3 ) E(m) ⊗ 1H .

The relation (6.4.5) follows immediately from (6.4.2). It is clear that ¬ defines a left H-module structure on M co(H) . Also, it follows that E(m) = E(p1 · m) · p2 ,

E(m) = X 1 · E(m) · S(X 2 )α X 3 , ∀ m ∈ M,

(6.4.6)

where E is the projection onto M co(H) defined in (6.3.7). By using (6.4.2) we get that the maps E : M co(H) → M co(H) and E : M co(H) → M co(H)

(6.4.7)

are inverse to each other. Note that in the case of a Hopf algebra, the maps E and E are equal to the identity on M co(H) = M co(H) . Coming back to the quasi-Hopf case we have that E : M co(H) → M co(H) is a left H-linear isomorphism since E(E(h · n))

=

E(h¬n)

(6.4.6),(6.4.2)

E(p1 h · n) · p2

(3.2.21),(6.4.3)

h1 · E(p1 · n) · p2 S(h2 )

(6.4.6),(6.3.8)

h → E(n),

= = =

for all h ∈ H and n ∈ M co(H) . We then have the following structure theorem for right quasi-Hopf H-bimodules. Theorem 6.20 The linear map

νM : M co(H) ⊗ H → M,

νM (n ⊗ h) = n · h, ∀ n ∈ M co(H) , h ∈ H,

(6.4.8)

is an isomorphism of right quasi-Hopf H-bimodules. Here M co(H) ⊗ H is a right quasi-Hopf H-bimodule with structures a · (n ⊗ h) · b = E(a1 · n) ⊗ a2 hb

and ρ (n ⊗ h) = E(x1 · n) ⊗ x2 h1 ⊗ x3 h2 ,

6.4 The Categories H MHH and H M

241

for all n ∈ N, a, h, b ∈ H. The inverse of ν is given by −1 νM (m) = E(m(0) ) ⊗ m(1) , ∀ m ∈ M.

(6.4.9)

∼ M co(H) are isomorphic as left H-modules, hence Proof We have seen that M co(H) = M co(H) ⊗ H ∼ = M co(H) ⊗ H as quasi-Hopf H-bimodules (in both cases, the structure is determined as in Example 6.6). From the structure theorem in Remark 6.17 it follows that M ∼ =M = M co(H) ⊗H as quasi-Hopf H-bimodules. Thus we find that M co(H) ⊗H ∼ as quasi-Hopf H-bimodules, and it is straightforward to verify that the connecting isomorphism is M co(H) ⊗ H m ⊗ h → E(m) ⊗ h → q1 · E(m) · S(q2 )h ∈ M, which is exactly νM since (6.4.6)

q1 · E(m) · S(q2 )h = q1 · E(p1 · m) · p2 S(q2 )h (6.4.3)

(3.2.23)

= E(q11 p1 · m) · q12 p2 S(q2 )h = E(m) · h = m · h.

This finishes our proof. We give more descriptions for the set of coinvariants of the second type. Proposition 6.21

We also have M co(H) = {n ∈ M | E(n(0) ) ⊗ n(1) = E(n) ⊗ 1H } = {n ∈ M | ρ (n) = E(x · n) · x ⊗ x }. 1

Proof

2

3

(6.4.10) (6.4.11)

If n ∈ M co(H) then E(n) = n, and therefore (6.4.4)

E(n(0) ) ⊗ n(1) = E(E(n)(0) ) ⊗ E(n)(1) = E(n) ⊗ 1H . (6.4.4)

Conversely, if E(n(0) ) ⊗ n(1) = E(n) ⊗ 1H then n = E(n(0) ) · n(1) = E(n) · 1H = E(n). Hence we have proved the equality in (6.4.10). We now prove (6.4.11). If n ∈ M such that ρ (n) = E(x1 · n) · x2 ⊗ x3 then (6.4.2)

(6.4.4)

E(n) = E(E(x1 · n) · x2 ) · x3 = E(n(0) ) · n(1) = n, and so n ∈ M co(H) . Conversely, if n ∈ M co(H) then

ρ (n) = (ν ⊗ IdH )ρMco(H) ⊗H (ν −1 (n)) = (ν ⊗ IdH )ρMco(H) ⊗H (n ⊗ 1H ) = E(x1 · n) · x2 ⊗ x3 , finishing the proof. The equivalence described in Remark 6.17 is a monoidal equivalence if we specialize and reconsider it as in Theorem 6.20. So the equivalence of categories described in Remark 6.17, specialized for A = H, is a monoidal equivalence. In what follows we denote by EM , E M : M → M the projection associated to a right quasi-Hopf H-bimodule M as in Definition 6.18 and (6.3.7), respectively.

242

Quasi-Hopf Bimodule Categories

Proposition 6.22 Let H be a quasi-Hopf algebra with bijective antipode and M, N ∈ H H MH .

Define the linear maps M co(H) ⊗ N co(H)

iM,N jM,N

M ⊗H N ,

iM,N (m ⊗ n) = EM (X 1 · m) ⊗H EN (X 2 · n) · X 3 , jM,N (m ⊗H n) = EM (m(0) ) ⊗ EN (m(1) · n), for all m ∈ M and n ∈ N. Then we have jM,N iM,N = IdMco(H) ⊗N co(H) and iM,N jM,N = EM⊗H N , where M ⊗H N is considered as object in H MHH with the structure from Proposition 6.8. Consequently, the image of iM,N is (M ⊗H N)co(H) , and so iM,N induces a left Hmodule isomorphism between M co(H) ⊗ N co(H) and (M ⊗H N)co(H) . Proof

First of all, jM,N is well defined since jM,N (m · h ⊗H n) = EM (m(0) · h1 ) ⊗H EN (m(1) h2 · n) (6.4.2)

= EM (m(0) ) ⊗ EN (m(1) · (h · n)) = jM,N (m ⊗H h · n),

for all m ∈ M, h ∈ H and n ∈ N. Furthermore, if m ∈ M co(H) and n ∈ N then (6.4.10)

jM,N (m ⊗H n) = EM (m(0) ) ⊗ EN (m(1) · n) = EM (m) ⊗ EN (1H · n) = m ⊗ EN (n), (6.4.12) and this fact allows to compute, for all m ∈ M co(H) and n ∈ N co(H) : jM,N iM,N (m ⊗ n) = jM,N (EM (X 1 · m) ⊗H EN (X 2 · n) · X 3 ) = EM (X 1 · m) ⊗ EN (EN (X 2 · n) · X 3 ) (6.4.2)

(6.4.4)

= EM (m) ⊗ EN2 (n) = EM (m) ⊗ EN (n) = m ⊗ n,

as stated. To prove the second equality in the statement, for all m ∈ M co(H) and n ∈ N we compute: EM⊗H N (m ⊗H n)

=

q1 · m(0) ⊗H n(0) · β S(q2 m(1) n(1) )

(6.4.11) 1

=

q · EM (x1 · m) · x2 ⊗H n(0) · β S(q2 x3 n(1) )

(6.4.3)

EM (q11 x1 · m) ⊗H q12 x2 · n(0) · β S(q2 x3 n(1) )

(5.5.17)

=

EM (X 1 · m) ⊗H q1 X12 · n(0) · β S(q2 X22 n(1) )X 3

=

EM (X 1 · m) ⊗H EN (X 2 · n) · X 3 .

=

So we have shown that EM⊗H N (m ⊗H n) = EM (X 1 · m) ⊗H EN (X 2 · n) · X 3 , ∀ m ∈ M co(H) , n ∈ N. (6.4.13) Therefore, for arbitrary m ∈ M and n ∈ N we have (6.4.4)

EM⊗H N (m ⊗H n) = EM⊗H N (EM (m(0) ) · m(1) ⊗H n) = EM⊗H N (EM (m(0) ) ⊗H m(1) · n) = EM (X 1 · EM (m(0) )) ⊗H EN (X 2 m(1) · n) · X 3

6.4 The Categories H MHH and H M

243

(6.4.2)

= EM (X 1 · EM (m(0) )) ⊗H EN (X 2 · EN (m(1) · n)) · X 3 = iM,N jM,N (m ⊗H n),

as desired. Now, from jM,N iM,N = IdMco(H) ⊗N co(H) we get that jM,N is surjective, and therefore that (M ⊗H N)co(H) = Im(EM⊗H N ) = Im(iM,N jM,N ) = Im(iM,N ). Since iM,N is injective it follows that its corestriction to (M ⊗H N)co(H) is a bijection. We will denote it by φ2,M,N : M co(H) ⊗ N coH → (M ⊗H N)co(H) . If ι : (M ⊗H N)co(H) → M ⊗H N is the inclusion map then EM⊗H N ι = ι and iM,N = −1 ιφ2,M,N . Thus if φ2,M,N := jM,N ι : (M ⊗H N)co(H) → M co(H) ⊗ N co(H) then −1 ιφ2,M,N φ2,M,N = iM,N jM,N ι = EM⊗H N ι = ι , −1 = Id(M⊗H N)co(H) . Similarly, and so φ2,M,N φ2,M,N −1 φ2,M,N φ2,M,N = jM,N ιφ2,M,N = jM,N iM,N = IdMco(H) ⊗N co(H) . −1 We conclude that φ2,M,N is bijective and φ2,M,N is its inverse. Finally, observe that

h · iM,N (m ⊗ n)

=

h · EM (X 1 · m) ⊗H EN (X 2 · n) · X 3

(6.4.3)

EM (h1 X 1 · m) · h2 ⊗H EN (X 2 · n) · X 3

(6.4.3)

EM (h1 X 1 · m) ⊗H EN (h(2,1) X 2 · n) · h(2,2) X 3

= =

(3.1.7),(6.4.2)

=

EM (X 1 · EM (h(1,1) · m)) ⊗H EN (X 2 · EN (h(1,2) · n)) · X 3 h2

=

iM,N (h(1,1) ¬m ⊗ h(1,2) ¬n) · h2 ,

(6.4.14)

for all h ∈ H, m ∈ M co(H) and n ∈ N co(H) . Thus φ2,M,N is left H-linear since

φ2,M,N (h¬(m ⊗ n)) = iM,N (h1 ¬m ⊗ h2 ¬n) =

iM,N jM,N iM,N (h1 ¬m ⊗ h2 ¬m)

=

EM⊗H N iM,N (h1 ¬m ⊗ h2 ¬m)

(6.4.2)

EM⊗H N (iM,N (h(1,1) ¬m ⊗ h(1,2) ¬n) · h2 )

(6.4.14)

=

EM⊗H N (h · iM,N (m ⊗ n))

=

h¬iM,N (m ⊗ n)

=

h¬φ2,M,N (m ⊗ n),

=

for all h ∈ H, m ∈ M co(H) and n ∈ N co(H) , as required. We now state and prove the main result of this section. Theorem 6.23 If H is a quasi-Hopf algebra with bijective antipode then the categories H MHH and H M are strong monoidally equivalent. Proof We show that the functor G : H MHH → H M defined by G(M) = M co(H) and G( f ) = f |Mco(H) produces the desired monoidal equivalence. Towards this end, we first show that G is a strong monoidal functor. If φ2,M,N are as in Proposition 6.22, we

244

Quasi-Hopf Bimodule Categories

prove that the first corresponding diagram for φ in Definition 1.22 is commutative. We compute, for all M, N, P ∈ H MHH :

φ2,M⊗H N,P ◦ (φ2,M,N ⊗ IdPco(H) )((m ⊗ n) ⊗ p) =

φ2,M⊗H N,P ((EM (X 1 · m) ⊗H EN (X 2 · n) · X 3 ) ⊗ p)

=

EM⊗H N (Y 1 · EM (X 1 · m) ⊗H EN (X 2 · n) · X 3 ) ⊗H EP (Y 2 · p) ·Y 3   1 1 EM⊗H N (EM (Y11 X 1 · m) ⊗H EN (Y(2,1) X 2 · n)) ·Y(2,2) X3

(6.4.3)

=

(6.4.2)

=

(6.4.13),(6.4.2)

=

(6.4.3)

=

⊗ H EP (Y 2 · p) ·Y 3

 EM⊗H N EM (Y11 · m) ⊗H EN (Y21 · n) ⊗H EP (Y 2 · p) ·Y 3 EM (X 1Y11 · m) ⊗H EN (X 2Y21 · n) · X 3 ⊗H EP (Y 2 · p) ·Y 3 EM (X 1Y11 · m) ⊗H EN (X 2Y21 · n) ⊗H EP (X13Y 2 · p) · X23Y 3 ,

for all m ∈ M co(H) , n ∈ N co(H) and p ∈ Pco(H) , and on the other hand,

φ2,M,N⊗H P ◦ (IdMco(H) ⊗ φ2,N,P ) ◦ aMco(H) ,N co(H) ,Pco(H) ((m ⊗ n) ⊗ p) φ2,M,N⊗H P ◦ (IdMco(H) ⊗ φ2,N,P )(EM (X 1 · m) ⊗ (EN (X 2 · n) ⊗ EP (X 3 · p)))

ϕ2,M,N⊗H P EM (X 1 · m) ⊗ [EN (Y 1 · EN (X 2 · n))  ⊗ H EP (Y 2 · EP (X 3 · p)) ·Y 3 ]

= = (6.4.2)

EM (Z 1 X 1 · m) ⊗H EN⊗H P (Z 2 · EN (Y 1 X 2 · n) ⊗H EP (Y 2 X 3 · p) ·Y 3 ) · Z 3

(6.4.3)

EM (Z 1 X 1 · m)

= =

(6.4.2)

=

(6.4.13),(6.4.2)

=

(3.1.9)

=

  2 2 Y 2 X 3 · p)) · Z(2,2) Y 3 · Z3 ⊗ H EN⊗H P (EN (Z12Y 1 X 2 · n) ⊗H EP (Z(2,1)

 EM (Z 1 X 1 · m) ⊗H EN⊗H P EN (Z12 X 2 · n) ⊗H EP (Z22 X 3 · p) · Z 3 EM (Z 1 X 1 · m) ⊗H EN (Y 1 Z12 X 2 · n) ⊗H EP (Y 2 Z22 X 3 · p) ·Y 3 Z 3 EM (X 1Y11 · m) ⊗H EN (X 2Y21 · n) ⊗H EP (X13Y 2 · p) · X23Y 3 ,

as claimed. Secondly, the unit object of H MHH is H, with the structure given by its multiplication and comultiplication. We have H co(H) = k1H since EH (h) = q1 h1 β S(q2 h2 ) = ε (h)X 1 β S(X 2 )α X 3 = ε (h)1H , ∀ h ∈ H. It follows that φ0 : k → G(H) = H co(H) , φ0 (κ ) = κ 1H , for all κ ∈ k, is a left H-module isomorphism and closes commutatively the two square diagrams in (1.3.2). So far we have shown that (G, φ0 , φ2 ) is a strong monoidal functor which at the same time defines an equivalence between the categories H MHH and H M . By Proposition 1.31 it follows that H MHH and H M are strong monoidally equivalent. Remark 6.24 The equivalence inverse of the functor G from the proof of Theorem 6.23 is the functor F = • ⊗ H : H M → H MHH , which coincides with the functor F defined in Remark 6.17 in the case when A = H. The functor F is strong monoidal via the structure given by ϕ2 determined, for all M, N ∈ H M , by the following com-

6.4 The Categories H MHH and H M

245

position of isomorphisms: F(M) ⊗H F(N) = (M ⊗H)⊗H (N ⊗H) ∼ = M ⊗(N ⊗H) ∼ = (M ⊗N)⊗H = F(M ⊗N), and ϕ0 = IdH : H → F(k) = k ⊗ H ∼ = H. Explicitly, we have that

ϕ2,M,N ((m ⊗ h) ⊗H (n ⊗ h )) = (x1 · m ⊗ x2 h1 · n) ⊗ x3 h2 h ,

(6.4.15)

for all m ∈ M, h, h ∈ H and n ∈ N. By working with alternative coinvariants instead of coinvariants of the second type we get a second strong monoidal equivalence between the categories H MHH and H M as follows. Corollary 6.25

Let H be a quasi-Hopf algebra with bijective antipode and con-

F G

H sider H M H MH the pair of functors defined in Remark 6.17, specialized for A = H. Then F, G are strong monoidal functors and they induce a strong monoidal equivalence between H MHH and H M .

Proof The functor G is naturally isomorphic to the functor G defined in the proof of Theorem 6.23. The natural isomorphism between them is given by the natural transformation   E = E M : G(M) = M co(H) → G(M) = M co(H ) . H M∈H MH

Note that the inverse natural transformation of E is E, and that F = F. Thus, by Theorem 6.23 it follows that F, G are strong monoidal functors, and that they provide a strong monoidal equivalence of categories. We end by pointing out that the strong monoidal structure of G is given by φ 2,M,N : M co(H) ⊗ N co(H) → (M ⊗H N)co(H) determined by the composition E ⊗E

φ2,M,N

E M⊗ N

H (M ⊗H N)co(H) , M co(H) ⊗ N co(H) M−→N M co(H) ⊗ N co(H) −→ (M ⊗H N)co(H) −→

and φ 0 : k → G(H) = kβ defined by φ 0 (κ ) = κβ , for all κ ∈ k. Explicitly, we have

φ 2,M,N (m ⊗ n) = q1 x11 · m · S(q2 x21 )x2 ⊗H n · S(x3 ),

(6.4.16)

for all M, N ∈ H MHH and m ∈ M co(H) , n ∈ N co(H) . Indeed, by taking into account the above definitions and structures we have that

φ 2,M,N (m ⊗ n) =

E M⊗H N φ2,M,N (EM (m) ⊗ EN (n))

=

E M⊗H N (EM (X 1 · EM (m)) ⊗H EN (X 2 · EN (n)) · X 3

(6.4.2)

=

E M⊗H N (EM (X 1 · m) ⊗H EN (X 2 · n) · X 3

=

EM (X 1 · m)(0) ⊗H EN (X 2 · n)(0) · X13 β S(EM (X 1 · m)(1) EN (X 2 · n)(1) X23 )

(3.2.1)

=

EM (m)(0) ⊗H EE(n) · S(EM (m)(1) )

246

Quasi-Hopf Bimodule Categories (6.4.11)

EM (x1 · EM (m)) · x2 ⊗H n · S(x3 )

(6.4.2)

=

EM (x1 · m) · x2 ⊗H n · S(x3 )

=

q1 x11 · m(0) · β S(q2 x21 m(1) )x2 ⊗H n · S(x3 )

=

q1 x11 · E M (m) · S(q2 x21 )x2 ⊗H n · S(x3 )

=

q1 x11 · m · S(q2 x21 )x2 ⊗H n · S(x3 ),

=

for all m ∈ M co(H) and n ∈ N co(H) , as stated.

6.5 A Structure Theorem for Comodule Algebras We shall see that the structure theorem for quasi-Hopf bimodules provides a structure theorem for algebras within categories of quasi-Hopf bimodules. We begin with a lemma of independent interest. Lemma 6.26 Let H be a quasi-bialgebra and H MHH the category of right quasiHopf bimodules over H equipped with the monoidal structure presented in Proposition 6.8. Then giving an algebra A in H MHH is equivalent to giving a triple (A, ρ , i) consisting of a k-algebra A, a k-linear map ρ : A → A ⊗ H and a k-algebra morphism i : H → A such that (A, ρ , Φρ := i(X 1 ) ⊗ X 2 ⊗ X 3 ) is a right H-comodule algebra and i is a right H-comodule morphism, that is, in addition

ρ (i(h)) = i(h1 ) ⊗ h2 , ∀ h ∈ H. Proof Assume that (A, m : A ⊗H A → A, i : H → A) is an algebra in H MHH . Since the forgetful functor from (H MHH , ⊗H , H) to (H MH , ⊗H , H) is strong monoidal we get that (A, m, i) is an algebra in (H MH , ⊗H , H), too (see Proposition 2.3). Otherwise stated, (A, m, i) is an H-ring. Thus, by Example 2.2(5) we obtain that A is a k-algebra with multiplication m = qH A,A m and unit 1A = i(1H ). Furthermore, the input H-bimodule structure of A is completely determined by h · a · h = i(h)ai(h ), for all h, h ∈ H and a ∈ A, and i : H → A becomes a k-algebra morphism. Now, since A is an object in H MHH we have a k-linear map ρ : A a → a0 ⊗a1 ∈ A ⊗ H such that ε (a1 )a0 = a and i(X 1 )a0,0 ⊗ X 2 a0,1 ⊗ X 3 a1 = a0 i(X 1 ) ⊗ a11 X 2 ⊗ a12 X 3 , ∀ a ∈ A, that is, (4.3.3) and (4.3.1) hold. Furthermore, ρ is an H-bimodule morphism, and so

ρ (i(h)ai(h )) = i(h1 )a0 i(h1 ) ⊗ h2 a1 h2 , ∀ a ∈ A and h, h ∈ H. Clearly, this implies ρ (i(h)) = i(h1 ) ⊗ h2 , for all h ∈ H. The latter equality allows us to show that Φρ := i(X 1 ) ⊗ X 2 ⊗ X 3 satisfies (4.3.2). (4.3.4) is automatic. It remains to prove that ρ is a k-algebra morphism. This follows easily from the fact that m and i are right H-colinear morphisms; we leave the verification of this detail to the reader. So we have shown that (A, ρ , Φρ ) is a right H-comodule algebra and i : H → A is a right H-comodule algebra morphism.

6.5 A Structure Theorem for Comodule Algebras

247

For the converse, assume that we have a datum (A, ρ , i) as in the statement. First, A becomes an H-bimodule via i, that is, h · a · h = i(h)ai(h ), for all h, h ∈ H and a ∈ A. Together with ρ this makes A an object in H MHH . Indeed, we now prove that ρ : A → A ⊗ H is an H-bimodule map. We compute:

ρ (h · a · h ) = ρ (i(h)ai(h )) = ρ (i(h))ρ (a)ρ (i(h )) = (i(h1 ) ⊗ h2 )(a0 ⊗ a1 )(i(h1 ) ⊗ h2 ) = i(h1 )a0 i(h1 ) ⊗ h2 a1 h2 = h1 · a0 · h1 ⊗ h2 a1 h2 = h · ρ (a) · h , as required. Obviously we have (IdA ⊗ ε ) ◦ ρ = IdA . Finally, it is easy to see that Φ · (ρ ⊗ IdA )(ρ (a)) = (IdA ⊗ Δ)(ρ (a)) · Φ, because this is the condition Φρ (ρ ⊗ IdA )(ρ (a)) = (IdA ⊗ Δ)(ρ (a))Φρ from the definition of a right H-comodule algebra, due to the fact that Φρ = i(X 1 ) ⊗ X 2 ⊗ X 3 . Hence with the above structure A is indeed a right quasi-Hopf H-bimodule. Since A is an associative unital k-algebra and i : H → A is a k-algebra morphism we get from Example 2.2(5) that (A, m, i) with m : A ⊗H A → A, m(a ⊗H a ) = aa , for all a, a ∈ A, is an algebra in (H MH , ⊗H , H). A simple inspection shows that (A, m, i) is, moreover, an algebra in H MHH , where H MHH has the monoidal structure from Proposition 6.8. Corollary 6.27 Let H be a quasi-bialgebra and A a left H-module algebra. Then A#H, the smash product between A and H, is an algebra in H MHH . Proof By Proposition 5.1, A#H is a k-algebra and j : H h → 1A #h ∈ A#H is a k-algebra map. Furthermore, with the structure as in Proposition 5.9, A#H becomes a right H-comodule algebra; note that its reassociator is just j(X 1 ) ⊗ X 2 ⊗ X 3 . Since

ρ ( j(h)) = ρ (1A #h) = x1 · 1A #x2 h1 ⊗ x3 h2 = 1A #h1 ⊗ h2 = j(h1 ) ⊗ h2 , for all h ∈ H, by Lemma 6.26 we conclude that A#H is an algebra in H MHH . We next show that, in the case when H is a quasi-Hopf algebra, any algebra A in H H MH is of the form presented in Corollary 6.27, for a certain algebra in H M . Unless otherwise specified, from now on H is a quasi-Hopf algebra and A is an algebra within H MHH with structure (ρ , i) as in Lemma 6.26. For the H-comodule algebra (A, ρ , Φρ ) we keep the same type of notation as in Section 4.3. Theorem 6.28 Let H be a quasi-Hopf algebra with bijective antipode and A an algebra in H MHH . Then there exists a left H-module algebra A such that A ∼ = A#H, as algebras in H MHH .

248

Quasi-Hopf Bimodule Categories

Proof We know from Corollary 6.25 that we have a strong monoidal functor G : H H H MH → H M . So to the algebra A in H MH there corresponds an algebra G(A) = co(H) A in H M , which will be denoted by A in what follows. By the definition of G in Remark 6.17 we get that A is a left H-module via the action given by h → a = h1 · a · S(h2 ) = i(h1 )ai(S(h2 )) := h i a, for all h ∈ H and a ∈ A ⊆ A. Keeping in mind the strong monoidal structure of G obtained in (6.4.16) and the proof of Proposition 2.3, we deduce that the multiplication of A in H M is a ∗ a = G(m)φ 2,A,A (a ⊗ a ) = (q1 x11 · a · S(q2 x21 )x2 )(a · S(x3 )) = i(q1 x11 )ai(S(q2 x21 )x2 )a i(S(x3 )), for all a, a ∈ A, while its unit is given by G(i)φ 0 (1k ) = G(i)(β ) = i(β ). But, using the definitions of qR , qL , (3.1.9) and (3.2.1) we deduce easily that X 1 ⊗ S(X 2 )q˜1 X13 ⊗ q˜2 X23 = q1 x11 ⊗ S(q2 x21 )x2 ⊗ x3 ,

(6.5.1)

and so we get that, for all a, a ∈ H, a ∗ a = i(X 1 )ai(S(X 2 )q˜1 X13 )a i(S(q˜2 X23 ) := a ◦ a .

(6.5.2)

The notation i and ◦ is imposed by analogy with the structure in Proposition 4.3. Hence, summing up, by (5.1.1) the multiplication in A#H is given by (a#h)(a #h ) = i(X 1 x11 )bi(S(y1 X 2 x21 )α y2 X13 x12 h(1,1) )b i(S(y3 X23 x22 h(1,2) ))#x3 h2 h , (6.5.3) for all a, a ∈ A and h, h ∈ H. On the other hand, by the structure theorem in Remark 6.17 we get that χ : A ⊗ H → A given by χ (a ⊗ h) = q1 · a · S(q2 )h = i(q1 )ai(S(q2 )h), for all a ∈ A and h ∈ H, is an isomorphism in H MHH with inverse χ −1 : A → A ⊗ H defined by

χ −1 (a) = E(a0 ) ⊗ a1 = a0,0 · β S(a0,1 ) ⊗ a1 = a0,0 i(β S(a0,1 )) ⊗ a1 , for all a ∈ A. So to end the proof it suffices to show that χ is an algebra morphism in H MHH , if it is considered as a morphism between A#H and A. To this end, we compute:

χ ((a#h)(a #h )) =

i(Z 1 X 1 x11 )ai(S(y1 X 2 x21 )α y2 X13 x12 h(1,1) )a i(S(Z 2 y3 X23 x22 h(1,2) )α Z 3 x3 h2 h )

(3.1.9),(3.1.7)

=

(3.2.1)

3 i(X 1 )ai(S(y1 z1 X 2 )α y2 z21 X(1,1) h(1,1) )a

6.6 Coalgebras in H MHH

249

3 i(S(y3 z22 X(1,2) h(1,2) )α z3 X23 h2 h ) (3.1.9),(3.2.1)

i(X 1 )ai(S(X 2 )α Y 1 (X 3 h)(1,1) )a i(S(Y 2 (X 3 h)(1,2) )α Y 3 (X 3 h)2 h )

(3.1.7),(3.2.1)

=

i(X 1 )ai(S(X 2 )α X 3 h)i(Y 1 )a i(S(Y 2 )α Y 3 h )

=

χ (a#h)χ (a #h ),

=

for all a, a ∈ A and h, h ∈ H. It follows that χ (i(β )#1H ) = 1A , hence χ is an algebra morphism, as stated. Observe also that

χ ( j(X 1 )) ⊗ X 2 ⊗ X 3 = χ (1A #X 1 ) ⊗ X 2 ⊗ X 3 = i(q1 )i(β )i(S(q2 )X 1 ) ⊗ X 2 ⊗ X 3 (3.2.2)

= i(X 1 ) ⊗ X 2 ⊗ X 3 ,

and this completes our proof. Remarks 6.29 (1) In the hypothesis of Theorem 6.28, since i : H → A is an algebra morphism, we have an algebra structure on A within H M ; see Proposition 4.3. It is easy to see that the algebra A in H M constructed in the proof of the cited theorem is actually a subalgebra of Ai in H M . (2) Let H be a quasi-Hopf algebra with bijective antipode, A a left H-module algebra and A := A#H; then, by Corollary 6.27, we have that A is an algebra in H H MH . In addition, Aco(H) = GF(A) = {p1 · a ⊗ p2 | a ∈ A} ∼ = A, an isomorphism of algebras in H M . Hence, the structure theorem allows us to recover the structure of A from the one of A#H; the details are left to the reader.

6.6 Coalgebras in H MHH We now move to the coalgebra case. As we pointed out several times, the quasi-Hopf algebra notion is not selfdual, thus the results of this section cannot be viewed as the formal dual of those proved for quasi-Hopf comodule algebras. But we should stress the fact that in both situations the key role is played by the monoidal equivalence between H MHH and H M . Recall from Example 2.14(5) that if A is a k-algebra then a coalgebra in A MA is called an A-coring. As we shall see next, any left module coalgebra C over a quasibialgebra H defines an H-coring structure on the k-vector space C ⊗ H. Proposition 6.30 Let H be a quasi-bialgebra and C a left H-module coalgebra. Then C ⊗ H is an H-coring, with structure given by h · (c ⊗ h ) · h = h1 · c ⊗ h2 h h , Δ(c ⊗ h) = (X · c1 ⊗ 1H ) ⊗H (X · c2 ⊗ X h) 1

2

3

and ε (c ⊗ h) = εC (c)h,

(6.6.1) (6.6.2)

250

Quasi-Hopf Bimodule Categories

for all c ∈ C and h, h , h ∈ H, where ΔC (c) := c1 ⊗ c2 , for all c ∈ C, is the comultiplication of the coalgebra C in H M , and · is the left action of H on C. Proof Clearly C ⊗ H is an H-bimodule with structure as in (6.6.1). Secondly, Δ is H-bilinear since Δ(h1 · c ⊗ h2 h h ) = (X 1 h(1,1) · c1 ⊗ 1H ) ⊗H (X 2 h(1,2) · c2 ⊗ X 3 h2 h h ) (3.1.7)

= (h1 X 1 · c1 ⊗ 1H ) ⊗H (h(2,1) X 2 · c2 ⊗ h(2,2) X 3 h h ) = (h1 X 1 · c1 ⊗ 1H ) ⊗H h2 · (X 2 · c2 ⊗ X 3 h h ) = (h1 X 1 · c1 ⊗ h2 ) ⊗H (X 2 · c2 ⊗ X 3 h h ) = h · Δ(c ⊗ h ) · h ,

for all c ∈ C and h, h , h ∈ H. It can be easily checked that ε is H-bilinear, too, and that ε is a counit for Δ. Also, the fact that Δ is coassociative follows from the coassociativity of ΔC in H M and the 3-cocycle property of the reassociator Φ of H. We leave the verification of these details to the reader. We call the structure in Proposition 6.30 an H-coring structure defined by a left H-module coalgebra. The next result describes the structure of a coalgebra within H MHH , in the case when H is a quasi-Hopf algebra with bijective antipode. Theorem 6.31 Let H be a quasi-Hopf algebra with bijective antipode. Then there exists a one-to-one correspondence between (i) coalgebra structures in H MHH ; (ii) coalgebra structures in H M ; (iii) H-coring structures defined by left H-module coalgebras. Proof The one-to-one correspondence between (i) and (ii) is established by the monoidal category equivalence between H MHH and H M . Up to an isomorphism, any coalgebra C in H MHH is of the form C ⊗ H for a suitable coalgebra C in H M . Once more, note that C ⊗ H is an object in H MHH via the structure determined by h · (c ⊗ h ) · h = h1 · c ⊗ h2 h h ,

ρC⊗H (c ⊗ h) = (x1 · c ⊗ x2 h1 ) ⊗ x3 h2 , for all c ∈ H and h, h , h ∈ H. Furthermore, by the strong monoidal structure of the functor F defined in Remark 6.24, we deduce that C ⊗ H is a coalgebra in H MHH with comultiplication and counit given by F(ΔC )

−1 ϕ2,C,C

Δ : C ⊗ H = F(C) −→ F(C ⊗C) −→ F(C) ⊗H F(C) = (C ⊗ H) ⊗H (C ⊗ H) F(εC )

ϕ −1

0 and ε : C ⊗ H = F(C) −→ F(k) −→ H,

6.7 Notes

251

where, as before, ΔC and εC are the comultiplication and the counit of the coalgebra C in H M . According to Theorem 6.23, we find that Δ and ε are given by (6.6.2). On the other hand, since the forgetful functor from H MHH to H MH is strong monoidal, it follows that a coalgebra in H MHH is nothing but an H-coring (C, ΔC , ε C ) for which the comultiplication ΔC : C → C ⊗H C and the counit ε C are right H-colinear maps. Since C ≡ C ⊗ H in H MHH , with C = Cco(H ) , a left H-module coalgebra, from the arguments presented above we conclude that ΔC and ε C are as in (6.6.2), and therefore right H-colinear maps. Thus the one-to-one correspondence between (i) and (iii) is established, too.

6.7 Notes Quasi-Hopf H-bimodules were introduced by Hausser and Nill in [109]. Afterwards, this concept was generalized in [63]. The structure theorems for generalized quasiHopf bimodules were proved in [63] as well, inspired at that time by the alternative structure theorem given in [47]. But the first structure theorem for quasi-Hopf bimodules is due to Hausser and Nill [109], as well as the monoidal equivalence between the category of quasi-Hopf H-bimodules and the category of H-modules. The duality theory for quasi-Hopf bimodules is based on the results published in [49], which gave a categorical flavour to some results obtained previously in [109]. Schauenburg noticed in [200] that H ∗ being a quasi-Hopf H-bimodule is a consequence of a categorical monoidal result proved by Pareigis in [182]. The content of Section 6.5 is taken from [44, 180], while that of Section 6.6 is taken from [44].

7 Finite-Dimensional Quasi-Hopf Algebras

The main goal of this chapter is to show that for a finite-dimensional quasi-Hopf algebra H the space of integrals in H, and the space of cointegrals on H, has dimension 1. We characterize semisimple and symmetric quasi-Hopf algebras with the help of integrals, and prove a formula for the fourth power of the antipode in terms of the modular elements by using the machinery provided by Frobenius algebras. The chapter ends with a freeness theorem stating that any finite-dimenisonal quasi-Hopf algebra is free over any quasi-Hopf subalgebra.

7.1 Frobenius Algebras Throughout this section A is a finite-dimensional algebra with unit 1A over a field k. By A∗ we denote the k-linear dual of A, that is A∗ = Homk (A, k), the set of k-linear maps from A to k. It can be easily checked that A∗ is an A-bimodule via the A-actions defined by a  a∗ , a  = a∗ , a a and a∗  a, a  = a∗ , aa , for all a∗ ∈ A∗ and a, a ∈ A. Also, the multiplication of A induces on A an A-bimodule structure as well. Proposition 7.1 For a finite-dimensional k-algebra A the following assertions are equivalent: (i) There exists a pair (φ , e) consisting of a k-linear map φ : A → k and an element e = e1 ⊗ e2 ∈ A ⊗ A (formal notation, summation implicitly understood) such that ae1 ⊗ e2 = e1 ⊗ e2 a , ∀ a ∈ A, and φ (e1 )e2 = φ (e2 )e1 = 1A ; (ii) A is isomorphic to A∗ as a right A-module; (iii) A is isomorphic to A∗ as a left A-module; (iv) A has a k-coalgebra structure (Δ, ε ) such that Δ : A → A ⊗ A is an A-bimodule morphism, where A ⊗ A is considered as an A-bimodule via the multiplication of A; (v) there exists a bilinear map Br : A×A → k which is right non-degenerate and associative, that is, Br (x, a) = 0, for all a ∈ A, implies x = 0, and Br (ab, c) = Br (a, bc), for all a, b, c ∈ A;

254

Finite-Dimensional Quasi-Hopf Algebras

(vi) there exists a bilinear map Bl : A × A → k which is left non-degenerate (i.e. Bl (a, x) = 0 for all a ∈ A implies x = 0) and associative; (vii) there exists a hyperplane (linear subspace of codimension 1) in A that does not contain either left or right non-zero ideals. Proof We sketch the proof, leaving the verification of the details to the reader. (i) ⇒ (ii). It can be easily seen that f : A → A∗ given by f (a) = φ  a, for all a ∈ A, is an isomorphism of right A-modules. Its inverse is f −1 : A∗ → A defined by f −1 (a∗ ) = a∗ (e1 )e2 , for all a∗ ∈ A∗ . (ii) ⇒ (i). Let {ai , ai }i be dual bases in A and A∗ . If f : A → A∗ is a right A-linear isomorphism then the pair ( f (1A ), ∑i ai ⊗ f −1 (ai )) obeys the required conditions. (ii) ⇒ (iii). If f : A → A∗ is an isomorphism of right A-modules then θ

f∗

A A∗∗ → A∗ g:A→

is an isomorphism of left A-modules, where θA is the canonical isomorphism (see also Corollary 1.76) and f ∗ is the transpose of f . The implication (iii) ⇒ (ii) can be proved in a similar manner. (ii) ⇒ (iv). Let f : A → A∗ be a right A-linear isomorphism. As, for instance, the left dual functor (−)∗ is monoidal, it follows that A∗ admits a coalgebra structure within the reverse monoidal category associated to k M , and therefore in k M , too. If we carry on A the coalgebra structure on A∗ through the isomorphism f we get that A is a k-coalgebra via the structure determined by Δ(a) = ∑ f (a), ai a j  f −1 (a j ) ⊗ f −1 (ai ) and ε (a) =  f (a), 1A , i, j

for all a ∈ A. From the right A-linearity of f we deduce that the above morphism Δ is an A-bimodule morphism, as required. (iv) ⇒ (i). If (A, Δ, ε ) is a k-coalgebra with Δ an A-bimodule morphism, then the pair (ε , Δ(1A )) satisfies the conditions in (i). (ii) ⇔ (v). For f : A → A∗ a right A-module isomorphism define Br : A × A → k, Br (a, b) = f (a)(b), for all a, b ∈ A. Then Br is an associative and right nondegenerate bilinear form. Conversely, given Br we have that f : A → A∗ , f (a)(b) = Br (a, b), for all a, b ∈ A, is a right A-linear isomorphism. (iii) ⇔ (vi). Similar to (ii) ⇔ (v): given g we define Bl : A × A → k by Bl (a, b) = g(b)(a), for all a, b ∈ A, and vice versa. (v) ⇒ (vii). Let Br be a right non-degenerate and associative bilinear map on A. Then A a → Br (1A , a) ∈ k is non-zero, and therefore its kernel H := {a ∈ A | Br (1A , a) = 0} is a hyperplane in A. If I is a right ideal in A contained in H then, for any x ∈ I, we have 0 = Br (1A , xa) = Br (x, a), for all a ∈ A. Now using that Br is right non-degenerate we get x = 0, and so I = 0. (vii) ⇒ (ii). Let H be a hyperplane in A. As k is a field one can find φ ∈ A∗ such that its kernel is H.

7.1 Frobenius Algebras

255

Assume that H does not contain non-zero right ideals. If f : A → A∗ is defined by f (a)(b) = φ (ab), for all a, b ∈ A, then ( f (a)  b)(c) = f (a)(bc) = φ (a(bc)) = φ ((ab)c) = f (ab)(c), for all a, b, c ∈ A, hence f is right A-linear. In addition, if x ∈ A is such that f (x) = 0 then φ (xa) = 0, for all a ∈ A, thus xA ⊆ H. Consequently, xA = 0, and so x = 0. Therefore f is injective, and so an isomorphism because A and A∗ have the same dimension over k. If A does not contain non-zero left ideals then in a manner similar to the one above one can show that f  : A → A∗ given by f  (a)(b) = φ (ba), for all a, b ∈ A, is left Alinear and an isomorphism. So we are done. Definition 7.2 If a finite-dimensional k-algebra A satisfies one, and therefore all, of the equivalent conditions in Proposition 7.1 we say that A is a Frobenius algebra. Furthermore, if (φ , e) is a Frobenius system for a Frobenius algebra A (i.e. a pair as in Proposition 7.1 (i)) we refer to φ as a Frobenius morphism for A, and call e a Frobenius element for A. Remark 7.3 Let f : A → A∗ be an isomorphism of left/right A-modules and g : A → A∗ the isomorphism of right/left A-modules corresponding to f as in the above proposition. Then the Frobenius systems defined by f and g coincide. Indeed, by the proof of Proposition 7.1 we have that g(a)(b) = ( f ∗ ◦ θA )(a)(b) = θA (a)( f (b)) = f (b)(a), for all a, b ∈ A. Thus, if f is left (resp. right) A-linear and φ = f (1A ) then f (b) = b  φ (resp. f (b) = φ  b), for all b ∈ A, and so g(a)(b) = φ (ab) = (φ  a)(b) (resp. g(a)(b) = φ (ba) = (a  φ )(b)), for all a, b ∈ A. We deduce that g(a) = φ  a (resp. g(a) = a  φ ), for all a ∈ A, thus g(1A ) = φ in both situations. This says that the Frobenius morphism associated to g is φ , as required. Now, if f is left A-linear then (φ , ∑i f −1 (ai ) ⊗ ai ) is the Frobenius system associated to f , while (φ , ∑i ai ⊗ g−1 (ai )) is the Frobenius system associated to g. Since g−1 (a∗ ) = θA−1 (a∗ ◦ f −1 ) = ∑ a∗ ( f −1 (ai ))ai , i

it follows that

∑ ai ⊗ g−1 (ai ) = ∑ ai ( f −1 (a j ))ai ⊗ a j = ∑ f −1 (a j ) ⊗ a j , i

i, j

j

as required. Likewise, if f is right A-linear then the Frobenius system induced by it is (φ , ∑i ai ⊗ f −1 (ai )), and the Frobenius system corresponding to g is (φ , ∑i g−1 (ai ) ⊗ ai ). According to the last computation they coincide, so the proof is complete. Examples 7.4 (1) If A, A are Frobenius algebras then so are A × A and A ⊗ A . More precisely, assume that Bl and Bl are left non-degenerate, associative bilinear maps on A and A , respectively. One can easily see that Bl : (A × A ) × (A × A ) → k

256

Finite-Dimensional Quasi-Hopf Algebras

given by Bl ((a, a ), (b, b )) = Bl (a, b) + Bl (a , b ), ∀ a, b ∈ A and a , b ∈ A , is a left non-degenerate, associative bilinear map on A × A . Thus A × A is a Frobenius algebra as well.

l : (A ⊗ A ) × (A ⊗ A ) → k by Similarly, if we define B

l (a ⊗ a , b ⊗ b ) = Bl (a, b)Bl (a , b ), ∀ a, b ∈ A and a , b ∈ A , B

l is a left non-degenerate, associative then a simple verification guarantees that B   bilinear map on A ⊗ A . Therefore A ⊗ A is a Frobenius algebra, as stated. (2) For any non-zero natural number n the n × n-matrix algebra Mn (k) is a Frobenius algebra. To see this, consider the pair (φ = Tr, e = ∑i, j Ei j ⊗ E ji ), where Tr : Mn (k) → k is the trace morphism and {Ei j | 1 ≤ i, j ≤ n} is the canonical basis of Mn (k). We have that Tr is a Frobenius morphism for Mn (k) since

∑ Tr(Ei j )E ji = ∑ δi, j E ji = ∑ Eii = In , i, j

i, j

i

and similarly ∑i, j Tr(E ji )Ei j = In . Here δi, j is the Kronecker delta and In is the unit matrix of Mn (k). Now, e is a Frobenius element for Mn (k) since, for any X = (xi j )1≤i, j≤n ∈ Mn (k),

∑ XEi j ⊗ E ji = ∑ i, j

xuv Euv Ei j ⊗ E ji

i, j,u,v

=

∑ xui Eu j ⊗ E ji = ∑

i, j,u

i, j,u,v

Eu j ⊗ xvi E ju Evi = ∑ Eu j ⊗ E ju X,

as required. (3) For any non-zero natural number n, A = this, denote by x the class of X modulo basis of A. If we define φ : A → k by  

φ

n

∑ a j x j−1

j=1

(X n )

j,u

k[X] (X n )

is a Frobenius algebra. To prove

and take {1, x, . . . , xn−1 } the canonical n

= an and e = ∑ xi−1 ⊗ xn−i i=1

we claim that (φ , e) is a Frobenius system for A. Indeed, one can easily see that n

n

i=1

i=1

∑ φ (xi−1 )xn−i = ∑ φ (xn−i )xi−1 = φ (xn−1 ) = 1.

Also, for any 1 ≤ j ≤ n − 1, we have n

x j e = ∑ x j+i−1 ⊗ xn−i i=1 j−1

=

n− j−1

∑ xt ⊗ x j−t−1 + ∑

t=0

t=0

x j+t ⊗ xn−t−1

7.1 Frobenius Algebras

257

n

= ∑ xi−1 ⊗ xn+ j−i = ex j , i=1

and so ae = ea, for all a ∈ A, because {1, x, . . . , xn−1 } is a basis for A. Let A be a Frobenius algebra and (φ , e) a Frobenius system for A. We have seen that f : A → A∗ defined by f (a) = φ  a, for all a ∈ A, is a right A-linear isomorphism. So for any a ∈ A there exists a unique element χ (a) ∈ A such that a  φ = φ  χ (a). In this way we have a well-defined map χ : A → A. Proposition 7.5 The map χ is an algebra automorphism of A. Furthermore, it can be explicitly computed in terms of the Frobenius system (φ , e = e1 ⊗ e2 ) of A as

χ (a) = φ (e1 a)e2 , ∀ a ∈ A. Proof

(7.1.1)

On the one hand, for all a, b ∈ A we have a  (b  φ ) = a  (φ  χ (b)) = (a  φ )  χ (b) = (φ  χ (a))  χ (b) = φ  χ (a)χ (b),

from which we deduce that a  (b  φ ) = f (χ (a)χ (b)). On the other hand, a  (b  φ ) = ab  φ = φ  χ (ab) = f (χ (ab)). By the injectivity of f it follows that χ is multiplicative. Clearly 1A  φ = φ = φ  1A , so χ (1A ) = 1A . We can now conclude that χ is an algebra endomorphism of A. It has also been proved that f  : A → A∗ given by f  (a) = a  φ is an isomorphism of left A-modules. Thus for any a ∈ A there exists a unique χ −1 (a) ∈ A such that φ  a = χ −1 (a)  φ , and so we have a well-defined map χ −1 : A → A. By the two definitions of χ and χ −1 we have f  (a) = a  φ = φ  χ (a) = χ −1 (χ (a))  φ = f  (χ −1 (χ (a))), f (a) = φ  a = χ −1 (a)  φ = φ  χ (χ −1 (a)) = f (χ (χ −1 (a))), for all a ∈ A. Since f , f  are injective maps it follows that χ and χ −1 are bijective inverses. Hence χ is an algebra automorphism of A, as desired. In addition, for all a, b ∈ A, we compute φ  φ (e1 a)e2 , b = φ (e1 a)φ (e2 b) = φ (be1 a)φ (e2 ) = φ (bφ (e2 )e1 a) = φ (ba) = a  φ , b, and therefore φ  φ (e1 a)e2 = a  φ , for all a ∈ A, proving (7.1.1). Definition 7.6 The Nakayama automorphism of a Frobenius algebra A corresponding to a Frobenius system (φ , e) is the algebra automorphism χ of A defined by a  φ = φ  χ (a), for all a ∈ A.

258

Finite-Dimensional Quasi-Hopf Algebras

Remark 7.7 The inverse of the Nakayama automorphism χ associated to a Frobenius algebra A with Frobenius system (φ , e) is given by

χ −1 (a) = φ (ae2 )e1 , ∀ a ∈ A.

(7.1.2)

Indeed, we compute (7.1.2)

χ χ −1 (a) = φ (ae2 )χ (e1 ) (7.1.1)

= φ (ae2 )φ (e1 e1 )e2 = φ (ae2 e1 )φ (e1 )e2 = φ (ae1 )e2 = φ (e1 )e2 a = a,

where e1 ⊗ e2 is another copy of e, and (7.1.1)

χ −1 χ (a) = φ (e1 a)χ −1 (e2 ) (7.1.2)

= φ (e1 a)φ (e2 e2 )e1 = φ (e2 e1 a)φ (e2 )e1 = φ (e2 a)e1 = φ (e2 )ae1 = a,

for all a ∈ A, as needed. Examples 7.8 (1) For Mn (k) with Frobenius system (Tr, ∑i, j Ei j ⊗ E ji ) as in Example 7.4(2) the Nakayama automorphism is the identity morphism of Mn (k). To see this we use (7.1.1) to compute

χ (X) = ∑ Tr(Ei j X)E ji = i, j

∑

Tr(xuv Ei j Euv )E ji =

i, j,u,v

∑ Tr(x jv Eiv )E ji = ∑ x ji E ji = X,

i, j,v

i, j

for all X = (xuv )1≤u,v≤n , as stated. k[X] (2) For A = (X n ) with the Frobenius system as in Example 7.4(3), the Nakayama automorphism is the identity map. This follows from n

n

i=1

i=1

χ (x j ) = ∑ φ (x j+i−1 )xn−i = ∑ δi,n− j xn−i = x j , for all 0 ≤ j ≤ n − 1, and the fact that {1, x, . . . , xn−1 } is a basis for A. We next show that a Frobenius system is unique up to an invertible element, in the following sense. Proposition 7.9 Let A be a Frobenius algebra with a Frobenius system (φ , e = e1 ⊗ e2 ). Then any other Frobenius system for A is of the form (φ  d, e1 ⊗ d −1 e2 ), for some invertible element d ∈ A with inverse d −1 , or equivalently of the form (d   φ , e1 d −1 ⊗ e2 ), for some invertible element d  ∈ A with inverse d −1 .

7.1 Frobenius Algebras

259

Proof If d ∈ A is an invertible element with inverse d −1 then one can easily check that both (φ  d, e1 ⊗ d −1 e2 ) and (d  φ , e1 d −1 ⊗ e2 ) are Frobenius systems for A. Conversely, let (ψ , b = b1 ⊗b2 ) be another Frobenius system for A. Then f , g : A → ∗ A given by f (a) = φ  a, and by g(a) = ψ  a, respectively, for all a ∈ A, are right A-linear isomorphisms. Furthermore, f −1 (a∗ ) = a∗ (e1 )e2 and g−1 (a∗ ) = a∗ (b1 )b2 , for all a∗ ∈ A∗ . For ψ ∈ A∗ there exists a unique d ∈ A such that f (d) = ψ , that is, φ  d = ψ . Similarly, for φ ∈ A∗ there exists a unique d −1 ∈ A such that g(d −1 ) = φ , that is, ψ  d −1 = φ . We get g(1A ) = ψ = φ  d = ψ  d −1 d = g(d −1 d), f (1A ) = φ = ψ  d −1 = φ  dd −1 = f (dd −1 ), and since f and g are injective maps it follows that d is invertible in A with inverse d −1 . For further use note that d and d −1 can be computed explicitly as d = ψ (e1 )e2

and d −1 = φ (b1 )b2 ,

(7.1.3)

since (φ  ψ (e1 )e2 )(a) = ψ (e1 )φ (e2 a) = ψ (ae1 φ (e2 )) = ψ (a), for all a ∈ A, and similarly ψ  φ (b1 )b2 = φ . Note now that f (a) = φ  a = ψ  d −1 a = g(d −1 a), ∀ a ∈ A, from which we obtain that f = g  d −1 or, equivalently, that g = f  d. It is clear at this point that g−1 (a∗ ) = d −1 f −1 (a∗ ), for all a∗ ∈ A∗ . Thus a∗ (b1 )b2 = a∗ (e1 )d −1 e2 , for all a∗ ∈ A∗ , and this is equivalent to b1 ⊗ b2 = e1 ⊗ d −1 e2 , as required. To land at the second possibility for (ψ , b) we use the Nakayama automorphism χ associated to the Frobenius system (φ , e). We obtain that ψ = φ  d = χ −1 (d)  φ , and therefore we set d  = χ −1 (d). If follows that d  is invertible with inverse d −1 = χ −1 (d −1 ), and so e1 d −1 ⊗ e2 = e1 χ −1 (d −1 ) ⊗ e2 (7.1.2)

= φ (d −1 e2 )e1 e1 ⊗ e2 = φ (d −1 e2 e1 )e1 ⊗ e2 = φ (e1 )e1 ⊗ e2 d −1 e2 = e1 ⊗ d −1 e2 = b1 ⊗ b2 ,

where e1 ⊗ e2 denotes a second copy of e. So our proof is complete. Corollary 7.10 Let A be a Frobenius algebra, χ the Nakayama automorphism associated to the Frobenius system (φ , e) of A and η the Nakayama automorphism corresponding to a second Frobenius system (ψ , b) for A. Then there exists an invertible element d in A such that χ (a) = d η (a)d −1 , for all a ∈ A.

260

Finite-Dimensional Quasi-Hopf Algebras

Proof By the previous result there exists an invertible element d ∈ A such that ψ = φ  d and b1 ⊗ b2 = e1 ⊗ d −1 e2 . We can compute: (7.1.1)

η (a) = ψ (b1 a)b2 = (φ  d)(e1 a)d −1 e2 = (a  φ )(de1 )d −1 e2 = (φ  χ (a))(de1 )d −1 e2 = φ (χ (a)de1 )d −1 e2 = φ (e1 )d −1 e2 χ (a)d = d −1 χ (a)d, for all a ∈ A, which is clearly equivalent to χ (a) = d η (a)d −1 , for all a ∈ A. We end this section by showing that any Frobenius algebra is a self injective module. This result will be used later on in the proof of the Nichols–Zoeller type theorem for quasi-Hopf algebras. First we need some preparatory work. Lemma 7.11 Let R and S be two rings and P an (R, S)-bimodule which is flat as  := HomS (P, M) is an a left R-module. If M is an injective right S-module then M  injective right R-module, where M is considered as a right R-module via the action  r ∈ R and p ∈ P. ( f · r)(p) = f (r · p), for all f ∈ M,  is exact, and this will Proof We show that the contravariant functor HomR (−, M)  imply that M is injective as a right R-module. Consider 0 → X → Y → Z → 0, a short exact sequence of right R-modules. Since P is flat as a left R-module we get that 0 → X ⊗ R P → Y ⊗R P → Z ⊗ R P → 0 is a short exact sequence of right S-modules. Now using that M is an injective right S-module we obtain that 0 → HomS (Z ⊗R P, M) → HomS (Y ⊗R P, M) → HomS (X ⊗R P, M) → 0 is a short exact sequence of abelian groups. For any right R-module U we have  = HomR (U, HomS (P, M)) ∼ HomR (U, M) = HomS (U ⊗R P, M), the isomorphism being produced by HomR (U, HomS (P, M)) λ → (x ⊗R p → λ (x)(p)) ∈ HomS (U ⊗R P, M). Its inverse is HomS (U ⊗R P, M) μ → (x → (p → μ (x ⊗R p))) ∈ HomR (U, HomS (P, M)).  → HomR (Y, M)  → HomR (X, M)  → 0 is a short It now follows that 0 → HomR (Z, M) exact sequence of abelian groups, so our proof is finished. Corollary 7.12 Let R be a k-algebra, P a projective left R-module and consider P as an (R, k)-bimodule in the obvious way. Then for any k-vector space M the right R-module Homk (P, M) is injective. Consequently, P∗ := Homk (P, k) is injective as a

7.2 Integral Theory

261

right R-module, where the R-action on P∗ is given by (p∗ · r)(p) = p∗ (r · p), for all p∗ ∈ P∗ , r ∈ R and p ∈ P. Proof Any projective module is flat, hence P is flat as a left R-module. Moreover, since k is a field any k-vector space is an injective k-module, so by the previous result we get that Homk (P, M) is an injective right R-module. The second assertion in the statement follows by taking M = k. One can now prove the result announced above. Proposition 7.13 A-module.

If A is a Frobenius algebra then A is injective as a left and right

Proof Since A is a Frobenius algebra we have that A ∼ = A∗ both as left and right A-modules. As A is a free left A-module, it is projective as a left A-module. By Corollary 7.12 we deduce that A∗ is injective as a right A-module, and therefore A is injective as a right A-module. By working now with the right-handed version we get that A is injective as a left A-module as well, so the proof is finished.

7.2 Integral Theory Unless otherwise specified, throughout this section H is a finite-dimensional quasiHopf algebra with an antipode S, {ei }i is a basis of H and {ei }i is the corresponding dual basis of H ∗ . In the first part of this section we show that the spaces of left and right integrals in H are one-dimensional. We then prove that S is bijective and that H is a Frobenius algebra. In the end we will see that the space of integrals of an infinite-dimensional quasi-Hopf algebra with bijective antipode is zero. Definition 7.14 Let H be a quasi-bialgebra. An element t ∈ H is called a left (resp.  right) integral in H if ht = ε (h)t (resp. th = ε (h)t), for all h ∈ H. We denote by lH  (resp. rH ) the space of left (resp. right) integrals in H. If there exists a non-zero left integral in H which is at the same time a right integral, then H is called unimodular. Examples 7.15 (1) For the two-dimensional quasi-Hopf algebra H(2) constructed in Example 3.26 it can be easily checked that t = 1 + g is both a non-zero left and right integral in H(2). Thus H(2) is a unimodular quasi-Hopf algebra. (2) Let H± (8) be the two eight-dimensional quasi-Hopf algebras considered in Example 3.30. If t = (1 + g)x3 then gt = g(1 + g)x3 = (g + 1)x3 = t = ε (g)t, xt = x(1 + g)x3 = (x + xg)x3 = (x − gx)x3 = (1 − g)x4 = 0 = ε (x)t, and so t is a non-zero left integral in H± (8), because g and x generate H± (8) as an algebra. In a similar manner one can show that r = (1 − g)x3 is a non-zero right

262

Finite-Dimensional Quasi-Hopf Algebras

integral in H± (8). Since the characteristic of k is not 2 it follows that H± (8) are not unimodular quasi-Hopf algebras. (3) By computations similar to the ones performed in the previous example one can show that t = (1 + g)x3 y3 is both a non-zero left and right integral in the 32dimensional quasi-Hopf algebra described in Example 3.32. Therefore H(32) is a unimodular quasi-Hopf algebra. One can construct a projection onto the space of left integrals as follows. Proposition 7.16 Let H be a finite-dimensional quasi-Hopf algebra with antipode S and for any h ∈ H define P(h) = ∑ei , β S(S(X 2 (ei )2 )α X 3 )hX 1 (ei )1 .

(7.2.1)

i



Then P(h) ∈ lH , for all h ∈ H, and ∑i ei , S(P(ei )β ) = 1. Consequently, at least one  of the elements P(ei ) is nonzero, and therefore lH = 0. Proof

We check that P(h) is a left integral in H, for all h ∈ H: aP(h) =

∑ei , β S(S(X 2 (ei )2 )α X 3 )haX 1 (ei )1 i

(3.2.1)

=

∑ei , β S(S(a(2,1) X 2 (ei )2 )α a(2,2) X 3 )ha1 X 1 (ei )1 i

(3.1.7)

=

∑ei , β S(S(X 2 (a1 ei )2 )α X 3 a2 )hX 1 (a1 ei )1 i

=

∑e j , a1 ei ei , β S(S(X 2 (e j )2 )α X 3 a2 )hX 1 (e j )1 i, j

=

∑e j , a1 β S(S(X 2 (e j )2 )α X 3 a2 )hX 1 (e j )1 j

= ε (a) ∑ei , β S(S(X 2 (ei )2 )α X 3 )hX 1 (ei )1 = ε (a)P(h), (7.2.1)

(3.2.1)

i

for all a ∈ H, as needed. We next compute that

∑e j , S(P(e j )β )

(7.2.1)

=

j

∑ei , β S(S(X 2 (ei )2 )α X 3 )e j e j , S(X 1 (ei )1 β ) i, j

=

∑ei , β S(α X 3 )S(X 1 (ei )1 β S((ei )2 )S(X 2 )) i

(3.2.1)

=

∑ ε (ei )ei , β S(X 1 β S(X 2 )α X 3 ) i

(3.2.2)

=

∑ ε (ei )ei , β  = ε (β ) = 1, i

as stated. 

The proof of the above proposition provides a k-linear map P : H → lH . If we   denote the inclusion of lH into H by i : lH → H, then a simple calculation shows that P ◦ i = Id. Moreover, by repeating the above arguments for the quasi-Hopf algebra  H op,cop instead of H we get that rH = 0, too, and this follows by proving the existence

7.2 Integral Theory

263

of a projection of H onto the space of right integrals in H that covers the natural inclusion. In order to prove the uniqueness of integrals for finite-dimensional quasi-Hopf algebras we need the following lemma. Lemma 7.17 Let t be a left integral in a quasi-Hopf algebra H. Then, for all h ∈ H, we have: hX 1t1 ⊗ S(X 2t2 )α X 3 = X 1t1 ⊗ S(X 2t2 )α X 3 h,

(7.2.2)

t1 ⊗ S(t2 ) = X t1 ⊗ S(X t2 )α X β = β X t1 ⊗ S(X t2 )α X . 1

Proof

2

3

1

2

3

For all h ∈ H we calculate, by using (3.2.1), (3.1.7) and t ∈

(7.2.3) H l

:

hX 1t1 ⊗ S(X 2t2 )α X 3 = h1 X 1t1 ⊗ S(h(2,1) X 2t2 )α h(2,2) X 3 = X 1 (h1t)1 ⊗ S(X 2 (h1t)2 )α X 3 h2 = X 1t1 ⊗ S(X 2t2 )α X 3 h. To prove the first equality in (7.2.3), we take X 1t1 ⊗ S(X 2t2 )α X 3 β . First we apply the 3-cocycle condition Φ ⊗ 1H = (IdH ⊗ Δ ⊗ IdH )(Φ−1 )(1H ⊗ Φ−1 )(IdH ⊗ IdH ⊗ Δ)(Φ)(Δ ⊗ IdH ⊗ IdH )(Φ) 

and then, successively using the fact that t ∈ lH , (3.2.1), (3.1.10), (3.1.11) and (3.2.2), we find the left-hand side of (7.2.3). The second equality in (7.2.3) follows from (7.2.2). We also need the dual structure of H. Recall that H ∗ has a natural multiplication ∗ ∗ ∗ ∗ 1 )g (h2 ), where h , g ∈ H and h ∈ H. Since H is finite dimensional, ∗ ˜ ε˜ ) given H is also equipped with a natural coassociative coalgebra structure (Δ, ∗  ∗  ∗ ∗ ∗ ∗ ˜ ), h ⊗ h  = h , hh  and ε˜ (h ) = h (1H ), where h ∈ H , h, h ∈ H and by Δ(h ,  : H ∗ ⊗ H → k denotes the dual pairing. As H is an algebra, on H ∗ we have the natural left and right H-actions h∗ g∗ , h = h∗ (h

h  h∗ , h  = h∗ , h h,

h∗  h, h  = h∗ , hh ,

where h, h ∈ H and h∗ ∈ H ∗ . This makes H ∗ into an H-bimodule. We also introduce S : H ∗ → H ∗ as the anti-coalgebra homomorphism dual to S, that is, S(h∗ ), h = h∗ , S(h), for all h∗ ∈ H ∗ , h ∈ H. As we have seen before, all these endow H ∗ with a dual quasi-Hopf algebra structure. Theorem 7.18 Let H be a finite-dimensional quasi-Hopf algebra, {ei }i a basis of  H with dual basis {ei }i of H ∗ , and define the linear map θ : lH ⊗ H ∗ → H by ∗

∗

θ (t ⊗ h ) = h (S(X t2 p )α X )X t1 p , ∀ t ∈ 2

2

3

1

1

 H l

, h∗ ∈ H ∗ ,

(7.2.4)

where pR = p1 ⊗ p2 is defined in (3.2.19). Then the following assertions hold:  (i) θ is an isomorphism of left H-modules, where lH ⊗ H ∗ is a left H-module via

264

Finite-Dimensional Quasi-Hopf Algebras 

h · (t ⊗ h∗ ) = t ⊗ h  h∗ , for all h ∈ H, t ∈ lH , h∗ ∈ H ∗ , and H is a left H-module  via left multiplication. Consequently, dimk lH = 1. The inverse of θ is given by

θ −1 (h) = ∑ P(ei h) ⊗ ei , ∀ h ∈ H,

(7.2.5)

i

where P is the projection onto the space of left integrals defined in (7.2.1). (ii) The antipode S is bijective.      (iii) S( lH ) = rH , S( rH ) = lH and dimk rH = 1. Proof have:

(i) First we show that θ and θ −1 are bijective inverses. For all h ∈ H we

θ (θ −1 (h))

∑e j , β S(S(X 2 (e j )2 )α X 3 )ei hei , S(Y 2 X21 (e j )(1,2) p2 )αY 3 

(7.2.4),(7.2.1)

=

i, j

Y 1 X11 (e j )(1,1) p1

∑e j , β S(Y 2 X21 p2 S(X 2 )α X 3 )αY 3 hY 1 X11 p1 e j

(3.2.21)

=

j

(3.2.19),(3.1.9)

Y 1 x1 Z 1 β S(Y 2 x2 X 1 Z12 β S(x13 X 2 Z22 )α x23 X 3 Z 3 )α Y 3 h

(3.2.1),(3.1.10)

Y 1 β S(Y 2 X 1 β S(X 2 )α X 3 )α Y 3 h = h.

= =

(3.1.11),(3.2.2)

For all t ∈

H l

and h∗ ∈ H ∗ we compute

θ −1 (θ (t ⊗ h∗ )) =

∑ h∗ (S(X 2t2 p2 )α X 3 )P(ei X 1t1 p1 ) ⊗ ei i

(7.2.1),(7.2.2)

=

∑ h∗ (S(X 2t2 p2 )α X 3 β S(S(Y 2 (e j )2 )αY 3 )ei ) i, j

e j , X 1t1 p1 Y 1 (e j )1 ⊗ ei ∑ h∗ (S(S(Y 2t(1,2) p12 )αY 3t2 p2 )ei )Y 1t(1,1) p11 ⊗ ei

(7.2.3)

=

i

(3.1.7),(3.2.1)

=

∑ h∗ (S(S(Y 2 p12 )αY 3 p2 )ei )tY 1 p11 ⊗ ei i

(3.2.19),(3.1.9)

=

∑ h∗ (S(S(y21 x1 X 2 )α y22 x2 X13 β S(y3 x3 X23 ))ei )ty1 X 1 ⊗ ei i

(3.2.1),(3.1.10),(3.1.11)

=

∑ h∗ (S(S(x1 )α x2 β S(x3 ))ei )t ⊗ ei i

(3.2.2)

=

∑ h∗ (ei )t ⊗ ei = t ⊗ h∗ . i



Since θ is a bijection and dimk H = dimk H ∗ is finite, it follows that dimk lH = 1. We  are left to show that θ is H-linear. For all h ∈ H, t ∈ lH and h∗ ∈ H ∗ we have: hθ (t ⊗ h∗ ) = h∗ (S(X 2t2 p2 )α X 3 )hX 1t1 p1 (7.2.2)

= h  h∗ , S(X 2t2 p2 )α X 3 X 1t1 p1 = θ (t ⊗ h  h∗ ).

7.2 Integral Theory

265

(ii) First we prove that S is bijective. H ∗ is finite dimensional, so it suffices to show  that S is injective. Let h∗ ∈ H ∗ be such that S(h∗ ) = 0, and take 0 = t ∈ lH . For all h ∈ H we have

θ (t ⊗ β S(h)  h∗ ) = β S(h)  h∗ , S(X 2t2 p2 )α X 3 X 1t1 p1 = h∗ , S(X 2t2 p2 )α X 3 β S(h)X 1t1 p1 (7.2.3)

= h∗ , S(ht2 p2 )t1 p1 = S(h∗ ), ht2 p2 t1 p1 = 0.

Since θ is bijective we obtain that t ⊗ β S(h)  h∗ = 0. Now, because t = 0 and  dimk lH = 1, it follows that β S(h)  h∗ = 0, for all h ∈ H. Therefore, by (3.2.2), for all h ∈ H we have h∗ (h ) = h∗ , h S(x1 )α x2 β S(x3 ) = β S(x3 )  h∗ , h S(x1 )α x2  = 0. ∗

∗

It is not hard to see that S : H ∗∗ → H ∗∗ , S (h∗∗ ) = h∗∗ ◦ S, for all h∗∗ ∈ H ∗∗ , is a bijective map. If we define ξ : H → H ∗∗ by ξ (h)(h∗ ) = h∗ (h), for all h ∈ H, h∗ ∈ H ∗ , then one can easily show that {ξ (ei )}i=1,n is a basis of H ∗∗ dual to the basis {ei }i=1,n of H ∗ , and it follows that ξ is bijective. Moreover, ξ −1 is given by ξ −1 (h∗∗ ) = ∗ ∑ h∗∗ (ei )ei , for all h∗∗ ∈ H ∗∗ . In addition, ξ −1 ◦ S ◦ ξ = S, so S is bijective. i



(iii) We have seen that S is an anti-algebra automorphism of H and dimk lH = 1.      This guarantees that S( lH ) = rH , S( rH ) = lH and dimk rH = 1, as required. 

Let H be a quasi-Hopf algebra and t ∈ lH . Since H is an associative algebra, th is also a left integral in H, for all h ∈ H, hence the space of left (right) integrals in H is a two-sided ideal. Moreover, if H is finite dimensional, then it follows from the uniqueness of the integrals in H that there exists μ ∈ H ∗ such that th = μ (h)t, ∀ t ∈

 H l

and h ∈ H.

(7.2.6)

One can easily see that μ : H → k is an algebra map, and so its inverse in the group Alg(H, k) defined in Remark 3.15 is μ −1 = μ ◦ S, which is also equal to μ ◦ S−1 . We call μ the modular element of H ∗ , or the distinguished grouplike element of H ∗ . Observe that μ = ε if and only if H is unimodular. Also, from the bijectivity of the antipode we get that hr = μ −1 (h)r = μ (S(h))r, ∀ r ∈

 H r

and h ∈ H.

(7.2.7)

Furthermore, by using the fact that the inverse of μ in the group Alg(H, k) is given either by μ −1 = μ ◦ S or by μ −1 = μ ◦ S−1 , one can see that (3.2.2)

μ (αβ )μ −1 (αβ ) = μ (X 1 β S(X 2 )α X 3 )μ −1 (S(x1 )α x2 β S(x3 )) = 1.

(7.2.8)

Examples 7.19 By using the non-zero integrals found in Examples 7.15 we get that for H(2) and H(32) the modular element is just the counit, while for H± (8) it is given by μ (1) = 1, μ (g) = −1 and μ (x) = 0, extended to a k-algebra morphism.

266

Finite-Dimensional Quasi-Hopf Algebras

Since a finite-dimensional quasi-Hopf algebra has bijective antipode we can make use of the elements qR , pR defined in (3.2.19). If for h∗ ∈ H ∗ and h ∈ H we define h∗ → h := h∗ (q2 h2 p2 )q1 h1 p1 then the bijective map θ defined in (7.2.4) takes the form θ (t ⊗ h∗ ) = h∗ ◦ S →t. Proposition 7.20 Any finite-dimensional quasi-Hopf algebra H is a Frobenius algebra. A Frobenius system for H is (φ , q1t1 p1 ⊗ S(q2t2 p2 )), where φ ∈ H ∗ is the unique map that satisfies

φ (q1t1 p1 )q2t2 p2 = 1H

or, equivalently, φ (S(q2t2 p2 ))q1t1 p1 = 1H ,

(7.2.9)

and where t is a non-zero left integral in H. Furthermore, the Nakayama automorphism corresponding to this Frobenius system is given by

χ (h) = μ (h1 )S2 (h2 ), ∀ h ∈ H,

(7.2.10)

where μ is the modular element of H ∗ . Proof

Theorem 7.18 implies that the map

ξ : H ∗ → H, ξ (h∗ ) = (h∗ ◦ S) → t := h∗ (S(q2t2 p2 ))q1t1 p1 , ∀ h∗ ∈ H ∗ , (7.2.11) is bijective. Moreover, ξ respects the natural left H-module structures of H and H ∗ , and so H and H ∗ are isomorphic as left H-modules. Thus H is a Frobenius algebra. A right H-module isomorphism between H ∗ and H can be obtained from ξ and the natural isomorphism θH−1 : H ∗∗ → H as in the proof of (iii) ⇒ (ii) of Proposition 7.1. Namely, there we have shown that ξ∗

- H ∗∗

ξ  : H∗

θH−1

- H

is a right H-module isomorphism. Explicitly,

ξ  (h∗ ) = θH−1 (ξ ∗ (h∗ )) = ∑ h∗ (ξ (ei ))ei = ∑h∗ , ei ◦ S →tei i

i

= ∑h∗ , ei (S(q2t2 p2 ))q1t1 p1 ei = h∗ (q1t1 p1 )S(q2t2 p2 ), i

for all h∗

∈ H ∗ . Now from Remark 7.3 we obtain that (φ , e) is a Frobenius system for

H, where φ ∈ H ∗ is the unique element in H ∗ satisfying ξ  (φ ) = 1H or, equivalently, ξ (φ ) = 1H and e = ∑i ei ⊗ ξ  (ei ). Since S is bijective it follows that a Frobenius morphism for H is the unique φ ∈ H ∗ that obeys one of the (equivalent) conditions in (7.2.9). Also, in this case the Frobenius element comes out as e = ∑ ei (q1t1 p1 )ei ⊗ S(q2t2 p2 ) = q1t1 p1 ⊗ S(q2t2 p2 ), i

as stated. Finally, to compute χ we use the formula in (7.1.1) to get (3.2.21)

χ (h) = φ (q1t1 p1 h)S(q2t2 p2 ) = φ (q1 (th1 )1 p1 )S(q2 (th1 )2 p2 S(h2 )) (7.2.6)

(7.2.9)

= μ (h1 )φ (q1t1 p1 )S(q2t2 p2 S(h2 )) = μ (h1 )S2 (h2 ),

for all h ∈ H, as desired.

7.2 Integral Theory

267

We end this section by showing that for an infinite-dimensional quasi-Hopf algebra with bijective antipode the space of integrals is zero. Lemma 7.21

Let H be a quasi-Hopf algebra with bijective antipode and define

Δ : H → H ⊗ H, Δ(h) = h1 ⊗ h2 := q1 h1 p1 ⊗ q2 h2 p2 , ∀ h ∈ H,

(7.2.12)

where pR = p1 ⊗ p2 and qR = q1 ⊗ q2 are defined by (3.2.19). If J is a non-zero two-sided ideal of H such that Δ(J) ⊆ J ⊗ H, then J = H. Proof

From (3.2.23), we easily deduce that (1 ⊗ S−1 (p2 ))Δ(p1 hq1 )(1 ⊗ S(q2 )) = Δ(h), ∀ h ∈ H.

This implies Δ(J) ⊆ J ⊗ H, since J is a two-sided ideal of H and Δ(J) ⊆ J ⊗ H. If ε (J) = 0, then for any h ∈ H we have h = ε (h1 )h2 ∈ ε (J)H = 0, so J = 0, a contradiction. Thus ε (J) = 0, and there exists a ∈ J with ε (a) = 1. By using (3.2.1), we obtain β = ε (a)β = a1 β S(a2 ) ∈ JH ⊆ J, so β ∈ J. By using (3.2.2) and the fact that J is a two-sided ideal of H, we find 1H = X 1 β S(X 2 )α X 3 ∈ J, and J = H. For h ∈ H and h∗ ∈ H ∗ , we have defined h∗ → h = h∗ (h2 )h1 . For a two-sided ideal I of H, we let H ∗ → I be the subspace of H generated by all the elements of the form h∗ → a, with h∗ ∈ H ∗ and a ∈ I. Lemma 7.22 Let H be a quasi-Hopf algebra with bijective antipode and I a nonzero two-sided ideal of H. Then J := H ∗ → I = H. Proof The statement follows from Lemma 7.21 if we can show that J is a non-zero two-sided ideal of H such that Δ(J) ⊆ J ⊗ H. Obviously ε → h = h, therefore I ⊆ J. For all h ∈ H, h∗ ∈ H ∗ , a ∈ I we have (h∗ → a)h

=

h∗ (q2 a2 p2 )q1 a1 p1 h

(3.2.21) ∗

=

h (q2 (ah1 )2 p2 S(h2 ))q1 (ah1 )1 p1

=

(S(h2 )  h∗ ) → (ah1 ),

and so J is a right ideal. J is also a left ideal, since h(h∗ → a)

=

h∗ (q2 a2 p2 )hq1 a1 p1

(3.2.21) ∗

=

h (S−1 (h2 )q2 (h1 a)2 p2 )q1 (h1 a)1 p1

=

(h∗  S−1 (h2 )) → (h1 a).

Let f = f 1 ⊗ f 2 be the Drinfeld element as in (3.2.15). By using (3.2.26), (3.1.7) and (3.2.25) one can show that h∗ → (g∗ → h) = [(g1 S(x3 )  h∗  S−1 ( f 2 X 3 ))(g2 S(x2 )  g∗  S−1 ( f 1 X 2 ))] → (X 1 ax1 ), for all h∗ , g∗ ∈ H ∗ and h ∈ H. I is a two-sided ideal of H, so the above equality shows that H ∗ → J ⊆ J. To prove that Δ(J) ⊆ J ⊗ H, we proceed as follows. Take

268

Finite-Dimensional Quasi-Hopf Algebras

a ∈ J, and write Δ(a) = ∑i ai ⊗ ai , where a1 , . . . , am ∈ J and am+1 , . . . , an are linearly independent modulo J. For any h∗ ∈ H ∗ , h∗ → a = ∑i h∗ (ai )ai ∈ J. The linear independence of am+1 , . . . , an modulo J implies that h∗ (ai ) = 0, and therefore ai = 0 (h∗ is arbitrary), for all i > m. We find that Δ(a) ∈ J ⊗ H, as required. As we next see, Theorem 7.18 has a converse. Theorem 7.23 If H is a quasi-Hopf algebra with an antipode S, then H is finite  dimensional if and only if S is bijective and lH = 0. Proof One implication follows from Theorem 7.18. Conversely, assume that S is  bijective and I = lH = 0. Then I is a non-zero two-sided ideal of H and Lemma 7.22  tells us that H ∗ →I = H. Thus there exist {h∗i }i=1,n ⊆ H ∗ and {ti }i=1,n ⊆ lH such that i aij ⊗ bij , for some {aij } j=1,ni ⊆ H 1H = ∑ni=1 h∗i → ti . For any i we have Δ(ti ) = ∑nj=1 i ∗ ∗ and {b j } j=1,ni ⊆ H. Therefore, for any h ∈ H and i = 1, n we have h∗ → ti = ni h∗ (bij )aij . For all h ∈ H we obtain that ∑ j=1 n

h = ∑ h(h∗i → ti ) i=1 n

= ∑ (h∗  S−1 (h2 )) → h1ti i=1 n

= ∑ (h∗  S−1 (h)) → ti

(since ti ∈

H l

, ∀ i = 1, n)

i=1 n ni

=∑

∑ h∗ (S−1 (h)bij )aij .

i=1 j=1

Thus H is a subspace of the span of {aij | i = 1, n, j = 1, ni }, and therefore it is finite dimensional. Corollary 7.24 If H is an infinite-dimensional quasi-Hopf algebra with bijective   antipode then lH = rH = 0. 



Proof The bijectivity of S implies S( lH ) = rH , and so dimk   Theorem 7.23, if H is infinite dimensional then lH = rH = 0.

H l

= dimk

H r

. By

7.3 Semisimple Quasi-Hopf Algebras We say that a quasi-Hopf algebra H is semisimple if it is semisimple as an algebra. Similarly, we say that H is separable if it is separable as an algebra. One of the characterizations of a separable algebra A is the following: there exists an element e = e1 ⊗ e2 ∈ A ⊗ A, called a separability element, satisfying e1 e2 = 1A and ae1 ⊗ e2 = e1 ⊗ e2 a, for all a ∈ A, where we again have suppressed summation and

7.3 Semisimple Quasi-Hopf Algebras

269

indices. The well-known example of separable algebra is given by the matrix algebra: for any i ∈ {1, . . . , n} the element ei = ∑nj=1 E ji ⊗ Ei j is a separability element for Mn (k). The result below measures how far a Frobenius algebra is from being separable. Proposition 7.25 Let A be a Frobenius algebra with Frobenius system (φ , e1 ⊗ e2 ). Then A is separable if and only if there exists a ∈ A such that e1 ae2 = 1A . Proof If there exists a ∈ A such that e1 ae2 = 1A it is clear that either e1 a ⊗ e2 or e1 ⊗ ae2 is a separability element for A. Conversely, if A is separable we consider f 1 ⊗ f 2 ∈ A ⊗ A a separability element for it and define a = φ ( f 2 ) f 1 . We have e1 a ⊗ e2 = φ ( f 2 )e1 f 1 ⊗ e2 = φ ( f 2 e1 ) f 1 ⊗ e2 = φ (e1 ) f 1 ⊗ e2 f 2 = f 1 ⊗ f 2 , where we used that both e1 ⊗ e2 and f 1 ⊗ f 2 satisfy the property of a Frobenius element. We then get e1 ae2 = f 1 f 2 = 1A , as needed. Note that, if we define a = φ ( f 1 ) f 2 , computations similar to those above prove that e1 ⊗ a e2 = f 1 ⊗ f 2 , and so e1 a e2 = 1A as well. By using integral theory we show that for finite-dimensional quasi-Hopf algebras the concepts of separable and semisimple are equivalent. Actually, we characterize both of them in terms of normalized integrals, providing in this way a Maschke type theorem for quasi-Hopf algebras. Let us start with the following general result. Proposition 7.26

Any separable algebra A over a field k is semisimple.

Proof Let e = e1 ⊗ e2 ∈ A ⊗ A be a separability element for A. Take M a left Amodule and N an A-submodule of it. Since k is a field there exists a k-linear map f : M → N that covers the natural inclusion i : N → M, that is, f (n) = n, for all n ∈ N. If we define f˜ : M → N, f˜(m) = e1 · f (e2 · m), ∀ m ∈ M, then by ae1 ⊗ e2 = e1 ⊗ e2 a, for all a ∈ A, it follows that f˜ is left A-linear. Also, the fact that e1 e2 = 1A implies f˜(n) = n, for all n ∈ N. Hence N is an A-direct summand of M and this proves that M is completely reducible. Since M was arbitrary we obtain that A is semisimple, as desired. Definition 7.27 We call a left (right) integral t in H normalized if ε (t) = 1. We call a Haar integral in H a normalized left integral which is at the same time a right integral. The equivalence between (i) and (ii) below is known as the Maschke theorem for quasi-Hopf algebras. Theorem 7.28 For a finite-dimensional quasi-Hopf algebra H the following assertions are equivalent:

270

Finite-Dimensional Quasi-Hopf Algebras

(i) H is semisimple; (ii) H has a normalized left or right integral; (iii) k is a projective left or right H-module via the H-action defined by ε ; (iv) H has a Haar integral; (v) H is a separable algebra. Proof (i) ⇒ (ii). Since H is semisimple there exists a left ideal I in H such that H = I ⊕ Ker(ε ), where Ker(ε ) is the kernel of ε ∈ H ∗ . For x ∈ Ker(ε ) and y ∈ I, we have xy ∈ Ker(ε ) ∩ I, hence xy = 0 = ε (x)y. Then, for any h ∈ H, h = (h − ε (h)1H ) + ε (h)1H , and since (h − ε (h)1H ) ∈ Ker(ε ) we get that hy = ε (h)y. So we have proved   that I ⊆ lH . As Ker(ε ) ∩ I = 0 it follows that ε ( lH ) = 0, for otherwise H = Ker(ε ), a contradiction. This guarantees the existence of a left integral t in H such that ε (t) = 1, as needed. (ii) ⇒ (iii). Let t ∈ H be a left (resp. right) normalized integral and define f : k → H by f (κ ) = κ t, for all κ ∈ k. Since k is viewed as a left (resp. right) H-module via ε it follows that f is a left (resp. right) H-module morphism obeying ε ◦ f = Idk . Thus k is isomorphic to a direct summand of the free H-module H, and therefore it is a projective left (resp. right) H-module. (iii) ⇒ (iv). Assume k is a projective left (resp. right) H-module. Clearly ε is left (resp. right) H-linear and surjective and from here we get that there exists a left (resp. right) H-linear morphism ϑ : k → H such that ε ◦ ϑ = Idk . Then t := ϑ (1k ) is a left (resp. right) integral since, for instance, ht = hϑ (1k ) = ϑ (h · 1k ) = ε (h)ϑ (1k ) = ε (h)t, for all h ∈ H. Moreover, ε (t) = ε (ϑ (1k )) = 1k , hence t is a normalized left (resp. right) integral in H. If μ is the modular element of H ∗ then by applying ε to the both sides of (7.2.6) (resp. (7.2.7)) we obtain that μ = ε (resp. μ ◦ S = ε ). But S is bijective and ε ◦ S = ε , so in both cases we obtain that μ = ε . This shows that t is a Haar integral in H, so the implication is proved. (iv) ⇒ (v). If t is a normalized left integral in H then both e1 = q1t1 β ⊗ S(q2t2 )

and e2 = q˜1t1 β ⊗ S(q˜2t2 )

are separability elements for H, where qR = q1 ⊗ q2 and qL = q˜1 ⊗ q˜2 are the elements defined in (3.2.19) and (3.2.20), respectively. For instance, e1 is a separability element because of (7.2.2) and since (3.2.1)

(3.2.2)

q1t1 β S(q2t2 ) = ε (t)q1 β S(q2 ) = X 1 β S(X 2 )α X 3 = 1H . Likewise, if r is a normalized right integral in H then either e1 = S(r1 p1 ) ⊗ α r2 p2 or e2 = S(r1 p˜1 ) ⊗ α r2 p˜2 is a separability element for H, where pR = p1 ⊗ p2 and pL = p˜1 ⊗ p˜2 are the elements defined in (3.2.19) and (3.2.20), respectively. We leave the verification of all these details to the reader. (v) ⇒ (i). This follows from Proposition 7.26.

7.3 Semisimple Quasi-Hopf Algebras

271

Remark 7.29 If H is a semisimple quasi-Hopf algebra with bijective antipode then it is finite dimensional. Indeed, Ker(ε ) is an ideal of H, and so there exists a left ideal I of H such that H = I ⊕ Ker(ε ). Then as in the proof of (i) ⇒ (ii) above one can   see that I ⊆ lH , and since Ker(ε ) has codimension 1 in H it follows that lH = 0. Combined with S bijective this implies that H is finite dimensional; see Theorem 7.23. Remark 7.30 Obviously, by (iv) in Theorem 7.28, a finite-dimensional semisimple quasi-Hopf algebra is unimodular. Examples 7.31 (1) The quasi-Hopf algebra H(2) from Example 3.26 is semisimple since l = 12 (1 + g) is a Haar integral for it. In this case the four separability elements considered in the proof of (iv) ⇒ (v) above are all equal to 12 (1 ⊗ 1 + g ⊗ g). To see this write Φ = 1 − 2p− ⊗ p− ⊗ p− under the form 1 3 Φ = 1 ⊗ 1 ⊗ 1 + (1 ⊗ 1 ⊗ g + 1 ⊗ g ⊗ 1 + g ⊗ 1 ⊗ 1) 4 4 1 1 − (1 ⊗ g ⊗ g + g ⊗ 1 ⊗ g + g ⊗ g ⊗ 1) + g ⊗ g ⊗ g, 4 4 to compute that 1 qR = X 1 ⊗ S−1 (α X 3 )X 2 = X 1 ⊗ X 2 gX 3 = (1 ⊗ g + g ⊗ g + 1 ⊗ 1 − g ⊗ 1) 2 and 1 pR = x1 ⊗ x2 β S(x3 ) = x1 ⊗ x2 x3 = (1 ⊗ 1 + 1 ⊗ g + g ⊗ 1 − g ⊗ g), 2 since, as can be easily checked, Φ−1 = Φ. Therefore 1 1 e1 = qR Δ(l) = qR (1 ⊗ 1 + g ⊗ g) = (1 ⊗ 1 + g ⊗ g), 2 2 and, likewise, 1 1 e1 = (1 ⊗ g)Δ(l)pR = (1 ⊗ g + g ⊗ 1)pR = (1 ⊗ 1 + g ⊗ g). 2 2 In a similar manner we compute that e2 = e2 = 12 (1 ⊗ 1 + g ⊗ g); the details are left to the reader. (2) The quasi-Hopf algebras presented in Example 3.30 are not semisimple since, as we have seen, they are not unimodular. (3) The quasi-Hopf algebra H(32) from Example 3.32 is unimodular but not semisimple because, according to Example 7.15(3), t = (1 + g)x3 y3 is a left and right non-zero integral in H(32) but ε (t) = 0. We next show that we always have e1 = e2 and e1 = e2 , and so for a semisimple quasi-Hopf algebra we can only construct one separability element from a fixed left (or right) normalized integral of it. To this end we need several formulas that will be intensively used from now on.

272

Finite-Dimensional Quasi-Hopf Algebras

Lemma 7.32 For a quasi-Hopf algebra H with bijective antipode define the elements U = U 1 ⊗U 2 and V = V 1 ⊗V 2 in H ⊗ H by and V := S−1 ( f 2 p2 ) ⊗ S−1 ( f 1 p1 ),

U := g1 S(q2 ) ⊗ g2 S(q1 )

(7.3.1)

where pR = p1 ⊗ p2 and qR = q1 ⊗ q2 are the elements defined in (3.2.19), and f = f 1 ⊗ f 2 is the Drinfeld twist considered in (3.2.15) with its inverse f −1 = g1 ⊗ g2 as in (3.2.16). Then the following relations hold: U[1H ⊗ S(h)] = Δ(S(h1 ))U[h2 ⊗ 1H ], ∀ h ∈ H, [1H ⊗ S

−1

(h)]V = [h2 ⊗ 1H ]V Δ(S 2 qR = [q˜ ⊗ 1H ]V Δ(S−1 (q˜1 )), pR = Δ(S( p˜1 ))U[ p˜2 ⊗ 1H ],

−1

(h1 )), ∀ h ∈ H,

(7.3.2) (7.3.3) (7.3.4) (7.3.5)

where qL = q˜1 ⊗ q˜2 and pL = p˜1 ⊗ p˜2 are the elements in H ⊗ H defined in (3.2.20). Proof We prove only (7.3.2) and (7.3.4); the other two can be obtained from these if we think of them in H op instead of H. Note only that when we pass from H to H op the roles of U and V , as well as of qR and pR and of qL and pL , interchange. To prove (7.3.2) we compute: Δ(S(h1 ))U[h2 ⊗ 1H ]

=

S(h1 )1 g1 S(q2 )h2 ⊗ S(h1 )2 g2 S(q1 )

(3.2.13) 1

=

g S(q2 h(1,2) )h2 ⊗ g2 S(q1 h(1,1) )

(3.2.21) 1

=

g S(q2 ) ⊗ g2 S(hq1 ) = U[1H ⊗ S(h)],

for all h ∈ H, as required. (7.3.4) follows from: [q˜2 ⊗ 1H ]V Δ(S−1 (q˜1 )) = (3.2.13)

=

q˜2 S−1 ( f 2 p2 )S−1 (q˜1 )1 ⊗ S−1 ( f 1 p1 )S−1 (q˜1 )2 q˜2 S−1 ( f 2 q˜12 p2 ) ⊗ S−1 ( f 1 q˜11 p1 )

(3.2.20),(3.2.13) 3 −1

=

x S

(S(x11 ) f 2 α2 x22 p2 ) ⊗ S−1 (S(x21 ) f 1 α1 x12 p1 )

(3.2.14)

x3 S−1 (S(x11 )γ 2 x22 p2 ) ⊗ S−1 (S(x21 )γ 1 x12 p1 )

(3.2.5)

S−1 (S(X 1 y11 x11 )α y3 x22 p2 S(x3 )) ⊗ S−1 (S(X 2 y12 x21 )α X 3 y2 x12 p1 )

= =

(3.2.19),(3.1.9)

S−1 (S(X 1 y1(1,1) x11 )α y2 x13 β S(y3 x23 )) ⊗ S−1 (S(X 2 y1(1,2) x21 )α X 3 y12 x2 )

(3.1.7),(3.2.1)

S−1 (S(y1 X 1 )α y2 β S(y3 )) ⊗ S−1 (S(X 2 )α X 3 )

= =

(3.2.2)

=

(3.2.19)

X 1 ⊗ S−1 (α X 3 )X 2 = qR .

So the proof is complete. Proposition 7.33 Let H be a finite-dimensional quasi-Hopf algebra, so its antipode S is bijective. If pR , qR and pL , qL are as in (3.2.19) and (3.2.20), respectively, then q1t1 ⊗ q2t2 = q˜1t1 ⊗ q˜2t2

and r1 p1 ⊗ r2 p2 = r1 p˜1 ⊗ r2 p˜2 ,

(7.3.6)

for any left integral t and right integral r in H. Consequently, when H is semisimple

7.4 Symmetric Quasi-Hopf Algebras

273

we have e1 = e2 and e1 = e2 , where e1 , e2 and e1 , e2 are the separability elements defined in the proof of Theorem 7.28. Proof We only prove the first relation, the second one is the first one regarded in H op instead of H.  The fact that t ∈ lH and (7.3.4) imply q1t1 ⊗ q2t2 = V 1t1 ⊗V 2t2 .

(7.3.7)

We use now the quasi-Hopf algebra structure of H cop to see that in H cop we have (qR )cop = q˜2 ⊗ q˜1 , (pR )cop = p˜2 ⊗ p˜1 and fcop = (S−1 ⊗ S−1 )( f ), and so Vcop = S( p˜1 ) f 2 ⊗ S( p˜2 ) f 1 . Thus, if we consider (7.3.7) in H cop instead of H, we obtain q˜1t1 ⊗ q˜2t2 = S( p˜2 ) f 1t1 ⊗ S( p˜1 ) f 2t2 . On the other hand, one can see that (3.2.20), (3.1.5), (3.2.17) and S−1 ( f 2 )β f 1 = S−1 (α ), which has been proved before, imply S( p˜2 ) f 1 ⊗ S( p˜1 ) f 2 = q1 g11 ⊗ S−1 (g2 )q2 g12 ,

(7.3.8)

where, as usual, we denote f −1 = g1 ⊗ g2 . From the above,  H we conclude that t ∈ q˜1t1 ⊗ q˜2t2 = q1 g11t1 ⊗ S−1 (g2 )q2 g12t2 = l q1t1 ⊗ q2t2 . The consequence now follows directly from the definitions of e1 , e2 , e1 and e2 .

7.4 Symmetric Quasi-Hopf Algebras In this section we characterize symmetric quasi-Hopf algebras by making use of the integral theory that we have developed so far. To this end we first check when a Frobenius algebra is symmetric and we develop an integral theory for augmented Frobenius algebras. These allow us to provide elegant proofs when we come back to the quasi-Hopf algebra setting. Definition 7.34 A symmetric algebra is a finite-dimensional k-algebra A that is isomorphic to A∗ as an A-bimodule. If follows that any symmetric algebra is a Frobenius algebra, and that the two notions coincide in the commutative case. To give some non-trivial examples we need first the following list of characterizations. The last two say how far a Frobenius algebra is from being symmetric. Proposition 7.35 Let A be a finite-dimensional k-algebra. Then the following are equivalent: (i) A is a symmetric algebra; (ii) there exists a bilinear map B : A × A → k which is non-degenerate (i.e. B is at the same time left and right non-degenerate), associative and symmetric (i.e. B(a, b) = B(b, a), for all a, b ∈ A);

274

Finite-Dimensional Quasi-Hopf Algebras

(iii) there exists a k-linear map φ : A → k that is a trace, that is, φ (ab) = φ (ba), for all a, b ∈ A, and such that Ker(φ ) does not contain non-zero left or right ideals; (iv) there exists a Frobenius system for A such that φ is a trace and e is symmetric, in the sense that e1 ⊗ e2 = e2 ⊗ e1 in A ⊗ A; (v) A is a Frobenius algebra for which the Nakayama automorphism is inner. Proof (i) ⇒ (ii). Let f : A → A∗ be an isomorphism of A-bimodules. Since f is right A-linear, as in the proof of (ii) ⇒ (v) in Proposition 7.1 we get that B : A×A → k given by B(a, b) = f (a)(b), for all a, b ∈ A, is associative and right non-degenerate. Using now the left A-linearity of f , which comes out as f (ab) = a  f (b), for all a, b ∈ A, we compute that B(ab, c) = f (ab)(c) = (a  f (b))(c) = f (b)(ca) = B(b, ca), ∀ a, b, c ∈ A. By taking c = 1A and using the associativity of B we obtain that B is symmetric, and so left-non-degenerate as well. (ii) ⇒ (i). Once we have B as in the statement define f : A → A∗ by f (a)(b) = B(a, b), for all a, b ∈ A. The proof of (v) ⇒ (ii) in Proposition 7.1 ensures that f is an isomorphism of right A-modules. It is also left A-linear since f (ba)(c) = B(ba, c) = B(c, ba) = B(cb, a) = B(a, cb) = f (a)(cb) = (b  f (a))(c), for all a, b, c ∈ A, as needed. (ii) ⇒ (iii). For B as in (ii) define φ ∈ A∗ by φ (a) = B(1A , a) = B(a, 1A ), for all a ∈ A. We have

φ (ab) = B(ab, 1A ) = B(a, b) = B(b, a) = B(1A , ba) = φ (ba), for all a, b ∈ A, so φ is a trace. Now let I ⊆ Ker (φ ) be a right ideal. Then φ (I) = 0, so B(I, 1A ) = 0. As I = IA we obtain B(I, A) = 0, and therefore I = 0, because B is non-degenerate. Similarly, if I ⊆ Ker (φ ) is a left ideal then I = 0. (iii) ⇒ (ii). If φ is as in (iii) we define B : A × A → k by B(a, b) = φ (ab), for all a, b ∈ A. Then B(ab, c) = φ ((ab)c) = φ (a(bc)) = B(a, bc), and this shows that B is associative. It is also symmetric since φ is a trace. Moreover, if a ∈ A is such that B(a, A) = 0 then φ (aA) = 0, hence a = 0 because aA is a right ideal in A. Thus B is non-degenerate, too. (i) ⇒ (iv). Let f : A → A∗ be an isomorphism of A-bimodules, and {ai }i a basis in A with corresponding dual basis {ai }i in A∗ . From the proof of (ii) ⇒ (i) in Proposition 7.1 we know that (φ := f (1A ), e = ∑i ai ⊗ f −1 (ai )) is a Frobenius system for A. The fact that f is left and right A-linear implies a  φ = a  f (1A ) = f (a) = f (1A )  a = φ  a, ∀ a ∈ A,

7.4 Symmetric Quasi-Hopf Algebras

275

and so φ (ab) = φ (ba), for all a, b ∈ A, which means that φ is a trace. Furthermore, if e1 ⊗ e2 is another copy of e then, for all a ∈ A, ae1 ⊗ e2 = φ (e1 )e2 ae1 ⊗ e2 = φ (ae1 e1 )e2 ⊗ e2 = φ (e1 ae1 )e2 ⊗ e2 = φ (e1 )e2 ⊗ e2 e1 a = e2 ⊗ φ (e1 )e2 e1 a = e2 ⊗ e1 a, where we used that φ is a trace. So e is a symmetric element. (iv) ⇒ (i). From the proof of (i) ⇒ (ii) in Proposition 7.1 we know that f : A → A∗ given by f (a) = φ  a, for all a ∈ A, is a right A-module isomorphism. The fact that φ is a trace is equivalent to a  φ = φ  a, for all a ∈ A, and this allows us to show that f is also left A-linear, because for all a, b ∈ A we have f (ba) = φ  ba = (φ  b)  a = (b  φ )  a = b  (φ  a) = b  f (a). (i) ⇒ (v). Assume again that f : A → A∗ is an A-bimodule isomorphism, so in particular A is a Frobenius algebra. Then we have seen that A admits a Frobenius system (φ , e) with e symmetric. If χ is its corresponding Nakayama automorphism then by (7.1.1) we deduce that χ (a) = φ (e1 a)e2 = φ (e2 a)e1 = φ (e2 )ae1 = a, for all a ∈ A, thus χ is inner. (v) ⇒ (i). Let (φ , e) be a Frobenius system such that the Nakayama automorphism defined by it is inner via, say, u ∈ A. In other words, u is invertible in A and χ (a) = φ (e1 a)e2 = uau−1 , for all a ∈ A. Now define g : A → A∗ by g(a) = φ  ua, for all a ∈ A. Since (φ  u, e1 ⊗ u−1 e2 ) is another Frobenius system for A (see Proposition 7.9) it follows that g is a right A-module isomorphism. In addition, a  g(b) = a  (φ  ub) = (a  φ )  ub = (φ  χ (a))  ub = φ  χ (a)ub = φ  uab = g(ab), for all a, b ∈ A, and so g is an isomorphism of A-bimodules. Remark 7.36 If A is a Frobenius algebra with Frobenius system (φ , e), then φ is a trace if and only if e is a symmetric element. Indeed, if φ is a trace then as in the proof of (i) ⇒ (iv) above we obtain that e is symmetric. Conversely, if e is symmetric then φ is a trace since

φ (ab) = φ (e1 )φ (ae2 b) = φ (be1 )φ (ae2 ) = φ (be2 )φ (ae1 ) = φ (be2 a)φ (e1 ) = φ (ba), for all a, b ∈ A, as required.

276

Finite-Dimensional Quasi-Hopf Algebras

We use the above characterizations to provide examples of symmetric algebras. Examples 7.37 (1) If A and A are symmetric algebras then so are A×A and A⊗A . More precisely, if B and B are non-degenerate, associative, symmetric bilinear maps

defined in exactly the same way as Bl and B

l on A and A , respectively, then B and B were defined in Example 7.4(1), are non-degenerate, associative, symmetric bilinear maps on A × A and A ⊗ A , respectively. (2) For any finite group G the group algebra k[G] is symmetric. To see this, consider φ (∑g αg g) = αe , for all α = ∑g αg g ∈ k[G], where e is the neutral element of G. Then φ is a trace since, for any other element β = ∑g βg g ∈ k[G], we have

φ (αβ ) = ∑ αg βg−1 = ∑ βh αh−1 = φ (β α ). g

h

Moreover, if I ⊆ Ker(φ ) is a non-zero right ideal take 0 = α = ∑g αg g ∈ I, so there exists g ∈ G such that αg = 0. We have 0 = α g−1 = αg e +

∑

αh hg−1 ∈ I ⊆ Ker(φ ),

h∈G\{g}

thus φ (α g−1 ) = 0. But the definition of φ says that φ (α g−1 ) = αg = 0, a contradiction. Hence I = 0. In a similar manner one can show that Ker(φ ) does not contain non-zero left ideals; we leave the details to the reader. (3) For any field k and non-zero natural number n the algebra Mn (k) is symmetric. The trace in this case is given by the usual trace of a matrix, that is, φ (A) = Tr(A) := ∑i aii , for any A = (ai j )1≤i, j≤n ∈ Mn (k). It is well-known that Tr(AB) = Tr(BA), for any A, B ∈ Mn (k), and this justifies, in general, the definition of a trace. If I ⊆ Ker(φ ) is a right ideal then Tr(AEi j ) = 0, for all A ∈ I and 1 ≤ i, j ≤ n, where {Ei j | 1 ≤ i, j ≤ n} is the canonical basis of Mn (k). As AEi j = ∑u aui Eu j we get that 0 = Tr(AEi j ) = a ji , for all 1 ≤ i, j ≤ n, hence A = 0. Similarly, one can shown that Ker(φ ) does not contain non-zero left ideals, therefore Mn (k) is a symmetric algebra. Our next aim is to find a necessary and sufficient condition for a finite-dimensional quasi-Hopf algebra H to be symmetric as an algebra. If H is symmetric then we will show that H is unimodular. Due to the “Frobenius” flavor of the proof, we develop first an integral theory for augmented Frobenius algebras. Definition 7.38 We say that a k-algebra is an augmented algebra if there exists an algebra morphism ε : A → k, called an augmentation morphism. If A is an augmented algebra with augmentation morphism ε then a left (resp. right) integral in A is an element t (resp. r) in A satisfying at = ε (a)t (resp. ra =   ε (a)r), for all a ∈ A. By lA (resp. rA ) we denote the space of left (resp. right) inte  grals in A, and call A unimodular if lA ∩ rA = 0.

7.4 Symmetric Quasi-Hopf Algebras Proposition 7.39

277

If A is a Frobenius augmented algebra then  A

dimk

l

 A

= dimk

= 1. r

Proof Let (φ , e) be a Frobenius system for A and f : A → A∗ the right A-module isomorphism induced by it, that is, f (a) = φ  a, for all a ∈ A. If ε ∈ A∗ is the augmentation morphism of A, there exists a unique element r ∈ A such that φ  r = ε . As f (r) = ε = 0 we get r = 0. Furthermore, for all a ∈ A, f (ra) = φ  ra = (φ  r)  a = ε  a = ε (a)ε = φ  ε (a)r = f (ε (a)r), 

and by using the fact that f is injective we conclude that 0 = r ∈ rA . To prove the uniqueness of right integrals in A we take r an arbitrary right integral in A and compute: (φ  r )(a) = φ (r a) = φ (r )ε (a) = (φ  φ (r )r)(a), ∀ a ∈ A. 

Thus f (r ) = f (φ (r )r), and so r = φ (r )r. We have proved that rA = kr with r = 0,  therefore dimk rA = 1.  To see that dimk lA = 1 we have to construct from (φ , e) a left A-module isomorphism f  : A → A∗ . As we have pointed out several times this can be done by taking ∗ θA - A∗∗ f - A∗ , where θA is the canonical isomorphism. It comes out f : A

that f  (a) = a  φ , for all a ∈ A, with inverse f −1 (a∗ ) = a∗ (e2 )e1 , for all a∗ ∈ A∗ . Now we have to repeat the above arguments for f  instead of f to conclude that  dimk lA = 1, as required.

As in the quasi-Hopf algebra case, one can show that for any Frobenius augmented   algebra A the spaces lA and rA are ideals in A. So, by the uniqueness of integrals in  A it follows that there exists μ  ∈ A∗ such that ar = μ  (a)r, for all r ∈ rA and a ∈ A. Note that, when A is a finite-dimensional quasi-Hopf algebra, μ  is nothing else than the convolution inverse of the modular element μ defined by (7.2.6). Proposition 7.40 Let A be a Frobenius augmented algebra with augmentation morphism ε and Nakayama automorphism χ associated to a Frobenius system (φ , e). If μ  is the map defined above then μ  is an algebra morphism and μ  ◦ χ = ε . Consequently, any symmetric augmented algebra is unimodular. Proof Let r be the non-zero right integral in A defined by φ  r = ε . Clearly μ  (1A ) = 1, and μ  is multiplicative since

μ  (ab)r = (ab)r = a(br) = μ  (b)ar = μ  (a)μ  (b)r and r is non-zero. Hence μ  is another augmentation morphism for A. To prove the second assertion observe first that (r  φ )(a) = φ (ar) = μ  (a)φ (r) = μ  (a)(φ  r)(1A ) = μ  (a)ε (1A ) = μ  (a),

278

Finite-Dimensional Quasi-Hopf Algebras

for all a ∈ A, and therefore μ  = r  φ . Then we compute:

μ  χ (a) = r  φ , χ (a) = φ  χ (r), χ (a) = φ , χ (r)χ (a) = φ , χ (ra) = ε (a)φ , χ (r) = ε (a)φ  χ (r), 1A  = ε (a)r  φ , 1A  = ε (a)φ (r) = ε (a)φ  r, 1A  = ε (a)ε (1A ) = ε (a), for all a ∈ A, where we have used that χ is multiplicative and the fact that r is a right integral in A. Thus μ  ◦ χ = ε . If A is, moreover, symmetric then χ is inner; see Proposition 7.35 (v). If u is invertible in A and such that χ (a) = uau−1 , for all a ∈ A, then (μ  ◦ χ )(a) = μ  (uau−1 ) = μ  (u)μ  (a)μ  (u−1 ) = μ  (a), for all a ∈ A. This means that ε = μ  ◦ χ = μ  . In other words, r is a non-zero left integral in A as well, and so A is unimodular. We are now in the position to apply all the above results to quasi-Hopf algebras. Theorem 7.41 A finite-dimensional quasi-Hopf algebra with antipode S is a symmetric algebra if and only if it is unimodular and S2 is an inner automorphism. Proof Let H be a finite-dimensional quasi-Hopf algebra with antipode S. From Proposition 7.20 we know that H is a Frobenius algebra with Nakayama automorphism χ given by χ (h) = μ (h1 )S2 (h2 ), for all h ∈ H, where μ is the modular element of H ∗ . Assume H is symmetric. As it is an augmented algebra as well, from the previous proposition it follows that H is unimodular. Hence μ = ε , and this implies χ = S2 . Now Proposition 7.35 (v) ensures that χ , and therefore S2 , is an inner automorphism of H, as required. Conversely, suppose that H is unimodular and S2 is inner. Then H is a Frobenius algebra with the Nakayama automorphism S2 , an inner automorphism. Thus H is symmetric; see Proposition 7.35. As we shall see later on, classes of symmetric quasi-Hopf algebras are given by involutory quasi-Hopf algebras, factorizable quasi-Hopf algebras and quasi-Hopf algebras of the type Dω (H). For the moment we restrict ourselves to the following: Example 7.42 The two-dimensional quasi-Hopf algebra H(2) is symmetric since it is unimodular and has the antipode defined by the identity morphism. More generally, we have the following result due to Eilenberg and Nakayama. In particular, it implies that any finite-dimensional semisimple (or, equivalently, separable) quasi-Hopf algebra is symmetric. Theorem 7.43 Any finite-dimensional semisimple algebra is symmetric.

7.5 Cointegral Theory

279

Proof By the Wedderburn theorem, A is isomorphic to Mn1 (D1 ) × · · · × Mnt (Dt ), where each D j is a finite-dimensional division algebra over k. Moreover, Mni (Di ) ∼ = Mni (k) ⊗ Di , for all 1 ≤ i ≤ t, thus, according to Examples 7.37 (1) and (3) it suffices to show that any finite-dimensional division k-algebra is symmetric. Let D be a finite-dimensional division k-algebra and [D, D] the commutator space of D. We show that [D, D] = D, and this will imply that any non-zero k-linear map φ : D → k whose kernel contains [D, D] satisfies the condition (iii) in Proposition 7.35. Indeed, φ is a trace since [D, D] ⊆ Ker(φ ), and Ker(φ ) does not contain nonzero left or right ideals since D is a division algebra over k. Take k the algebraic closure of k and Dk := D ⊗k k. Then Dk ∼ = Mm (k), for a certain non-zero natural number m. The trace morphism Tr : Mm (k) → k is nonzero and [Mm (k), Mm (k)] ⊆ Ker(Tr), and therefore [Dk , Dk ] = Dk . This together with [D, D]k := [D, D] ⊗k k ⊆ [Dk , Dk ] implies [D, D] = D, as needed. Corollary 7.44 If H is a finite-dimensional semisimple quasi-Hopf algebra with antipode S then S2 is an inner automorphism of H.

7.5 Cointegral Theory Unless otherwise specified, throughout this section H is a finite-dimensional quasiHopf algebra with antipode S, and {ei }i is a basis in H with dual basis {ei }i in H ∗ . We know that S is bijective, thus we will make use of the elements pR , qR from (3.2.19) or pL , qL from (3.2.20) without any restriction. Also, we will freely use the dual quasi-Hopf algebra structure of H ∗ , the linear dual of H. We next show that H ∗ is a right quasi-Hopf H-bimodule, and that the corresponding space of coinvariants has dimension 1. As we shall see, these coinvariants of H ∗ play an important role in the structure of both H ∗ (by generating it as a left Hmodule) and H (by defining a Frobenius morphism for it). We end by defining the set of alternative cointegrals on H, elements in H ∗ that can be defined even in the infinite-dimensional case. We start with the following technical result. Lemma 7.45 Let H be an arbitrary quasi-Hopf algebra with bijective antipode S and let U,V ∈ H ⊗ H be the elements defined in (7.3.1). Then we have: Φ−1 (IdH ⊗ Δ)(U)(1H ⊗U) = (Δ ⊗ IdH )(Δ(S(X 1 ))U)(X 2 ⊗ X 3 ⊗ 1H ), (Δ ⊗ IdH )(V )Φ

−1

= (X ⊗ X ⊗ 1H )(1H ⊗V )(IdH ⊗ Δ)(V Δ(S 2

3

−1

1

(X ))).

(7.5.1) (7.5.2)

Proof In H op the roles of U and V interchange, and (7.5.2) reduces to (7.5.1) in H op , so it suffices to prove only (7.5.1). To this end denote by Q1 ⊗ Q2 another copy of qR , by G1 ⊗ G2 another copy of f −1 , and then compute that Φ−1 (IdH ⊗ Δ)(U)(1H ⊗U) (7.3.1)

=

x1 g1 S(q2 ) ⊗ x2 g21 S(q1 )1U 1 ⊗ x3 g22 S(q1 )2U 2

280

Finite-Dimensional Quasi-Hopf Algebras (7.3.1),(3.2.13) 1 1

=

(3.2.26)

=

x g S(q2 ) ⊗ x2 g21 G1 S(Q2 q12 ) ⊗ x3 g22 G2 S(Q1 q11 ) 1 1 x1 g1 S(q22 X(2,2) Y 3 ) f 1 X 2 ⊗ x2 g21 G1 S(q21 X(2,1) Y 2) f 2X 3

⊗ x3 g22 G2 S(q1 X11Y 1 ) (3.1.5),(3.2.17) 1 1 1 1 = g1 G S(q22 X(2,2) ) f 1 X 2 ⊗ g12 G2 S(q21 X(2,1) ) f 2 X 3 ⊗ g2 S(q1 X11 ) (3.2.13) = (Δ(g1 S(q2 X21 )) ⊗ g2 S(q1 X11 ))(X 2 ⊗ X 3 ⊗ 1H ) (3.2.13),(7.3.1) = (Δ(S(X 1 )1U 1 ) ⊗ S(X 1 )2U 2 )(X 2 ⊗ X 3 ⊗ 1H ) = (Δ ⊗ IdH )(Δ(S(X 1 ))U)(X 2 ⊗ X 3 ⊗ 1H ), as required. By ∗ we denote the multiplication ∗ : H ∗ ⊗ H ∗ → H ∗ given by h∗ ∗ g∗ , h = h∗ (V 1 h1U 1 )g∗ (V 2 h2U 2 ), ∀ h∗ , g∗ ∈ H ∗ , h ∈ H.

(7.5.3)

Proposition 7.46 If H is a finite-dimensional quasi-Hopf algebra then H ∗ ∈ H MHH with the following structure: • H ∗ is an H-bimodule via the H-actions induced by S, that is, h · h∗ · h = S(h )  h∗  S−1 (h), ∀ h, h ∈ H, h∗ ∈ H ∗ ;

(7.5.4)

• H coacts from the right on H ∗ by n

ρ (h∗ ) = ∑ ei ∗ h∗ ⊗ ei , ∀ h∗ ∈ H ∗ ,

(7.5.5)

i=1

where ∗ is the multiplication on H ∗ defined above. Proof This follows from Proposition 6.10, specialized for C = H. Note that the structures above come from the right dual structure ∗ H of the coalgebra H in H MH , viewed in a canonical way as an object in H H MH . Definition 7.47 The coinvariants (of the second type) λ ∈ H ∗co(H) are called left cointegrals on H and the space of left cointegrals is denoted by L . For an algebra A, we call an element of A∗ non-degenerate if its kernel does not contain non-zero left or right ideals. Theorem 7.48 Let H be a finite-dimensional quasi-Hopf algebra. Then dimk L = 1 and all non-zero left cointegrals on H are non-degenerate. Proof

Theorem 6.20 implies that L ⊗ H λ ⊗ h → λ · h = S(h)  λ ∈ H ∗

(7.5.6)

is an isomorphism of right quasi-Hopf H-bimodules. As dimk H = dimk H ∗ is finite it follows that dimk L = 1. Let λ ∈ L be non-zero. If I is a left ideal in H such that λ (I) = 0 then λ (HI) = 0, and so I  λ = 0, from which we conclude that I = 0. Likewise, if I is a right ideal

7.5 Cointegral Theory

281

in H such that λ (I) = 0 then (H  λ )(I) = 0, thus h∗ (I) = 0, for all h∗ ∈ H ∗ because S is bijective. Therefore I = 0, hence λ is non-degenerate. According to Proposition 6.21 and (7.5.4), an element λ ∈ H ∗ is a left cointegral on H if and only if

∑ ei ∗ λ ⊗ ei = S(x2 )  E(λ  S−1 (x1 )) ⊗ x3 , i

which is equivalent to

λ (V 2 h2U 2 )V 1 h1U 1 = E(λ  S−1 (x1 )), hS(x2 )x3 , ∀ h ∈ H.

(7.5.7)

Here E : H ∗ → L is the projection from Definition 6.18, specialized for our context. More precisely, E(h∗ ), h = ∑ei ∗ h∗ , S−1 (q1 )hS(β S(q2 ei )),

(7.5.8)

i

for all h∗ ∈ H ∗ and h ∈ H. It can be expressed in terms of the projection P onto the space of left integrals defined in (7.2.1) as follows. Lemma 7.49 Let H be a finite-dimensional quasi-Hopf algebra and E, P the two projections mentioned above. Then E(h∗ ), h = h∗ , S−1 (P(S(h))),

(7.5.9)

for all h∗ ∈ H ∗ and h ∈ H. In particular, E(h∗  S−1 (h)) = μ (h)E(h∗ ), ∀ h∗ ∈ H ∗ and h ∈ H,

(7.5.10)

which implies

λ (S−1 (h)h ) = μ (h1 )λ (h S(h2 )), ∀ λ ∈ L and h, h ∈ H.

(7.5.11)

As usual, μ ∈ H ∗ is the modular element of H ∗ . Proof Take f = f 1 ⊗ f 2 = F 1 ⊗ F 2 and f −1 = g1 ⊗ g2 = G1 ⊗ G2 as in (3.2.15) and (3.2.16). As we have seen before, we have g1 S(g2 α ) = β , S(β f 1 ) f 2 = α ,

f 1 β S( f 2 ) = S(α ),

(7.5.12)

and so we compute, for h∗ ∈ H ∗ and h ∈ H: E(h∗ ), h =

∑ei ∗ h∗ , S−1 (q1 )hS2 (q2 ei )S(β ) i

(7.3.1),(3.2.13)

=

∑e j , hS2 (q2 S−1 ( f 2 q12 p2 )(e j )1U 1 )S(β )h∗ , S−1 ( f 1 q11 p1 )(e j )2U 2  j

(3.2.23),(7.3.1)

=

(3.2.19)

∑e j , hS2 (S−1 ( f 2 )(e j )1 g1 S(X 2 )α X 3 S−1 (β )) j

h∗ , S−1 ( f 1 )(e j )2 g2 S(X 1 )

282

Finite-Dimensional Quasi-Hopf Algebras

∑e j , hS2 (S−1 ( f 2 )(e j )1 F12 X 2 g12 G2 α S−1 (β F 1 X 1 g11 G1 ))

(3.1.5)

=

(3.2.17)

j

h∗ , S−1 ( f 1 )(e j )2 F22 X 3 g2  (7.5.12),(3.2.1)

=

∑ei , hS2 (S−1 ( f 2 )(ei )1 X 2 )S(β F 1 X 1 β )F 2 h∗ , S−1 ( f 1 )(ei )2 X 3  i

(7.5.12),(3.2.13)

=

= ∑ei , hS2 (S−1 (S(ei )2 )S−1 ( f 2 )X 2 S−1 (S−1 (α )X 1 β )) i

h∗ , S−1 (S(ei )1 )S−1 ( f 1 )X 3 

∑ei , hS(S(ei )2 )S(β F 1 f12 X 2 g12 )F 2 f22 X 3 g2 α 

(3.1.5)

=

(3.2.17)

i

h∗ , S−1 (S(ei )1 )S−1 ( f 1 X 1 g11 ) (7.5.12),(3.2.1)

=

∑e j , g1 S(hS((e j )2 )S(X 2 )α X 3 g2 α )h∗ , S−1 (X 1 (e j )1 ) j

(7.5.12),(3.2.19)

=

(7.2.1)

h∗ , S−1 (P(S(h))),

as desired. Now, the relation (7.5.10) follows easily from (7.5.9) and the fact that  P(h) ∈ lH , for all h ∈ H. To prove (7.5.11), let qR be the element defined by (3.2.19). For all h, h ∈ H and λ ∈ L we have:

λ (S−1 (h)h )

= (7.5.9),(7.2.1)

=

E(λ ), S−1 (h)h 

∑ei , β S2 (q2 (ei )2 )S(h )hλ , S−1 (q1 (ei )1 ) i

=

∑e j , β S2 (q2 (e j )2 h2 )S(h )λ , S−1 (q1 (e j )1 h1 ) j

(7.2.1),(7.5.9)

=

(7.5.10)

E(λ  S−1 (h1 )), h S(h2 ) = μ (h1 )λ (h S(h2 )),

as claimed, so the proof is finished. Lemma 7.49 and (7.5.7) say that λ ∈ H ∗ is a left cointegral on H if and only if

λ (V 2 h2U 2 )V 1 h1U 1 = μ (x1 )λ (hS(x2 ))x3 , ∀ h ∈ H.

(7.5.13)

In what follows we will give other characterizations for a left cointegral on H. We first need the following result. Lemma 7.50 If φ is the Frobenius morphism for H defined in Proposition 7.20 then λ := φ ◦ S is a non-zero left cointegral on H. Proof λ is non-zero since ξ (λ ◦ S−1 ) = ξ (φ ) = 1H , where ξ is the isomorphism considered in the proof of Proposition 7.20. We next show that

λ0 (q2t2 p2 )q1t1 p1 = μ (β )λ0 (t)1H , ∀ λ0 ∈ L and t ∈

 H

, l

where μ is the modular element of H ∗ defined in (7.2.6). We have

λ0 (q2 h2 p2 )q1 h1 p1 (7.3.4),(7.3.5)

=

λ0 (V 2 [S−1 (q˜1 )hS( p˜1 )]2U 2 )q˜2V 1 [S−1 (q˜1 )hS( p˜1 )]1U 1 p˜2

(7.5.14)

7.5 Cointegral Theory (7.5.13)

=

283

μ (x1 )λ0 (S−1 (q˜1 )hS(x2 p˜1 ))q˜2 x3 p˜2 ,

for all h ∈ H. Thus we have shown that

λ0 (q2 h2 p2 )q1 h1 p1 = μ (x1 )λ0 (S−1 (q˜1 )hS(x2 p˜1 ))q˜2 x3 p˜2 , ∀ h ∈ H.

(7.5.15)

Specializing the above formula for h = t, a left integral in H, we obtain that (3.2.20)

λ0 (q2t2 p2 )q1t1 p1

=

(7.2.6),(3.2.20)

μ (x1 )λ0 (tS(x2 p˜1 ))x3 p˜2

=

μ (x1 )μ (X 1 β S(X 2 ))μ (S(x2 ))λ0 (t)x3 X 3

=

μ (β )λ0 (t)1H ,

as desired. Observe now that (7.5.14) implies

ξ (λ0 ◦ S−1 ) = μ (β )λ0 (t)1H = ξ (μ (β )λ0 (t)φ ), and so λ0 = μ (β )λ0 (t)φ ◦ S = μ (β )λ0 (t)λ , for all λ0 ∈ L . Taking a non-zero λ0 in L and using the uniqueness of left cointegrals on H we conclude that λ is a non-zero left cointegral on H, as needed. Theorem 7.51 For a finite-dimensional quasi-Hopf algebra H and λ ∈ H ∗ , the following assertions are equivalent: (i) λ is a non-zero left cointegral on H; (ii) λ (t) = 0 and λ (q2t2 p2 )q1t1 p1 = μ (β )λ (t)1H , for any left integral t = 0 in H;  (iii) λ (t) = 0 and λ (ht2 p2 )t1 p1 = μ (β )λ (t)β S(h), for all t ∈ lH \{0} and h ∈ H. Furthermore, if β has a left inverse in H then (i)–(iii) above are also equivalent to  (iv) λ (t) = 0 and λ (t2 p2 )t1 p1 = μ (β )λ (t)β , for all t ∈ lH \{0}. Here pR and qR are the elements defined in (3.2.19). Proof (i) ⇒ (ii). Take t a non-zero left integral in H. If λ (t) = 0 then λ (Ht) = 0, and so Ker(λ ) contains a non-zero left ideal, a contradiction (see Theorem 7.48). Thus λ (t) = 0. The rest of the proof follows from (7.5.14). (ii) ⇒ (i). Take t a non-zero left integral in H. The fact that λ (t) = 0 implies that λ is non-zero. We know from Remark 3.15 that μ (β ) = 0, and so we can write (μ (β )λ (t))−1 λ (q2t2 p2 )q1t1 p1 = 1H , which means that ξ ((μ (β )λ (t))−1 λ ◦ S−1 ) = 1H = ξ (φ ). As ξ is an injective map, Lemma 7.50 implies that (μ (β )λ (t))−1 λ , and so λ as well, is a left cointegral on H. (i) ⇒ (iii). By (3.2.21) and (7.2.6) it follows that t1 p1 h ⊗ t2 p2 = μ (h1 )t1 p1 ⊗ t2 p2 S(h2 ), ∀ h ∈ H.

(7.5.16)

Also, as an immediate consequence of (7.2.3), we obtain t1 ⊗ t2 = β q1t1 ⊗ q2t2 = q1t1 ⊗ S−1 (β )q2t2 . These formulas allow us to compute:

λ (ht2 p2 )t1 p1

=

λ , S−1 (S(h))t2 p2 t1 p1

(7.5.17)

284

Finite-Dimensional Quasi-Hopf Algebras (7.5.11)

μ (S(h)1 )λ ,t2 p2 S(S(h)2 )t1 p1

(7.5.16)

λ (t2 p2 )t1 p1 S(h)

= =

(7.5.17),(7.5.14)

=

μ (β )λ (t)β S(h),

for all h ∈ H, as required. (iii) ⇒ (ii). We use (iii) to compute

λ (q2t2 p2 )q1t1 p1

=

μ (β )λ (t)q1 β S(q2 )

(3.2.19)

μ (β )λ (t)X 1 β S(X 2 )α X 3

(3.2.2)

μ (β )λ (t)1H ,

= =

as desired. It is clear that (iii) implies (iv) in general, without the extra hypothesis on β . For the converse, we need β to be left invertible in H since (7.5.17)

(iv)

λ (q2t2 p2 )β q1t1 p1 = λ (t2 p2 )t1 p1 = μ (β )λ (t)β . Thus the relation in (ii) holds, and so the one in (iii) also holds. Proposition 7.52 A cointegral λ on H is non-zero if and only if λ (t) = 0 for some non-zero left integral t in H. Proof We prove the opposite statement. Namely, λ = 0 if and only if λ (t) = 0 for some non-zero left integral t in H. The direct implication is obvious. For the converse, if λ (t) = 0 then λ is zero on a non-zero left ideal of H, which implies λ = 0 by Theorem 7.48. The above characterizations show how to find left cointegrals on a finite-dimensional quasi-Hopf algebra H. First we have to find a non-zero left integral in H. Secondly, working maybe with dual bases, we have to determine the element λ ∈ H ∗ that satisfies, for instance, (ii) of the above theorem. The simplest condition that we have is (iv) but it can be applied only in the case when β is invertible. Nevertheless, this situation occurs in all the forthcoming examples in this section. Notice that when H is unimodular and α , β are invertible, (iv) simplifies as follows: λ (t2 )t1 = λ (t)β S−1 (α ), and this is because μ = ε , and t, which this time is a right integral as well, obeys (3.2.23)

(7.5.16)

t1 ⊗ t2 = t1 q11 p1 ⊗ t2 q12 p2 S(q2 ) = t1 p1 ⊗ t2 p2 α = t1 p1 S−1 (α ) ⊗ t2 p2 . We next apply these observations to H(2) and H± (8), respectively. Example 7.53 For the quasi-Hopf algebra H(2) defined in Example 3.26, denote by {P1 , Pg } the dual basis in H(2)∗ corresponding to the basis {1, g} of H(2). Then Pg is a (non-zero) left cointegral on H(2). Proof For H(2) we have β = 1, α = g is invertible and H(2) is unimodular (see Example 7.15(1)), a left and right non-zero integral being given by t = 1 + g. Thus, finding all the left cointegrals on H(2) is equivalent to finding all those elements

7.5 Cointegral Theory

285

λ ∈ H(2)∗ satisfying λ (1)1 + λ (g)g = λ (1 + g)g. It is clear that we must have λ (1) = 0, and so L = kPg . Note that Pg (t) = 1, so Pg is non-zero, as needed. Example 7.54 The spaces of left cointegrals for H± (8), the quasi-Hopf algebras from Example 3.30, are both equal to kPx3 . Here we have denoted by {Pgi x j | 0 ≤ i ≤ 1, 0 ≤ j ≤ 3} the dual basis corresponding to the canonical basis {gi x j | 0 ≤ i ≤ 1, 0 ≤ j ≤ 3} of H± (8). Proof This time the computations are much more complicated. Recall first that t = (1 + g)x3 = x3 (1 − g) is a non-zero left integral in H± (8) which is not a right integral; see Example 7.15(2). Denote ω := 12 (1 ± i) and let ω = 12 (1 ∓ i) be its conjugate. In order to compute Δ(t) we rewrite Δ(x) as Δ(x) = ω x ⊗ 1 + ω x ⊗ g + p+ ⊗ x + p− ⊗ gx, to compute that Δ(x2 ) = x2 ⊗ g + g ⊗ x2 + (p+ ± ip− )x ⊗ x + (p− ± ip+ )x ⊗ gx, from which we obtain Δ(x3 ) = ω x3 ⊗ 1 + ω x3 ⊗ g ± ip− x2 ⊗ x + ω gx ⊗ x2 + p+ ⊗ x3 ± ip+ x2 ⊗ gx − ω gx ⊗ gx2 − p− ⊗ gx3 . We have Δ(t)pR = Δ(x3 )Δ(1 − g)pR . By taking the formula of pR from Example 7.31(1) and using Δ(1 − g)pR = 1 ⊗ 1 − g ⊗ g we conclude that Δ(t)pR = (ω + ω g)x3 ⊗ 1 + (ω + ω g)x3 ⊗ g ± ix2 ⊗ x + (ω g − ω )x ⊗x2 + 1 ⊗ x3 ± igx2 ⊗ gx − (ω g − ω )x ⊗ gx2 + g ⊗ gx3 . Now take λ = ∑i, j ci j Pgi x j be an element of H ∗ . It follows that λ satisfies (iv) in the statement of Theorem 7.51 if and only if c01 = c11 = c13 = 0 and the following relations hold:

ω c12 − ω c02 = 0,

ω c00 + ω c10 = 0,

ω c02 − ω c12 = 0 and ω c00 + ω c10 = 0.

We get c00 = c02 = c10 = c12 = 0 as well, and so λ = c03 Px3 . We thus have L = kPx3 , since Px3 (t) = 1. Other characterizations for left cointegrals are included in the result below. This time they involve all the elements and not only the “generator,” that is, an integral. Proposition 7.55 Let H be a finite-dimensional quasi-Hopf algebra and μ the modular element of H ∗ . For λ ∈ H ∗ , the following statements are equivalent: (i) λ is a left cointegral on H; (ii) for all h ∈ H, we have

λ (S−1 ( f 1 )h2 S−1 (q1 g1 ))S−1 ( f 2 )h1 S−1 (q2 g2 ) = μ (q11 x1 )λ , hS−1 ( f 1 )g2 S(q12 x2 )q2 x3 S−1 (S−1 ( f 2 )g1 );

(7.5.18)

286

Finite-Dimensional Quasi-Hopf Algebras

(iii) for all h ∈ H, we have

λ (S−1 ( f 1 )h2U 2 )S−1 ( f 2 )h1U 1 = μ (q11 x1 )λ , hS(q12 x2 )q2 x3 . Here f = f 1 ⊗ f 2 , qR = q1 ⊗ q2 and U = U 1 ⊗U 2 are defined by (3.2.15), (3.2.19) and (7.3.1), respectively, and f −1 = g1 ⊗ g2 is as in (3.2.16). Proof (i) ⇒ (ii). Suppose that λ is a left cointegral. As before, we write f = f 1 ⊗ f 2 = F 1 ⊗ F 2 = F1 ⊗ F2 , f −1 = g1 ⊗ g2 = G1 ⊗ G2 = G1 ⊗ G2 , qR = q1 ⊗ q2 and pR = p1 ⊗ p2 . We compute:

μ (q11 x1 )λ , hS−1 ( f 1 )g2 S(q12 x2 )q2 x3 S−1 (S−1 ( f 2 )g1 ) (7.5.10)

λ ,V 2 [S−1 (q1 )hS−1 ( f 1 )g2 ]2U 2 q2V 1 [S−1 (q1 )hS−1 ( f 1 )g2 ]1U 1

=

(7.5.13)

S−1 (S−1 ( f 2 )g1 )

(7.3.1)

λ , S−1 (F1 q11 p1 )h2 S−1 (F 1 f11 G1 )g22U 2 q2 S−1 (F2 q12 p2 )h1

=

(3.2.13)

S−1 (F 2 f21 G2 )g21U 1 S−1 (S−1 ( f 2 )g1 )

(3.2.23)

λ , S−1 (S(h)1 )S−1 (F 1 f11 )g22 G2 S(X 1 )S−1 (S(h)2 )S−1 (F 2 f21 )

=

(3.2.13)

g21 G1 S(X 2 )α S−1 (S−1 ( f 2 )g1 S(X 3 ))

(3.1.5)

λ , S−1 (S(h)1 )S−1 (S(X 3 )F 1 f11 )g2 S−1 (S(h)2 )S−1 (S(X 2 )F 2 f21 )

=

(3.2.17)

g12 G2 α S−1 (S−1 (S(X 1 ) f 2 )g11 G1 )

(7.5.12)

λ , S−1 (S(h)1 )S−1 (S(X 3 )F 1 f11 )S−1 (S(h)2 )

=

(3.2.1)

S−1 (S−1 (S(X 1 ) f 2 )β S(X 2 )F 2 f21 )

(3.1.5),(3.2.17)

λ , S−1 (S(h)1 )S−1 ( f 1 X 1 )S−1 (S(h)2 )S−1 (S−1 (F 2 f22 X 3 )β F 1 f12 X 2 )

(7.5.12),(3.2.1)

λ , S−1 (X 1 S(h)1 )S−1 (S−1 (α X 3 )X 2 S(h)2 )

(3.2.19),(3.2.13)

λ , S−1 ( f 1 )h2 S−1 (q1 g1 )S−1 ( f 2 )h1 S−1 (q2 g2 ).

= = =

(ii) ⇒ (i). Assume that λ ∈ H ∗ satisfies (7.5.18). It follows from (3.2.13) that

λ (S−1 (q1 h1 ))q2 h2 = μ (q11 x1 )λ , S−1 ( f 1 h)g2 S(q12 x2 )S−1 ( f 2 )g1 S(q2 x3 ), (7.5.19) for all h ∈ H, and λ , S−1 (P(S(h))) (7.2.1),(7.5.19)

μ (q11 x1 )λ , S−1 ( f 1 β S( f 2 )S2 (g1 )S3 (q2 x3 )S(h))g2 S(q12 x2 )

=

(7.5.12)

μ (q11 x1 )λ , hS2 (q2 x3 )S(g1 )α g2 S(q12 x2 )

=

(7.5.12),(3.2.19),(3.2.23)

=

λ (h).

It follows from Lemma 7.49 that E(λ ) = λ , so λ ∈ L . (iii) ⇒ (i). Repeating the computations made in the first part of the proof of Lemma 7.49, we find that the projection E (see (7.5.8)) is given by n

E(h∗ ), h = ∑ ei , hS( f 2 )S2 ((ei )1U 1 )S(β )h∗ , S−1 ( f 1 )(ei )2U 2 ,

(7.5.20)

i=1

for all h∗ ∈ H, h ∈ H. Using (7.5.20), we can compute that E(λ ) = λ , so λ ∈ L .

7.5 Cointegral Theory

287

(i) ⇒ (iii). Assume that λ ∈ L . We calculate:

μ (q11 x1 )λ , hS(q12 x2 )q2 x3 (7.5.11),(7.5.13)

=

(7.3.1),(3.2.13),(3.2.23)

=

λ ,V 2 [S−1 (q1 )h]2U 2 q2V 1 [S−1 (q1 )h]1U 1 λ , S−1 ( f 1 )h2U 2 S−1 ( f 2 )h1U 1

and the proof is complete. The notion of right cointegral on a finite-dimensional quasi-Hopf algebra H can be introduced, too. In case of integrals we have that r is a right integral in H if and only if it is a left integral in H op . By analogy, we say that Λ ∈ H ∗ is a right cointegral on H if and only if it is a left cointegral on H cop . Since H cop is also a finite-dimensional quasi-Hopf algebra all the results obtained for left cointegrals have counterparts for right cointegrals: all we have to do is to transfer them through the “cop” machinery. For instance, in H cop we have Ucop = S−1 (q˜1 g1 ) ⊗ S−1 (q˜2 g2 ), Vcop = S( p˜1 ) f 2 ⊗ S( p˜2 ) f 1 and μcop = μ . Thus Λ is a right cointegral on H if and only if Λ(S( p˜2 ) f 1 h1 S−1 (q˜2 g2 ))S( p˜1 ) f 2 h2 S−1 (q˜1 g1 ) = μ (X 3 )Λ(hS−1 (X 2 ))X 1 , ∀ h ∈ H. The H cop -version of Theorem 7.51 is the following: Theorem 7.56 For a finite-dimensional quasi-Hopf algebra H and Λ ∈ H ∗ , the following assertions are equivalent: (i) Λ is a non-zero right cointegral on H;  (ii) Λ(t) = 0 and Λ(q˜1t1 p˜1 )q˜2t2 p˜2 = μ −1 (β )Λ(t)1H , for all 0 = t ∈ lH ;  (iii) Λ(t) = 0 and Λ(ht1 p˜1 )t2 p˜2 = μ −1 (β )Λ(t)S−1 (hβ ), for all 0 = t ∈ lH and h ∈ H. Furthermore, if β is invertible then (i)–(iii) above are also equivalent to  (iv) Λ(t) = 0 and Λ(t1 p˜1 )t2 p˜2 = μ −1 (β )Λ(t)S−1 (β ), for all 0 = t ∈ lH . Also, for a right cointegral Λ on H we have Λ(S(h)h ) = μ (h2 )Λ(h S−1 (h1 )), ∀ h, h ∈ H. The space of right cointegrals on H will be denoted by R. Then H ⊗ R h ⊗ Λ → S−1 (h)  Λ ∈ H ∗ is an isomorphism of left quasi-Hopf H-bimodules, and so any non-zero right cointegral is non-degenerate and dimk R = 1, according to the “cop” version of Theorem 7.48. We do not mention here the structure of H ∗ in H H MH but we will explain in the notes of this chapter a natural monoidal way in which one can obtain it. Also, in the next section we will see connections between left and right cointegrals given by the antipode. We end with few comments about the infinite-dimensional case. As we have seen, the concept of cointegral only makes sense in the case where H is finite dimensional: indeed, we need dual bases in H and H ∗ in order to make H ∗ into a right quasi-Hopf

288

Finite-Dimensional Quasi-Hopf Algebras

bimodule. Also, the equivalent characterizations from Theorem 7.51 and Proposition 7.55 make no sense in the infinite-dimensional case, as they involve the modular element and/or left integrals, which can only be defined in the finite-dimensional case. To avoid this we can make use of the second structure theorem for quasi-Hopf bimodules presented in Chapter 6. Suppose first that H is finite dimensional and consider H ∗ ∈ H MHH exactly as in the statement of Proposition 7.46. The coinvariants λ ∈ H ∗co(H) are called left alternative cointegrals on H, and the space of left alternative cointegrals is denoted by L = H ∗co(H) . From the second structure theorem for quasi-Hopf bimodules we obtain immediately that dimk L = 1, assuming H is finite dimensional. Furthermore, from the results obtained in Chapter 6 we have L ∼ =L as left H-modules. By applying (3.1.9) and (3.2.1), we find that λ ∈ H ∗ is an alternative left cointegral if and only if

λ (V 2 h2U 2 )V 1 h1U 1 = λ (S−1 (X11 p1 )hS(S(X 3 ) f 1 ))X21 p2 S(X 2 ) f 2

(7.5.21)

for all h ∈ H. Then (7.5.21) can be used to extend the definition of left alternative cointegrals to infinite-dimensional quasi-Hopf algebras with bijective antipode. The same can be done in the case of right cointegrals.

7.6 Integrals, Cointegrals and the Fourth Power of the Antipode We will present various Frobenius systems for a finite-dimensional quasi-Hopf algebra H in terms of integrals and cointegrals in and on H, respectively. Combined with the uniqueness of a Frobenius system this yields a formula for the fourth power of the antipode of H, known as the Radford S4 formula. Lemma 7.57 Let A be a Frobenius algebra and S : A → A an anti-algebra automorphism. If (φ , e = e1 ⊗ e2 ) is a Frobenius system for A then (φ ◦ S, S−1 (e2 ) ⊗ S−1 (e1 )) is a Frobenius system for A, too. In addition, if χ is the Nakayama automorphism associated to (φ , e) then S−1 ◦ χ −1 ◦ S is the Nakayama automorphism corresponding to (φ ◦ S, S−1 (e2 ) ⊗ S−1 (e1 )). Proof To see that (φ ◦ S, S−1 (e2 ) ⊗ S−1 (e1 )) is a Frobenius system for A is straightforward, so we leave the verification of the details to the reader. Now, if we denote by η the Nakayama automorphism defined by this new Frobenius system then (7.1.1)

(7.1.2)

η (a) = (φ ◦ S)(S−1 (e2 )a)S−1 (e1 ) = φ (S(a)e2 )S−1 (e1 ) = S−1 (χ −1 (S(a))), for all a ∈ A, as needed. From now on H is a finite-dimensional quasi-Hopf algebra. From Proposition 7.20 we know that (φ , q1t1 p1 ⊗ S(q2t2 p2 )) is a Frobenius system for H, where t is a nonzero left integral in H and φ ∈ H ∗ is the unique map satisfying one of the equivalent conditions in (7.2.9). By Lemma 7.50, λ := φ ◦ S is a non-zero left cointegral on H,

7.6 Integrals, Cointegrals and the Fourth Power of the Antipode

289

and so this Frobenius system can be rewritten as (λ ◦ S−1 , q1t1 p1 ⊗ S(q2t2 p2 )), with  0 = t ∈ lH and λ ∈ L the unique map satisfying λ (S−1 (q1t1 p1 ))q2t2 p2 = 1H or, equivalently, λ (q2t2 p2 )q1t1 p1 = 1H . By Theorem 7.51 this reduces to μ (β )λ (t) = 1, and we shall see that this is equivalent to λ (S−1 (t)) = 1. We apply the previous lemma to the above Frobenius system for H. In this way we get that (λ , q2t2 p2 ⊗ S−1 (q1t1 p1 )) is another Frobenius system for H. From Proposition 7.9 there exists an invertible element g in H such that

λ ◦ S−1 = λ  g and q1t1 p1 ⊗ S(q2t2 p2 ) = q2t2 p2 ⊗ g−1 S−1 (q1t1 p1 ).

(7.6.1)

Furthermore, by (7.1.3) we have g = λ (S−1 (q2t2 p2 ))S−1 (q1t1 p1 )

and g−1 = λ (q1t1 p1 )S(q2t2 p2 ).

(7.6.2)

Definition 7.58 The invertible element g of H defined by (7.6.2) is called the modular element of H. Clearly, g is independent of the choice of the pair (λ ,t) with the above properties. Proposition 7.59 Let H be a finite-dimensional quasi-Hopf algebra and (λ ,t) ∈  L × lH satisfying λ (S−1 (t)) = 1. Then (λ ◦ S−1 , q1t1 p1 ⊗ S(q2t2 p2 )) is a Frobenius system for H. 

Proof All we have to prove is that giving a pair (λ ,t) ∈ L × lH such that, for  instance, λ (S−1 (q1t1 p1 ))q2t2 p2 = 1H is equivalent to giving a pair (λ ,t) ∈ L × lH such that λ (S−1 (t)) = 1. Then everything follows because of Proposition 7.20. Indeed, if λ (S−1 (q1t1 p1 ))q2t2 p2 = 1H then by applying ε to both sides we get λ (S−1 (t)) = 1. Conversely, if λ (S−1 (t)) = 1 then 1 = λ (S−1 (t)) = (λ  g)(t) = λ (gt) = ε (g)λ (t) = μ (β )λ (S−1 (t))λ (t) = μ (β )λ (t). Then Theorem 7.51 says that λ (q2t2 p2 )q1t1 p1 = 1H and as we mentioned several times this is equivalent to λ (S−1 (q1t1 p1 ))q2t2 p2 = 1H , as required. 

Remark 7.60 If (λ ,t) ∈ L × lH is such that λ (S−1 (t)) = 1 then the inverse of ξ : H ∗ → H considered in the proof of Proposition 7.20 is given by ξ −1 (h) = h  λ ◦ S−1 , for all h ∈ H. This is because ξ is left H-linear and so ξ (h  λ ◦ S−1 ) = hξ (λ ◦ S−1 ) = hξ (φ ) = h, for all h ∈ H. Thus the pair (λ ,t) also has the property that t  λ ◦ S−1 = ε (since ξ (ε ) = t). Note also that λ ◦ S−1 = λ  g implies

λ ◦ S = (λ ◦ S−1  g−1 ) ◦ S = S−1 (g−1 )  λ .

(7.6.3)

Another Frobenius system for H can be obtained by working with H cop instead of H. This allows to find a bijection between the space of left and right cointegrals. Proposition 7.61 Let H be a finite-dimensional quasi-Hopf algebra, t a non-zero left integral in H and λ ∈ L and Λ ∈ R such that λ (S−1 (t)) = 1 and Λ(S(t)) = 1.

290

Finite-Dimensional Quasi-Hopf Algebras

Then u := μ (V 1 )S2 (V 2 ) is invertible in H and λ ◦ S−1 = Λ  u, where the element V = V 1 ⊗V 2 is given by (7.3.1). Consequently, L λ → λ ◦ S−1  u−1 ∈ R is a well-defined bijection. Proof The Frobenius system (λ ◦ S−1 , q1t1 p1 ⊗ S(q2t2 p2 )) of H turns into (Λ ◦ S, q˜2t2 p˜2 ⊗ S−1 (q˜1t1 p˜1 )) for H cop , where Λ is the unique right cointegral on H such that Λ(S(t)) = 1. By Lemma 7.57 we have that (Λ, q˜1t1 p˜1 ⊗ S(q˜1t2 p˜1 )) is also a Frobenius system for H cop , and therefore for H as well. Now, according to Proposition 7.9, the Frobenius systems (λ ◦ S−1 , q1t1 p1 ⊗ S(q2t2 p2 )) and (Λ, q˜1t1 p˜1 ⊗ S(q˜2t2 p˜2 )) are related through an invertible element u ∈ H satisfying

λ ◦ S−1 = Λ  u,

q1t1 p1 ⊗ S(q2t2 p2 ) = q˜1t1 p˜1 ⊗ u−1 S(q˜2t2 p˜2 ).

(7.6.4)

Thus to prove the first assertion it suffices to show that u = μ (V 1 )S2 (V 2 ). To this end, we first use (7.1.3) to find u = λ (S−1 (q˜1t1 p˜1 ))S(q˜2t2 p˜2 ). We have (3.2.20)

f 1 p˜1 ⊗ f 2 p˜2

=

f 1 X 2 S−1 (X 1 β ) ⊗ f 2 X 3

(7.5.12),(3.2.1)

f 1 X 1 g21 G2 α S−1 (X 1 g11 G1 ) ⊗ f 2 X 3 g2

(3.1.5),(3.2.17)

f 1 g21 G1 S(X 2 )α X 3 S−1 (g1 ) ⊗ f 2 g22 G2 S(X 1 )

(3.2.13),(3.2.19)

S(q2 S−1 (g2 )2 )S−1 (g1 ) ⊗ S(q1 S−1 (g2 )1 ),

= = =

where f = f 1 ⊗ f 2 = F 1 ⊗ F 2 and f −1 = g1 ⊗ g2 = G1 ⊗ G2 , respectively. Also, by (3.2.21) it follows that any right integral r in H satisfies r1 p1 h ⊗ r2 p2 = r1 p1 ⊗ r2 p2 S(h), ∀ h ∈ H. In particular, if we set r := S−1 (t) ∈ t1 p˜1 ⊗ t2 p˜2

(3.2.13)

H r

(7.6.5)

then

=

g1 S(r2 ) f 1 p˜1 ⊗ g2 S(r1 ) f 2 p˜2

=

g1 S(q2 S−1 (G2 )2 r2 )S−1 (G1 ) ⊗ g2 S(q1 S−1 (G2 )1 r1 )

(7.2.7),(3.2.16)

=

(3.2.6)

μ (G2 )α1 δ 1 S(q2 r2 )S−1 (G1 ) ⊗ α2 δ 2 S(q1 r1 ) μ (G2 )α1 β S(q2 r2 X 3 )S−1 (G1 ) ⊗ α2 X 1 β S(q1 r1 X 2 )

=

(3.2.20),(7.3.6)

μ (G2 )α1 β S(q2 r2 p2 )S−1 (G1 ) ⊗ α2 S(q1 r1 p1 )

(7.6.5)

μ (G2 )α1 β S(q2 r2 p2 α2 )S−1 (G1 ) ⊗ S(q1 r1 p1 )

(3.2.1)

μ (G2 )β S(q2 r2 p2 )S−1 (G1 ) ⊗ S(q1 r1 p1 ),

= = =

and therefore q˜1t1 p˜1 ⊗ q˜2t2 p˜2

= (7.6.5)

=

μ (G2 )q˜1 β S(q2 r2 p2 )S−1 (G1 ) ⊗ q˜2 S(q1 r1 p1 ) μ (G2 )q˜1 β S(q2 r2 p2 q˜2 )S−1 (G1 ) ⊗ S(q1 r1 p1 )

7.6 Integrals, Cointegrals and the Fourth Power of the Antipode (3.2.20),(3.2.2)

=

291

μ (G2 )S(q2 r2 p2 )S−1 (G1 ) ⊗ S(q1 r1 p1 ).

From here we compute u

μ (G2 )λ (S−2 (G1 )q2 r2 p2 )S2 (q1 r1 p1 )

= (7.5.11)

μ (g2 )μ (S−1 (g1 )1 )λ (q2 r2 p2 S(S−1 (g1 )2 ))S2 (q1 r1 p1 )

(7.6.5)

μ (g2 )μ (S−1 (g1 )1 )λ (q2 r2 p2 )S2 (q1 r1 p1 S−1 (g1 )2 ).

= =

Now, since

λ (q2 r2 p2 )q1 r1 p1 (r ∈

H r

(7.3.4),(7.3.5)

, (7.2.7))

=

λ (V 2 [S−1 (q˜1 )rS( p˜1 )]2U 2 )q˜2V 1 [S−1 (q˜1 )rS( p˜1 )]1U 1 p˜2

=

μ (q˜1 )λ (V 2 r2U 2 )q˜2V 1 r1U 1

(7.5.13)

=

μ (q˜1 )μ (x1 )λ (rS(x2 ))q˜2 x3

=

μ (q˜1 )λ (S−1 (t))q˜2

=

μ (q˜1 )q˜2 ,

we get that u

=

μ (g2 q˜1 )μ (S−1 (g1 )1 )S2 (q˜2 S−1 (g1 )2 )

(3.2.20),(3.2.13)

μ (g2 S(x1 )α S−1 ( f 2 g12 G2 S(x2 )))S( f 1 g11 G1 S(x3 ))

(3.1.5),(3.2.17)

μ (x3 S−1 ( f 2 x2 β ))S( f 1 x1 ) = μ −1 ( f 2 p2 )S( f 1 p1 )

= =

(7.3.1)

=

(3.2.19)

μ (V 1 )S2 (V 2 ),

as stated. Using (7.1.3) or a simple observation guarantees that u−1 = μ −1 (q12 g2 S(q2 ))q11 g1 .

(7.6.6)

From the uniqueness of left and right cointegrals on H it is clear at this point that L λ → λ ◦ S−1  u−1 ∈ R is well defined. It is also bijective with the inverse defined by R Λ → (Λ  u) ◦ S ∈ L . So our proof is complete. Corollary 7.62 If H is a finite-dimensional unimodular quasi-Hopf algebra and S −1 is the antipode of H ∗ then S (L ) = R. Proof μ = ε implies u = 1, and so λ ◦ S−1 = Λ. Everything follows now from the uniqueness of left and right cointegrals on H. 

Corollary 7.63 Consider (Λ,t) ∈ R × lH such that Λ(S(t)) = 1. Then Λ ◦ S = Λ  uS2 (S−1 (u−1 )  μ )g−1 , where g is the modular element of H, and where we define h  h∗ := h∗ (h1 )h2 , for all h∗ ∈ H ∗ and h ∈ H. 

Proof If (λ ,t) ∈ L × lH satisfies λ (S−1 (t)) = 1 we know that λ ◦ S−1 = λ   g. If we think of this relation in H cop then for any pair (Λ,t) ∈ R × lH such that

292

Finite-Dimensional Quasi-Hopf Algebras

Λ(S(t)) = 1 we have Λ ◦ S = Λ  gcop with gcop = Λ(S(q˜1t1 p˜1 ))S(q˜2t2 p˜2 ). We next

show that gcop = uS2 (S−1 (u−1 )  μ )g−1 and this would finish the proof.

By λ ◦ S−1 = Λ  u we get Λ ◦ S = (λ ◦ S−1  u−1 ) ◦ S = S−1 (u−1 )  λ , so =

gcop

λ (q˜1t1 p˜1 S−1 (u−1 ))S(q˜2t2 p˜2 )

(7.6.4)

λ (q1t1 p1 S−1 (u−1 ))uS(q2t2 p2 )

(7.5.16)

μ (S−1 (u−1 )1 )λ (q1t1 p1 )uS2 (S−1 (u−1 )2 )S(q2t2 p2 )

(7.6.2)

uS2 (S−1 (u−1 )  μ )g−1 ,

= = =

as desired. Notice that gcop = g−1 in the case when H is unimodular. Corollary 7.64 For a finite-dimensional quasi-Hopf algebra H assume that L = R. Then g = μ (β )μ −1 (β )−1 u. Consequently, if H is unimodular and admits a nonzero left cointegral that is at the same time right cointegral then g = 1H . Proof Since dimk L = dimk R = 1 it follows that L = R if and only if L ∩ R = 0.  Let 0 = T ∈ L ∩ R and t ∈ lH such that T (S−1 (t)) = 1. Then T ◦ S−1 = Λ  u, for some non-zero Λ ∈ R. But Λ = cT for a certain c ∈ k, so T ◦ S−1 = cT  u. We have μ (β )T (t) = 1, hence 1 = T (S−1 (t)) = cT (ut) = cε (u)T (t) = cμ −1 (β )μ (β )−1 . We get c = μ (β )μ −1 (β )−1 and therefore g = T (S−1 (q2t2 p2 ))S−1 (q1t1 p1 ) = cT (uq2t2 p2 )S−1 (q1t1 p1 ) (7.2.2)

= cT (q2t2 p2 )S−1 (q1t1 p1 )u = μ (β )μ −1 (β )−1 u,

as stated. Corollary 7.65 The inverse of the Nakayama isomorphism χ introduced in Proposition 7.20 is χ −1 (h) = μ (S−1 (uhu−1 )2 )S−1 (S−1 (uhu−1 )1 ), for all h ∈ H, where u is the element of H introduced in Proposition 7.61. Consequently,

μ −1 (q˜1 h1 p˜1 )λ  S−1 (q˜2 h2 p˜2 ) = μ −1 (α )μ (β )S(h)  λ , for all λ ∈ L and h ∈ H. Proof

It follows from Proposition 7.61 that

λ (q˜2t2 p˜2 )q˜1t1 p˜1

= (3.2.22)

=

Λ(uS(q˜2t2 p˜2 ))q˜1t1 p˜1

μ (S−1 (u)2 )Λ(S(q˜2t2 p˜2 ))q˜1t1 p˜1 S−1 (S−1 (u)1 ).

Hence, by (7.1.1) we obtain that

χ −1 (h) = φ (hS(q2t2 p2 ))q1t1 p1 = λ (q2t2 p2 S−1 (h))q1t1 p1 = (3.2.22)

λ (q˜2t2 p˜2 S−1 (hu−1 ))q˜1t1 p˜1

=

μ (S−1 (hu−1 )2 )λ (q˜2t2 p˜2 )q˜1t1 p˜1 S−1 (S−1 (hu−1 )1 )

=

μ (S−1 (uhu−1 )2 )S−1 (S−1 (uau−1 )1 ). QED

(7.6.7)

7.6 Integrals, Cointegrals and the Fourth Power of the Antipode

293

The above formula allows us to compute: S−1 χ −1 S2 (h) =

μ (S−1 (u−1 )2 S(h)2 S−1 (u)2 )S−2 (S−1 (u−1 )1 S(h)1 S−1 (u)1 )

=

μ −1 (q12 g2 S(q2 ))μ (V 1 )μ (q1(1,2) g12 S(V 2 h)2 )S−2 (q1(1,1) g11 S(V 2 h)1 )

(3.2.13),(3.2.17)

=

(3.1.7) (7.3.1),(3.2.13)

=

(3.2.17)

μ −1 (x3 G2 S(q2 X 1 ))μ (V 1 )μ (x2 G1 S(V12 h1 X 2 ) f 2 ) S−2 (x1 q1 S(V22 h2 X 3 ) f 1 )

μ −1 (x3 G2 S(q2 X 1 ))μ (S−1 (F 2Y 3 p2 )y1 )μ (x2 G1 S(y2 h1 X 2 )F 1Y 2 p12 ) S−2 (x1 q1 S(y3 h2 X 3 )Y 1 p11 )

(3.2.19),(5.5.16)

μ (X 1 β )μ (S−1 (F 2 p2 )y1 )μ (S(y2 h1 X 2 )F 1 p1 )S−1 (y3 h2 X 3 )

(3.2.20),(7.5.12)

(μ −1 (α )μ (β ))−1 μ −1 (q˜1 h1 p˜1 )S−1 (q˜2 h2 p˜2 ),

= =

for all h ∈ H. Consequently, S(h)  λ = S(h)  φ ◦ S = (φ  S2 (h)) ◦ S = (χ −1 S2 (h)  φ ) ◦ S = (χ −1 S2 (h)  λ ◦ S−1 ) ◦ S = λ  S−1 χ −1 S2 (h) = (μ −1 (α )μ (β ))−1 μ −1 (q˜1 h1 p˜1 )λ  S−1 (q˜2 h2 p˜2 ), finishing our proof. The formulas that we have just proved indicate how to find right cointegrals when we know a left cointegral, and vice versa. Example 7.66

For H(2), Pg is both a left and right cointegral and g = 1.

Proof H(2) is unimodular and has the antipode defined by the identity map. Thus in this particular case the formula λ ◦ S−1 = Λ  u reduces to λ = Λ, and so L = R. From Example 7.53 we deduce that Pg is a left and right non-zero cointegral on H(2), and from Corollary 7.64 we get g = 1. Example 7.67 For H± (8) we have R = k(ω Px3 + ω Pgx3 ), g = ω 1 + ω g and g−1 = ω 1 + ω g. Proof To find a right cointegral on H± (8) we compute λ ◦ S−1 and the element u. Then λ ◦ S−1  u−1 will be a non-zero right cointegral on H± (8). Consider the left integral t = (1 + g)x3 and take λ = cPx3 with c ∈ k that has to be determined such that λ (S−1 (t)) = 1. Actually, since β = 1 we need to find that unique c ∈ k such that λ (t) = 1 and it then follows that we should have c = 1, and thus λ = Px3 . We use now (p+ ± ip− )(p+ ∓ ip− ) = 1 to see that S−1 (x) = −(p+ ∓ ip− )x, and S−1 (x2 ) = ∓ix2 ,

S−1 (x3 ) = ±i(p+ ∓ ip− )x3 ,

S−1 (gx2 ) = ∓igx2 ,

S−1 (gx) = (p+ ± ip− )x,

S−1 (gx3 ) = ∓i(p+ ± ip− )x3 .

294

Finite-Dimensional Quasi-Hopf Algebras

In particular, we get λ (S−1 (gi x j )) = 0 except in the following two cases where 1 λ (S−1 (x3 )) = ±iλ (p+ x3 ) + λ (p− x3 ) = (1 ± i) = ω , 2 1 −1 3 3 3 λ (S (gx )) = ∓iλ (p+ x ) + λ (p− x ) = (∓i + 1) = ω . 2 In other words, we have λ ◦ S−1 = ω Px3 + ω Pgx3 . It can be easily checked that f = f −1 = pR in the case when H = H± (8). We conclude that u = 1, even if H is not unimodular. Thus ω Px3 + ω Pgx3 is a right non-zero cointegral on H± (8). We end by computing g. Since β = 1, formula (7.2.3) implies q2t2 p2 ⊗ q1t1 p1 = t1 p1 ⊗t2 p2 , hence g = λ (S−1 (t2 p2 ))S−1 (t1 p1 ). By the expression of Δ(t)pR found in Example 7.54 we then obtain g = λ (S−1 (x3 ))1 + λ (S−1 (gx3 ))g = ω 1 + ω g, as desired. A simple inspection shows that g−1 = ω 1 + ω g. We next indicate a connection between λ ◦ S and Λ. Proposition 7.68 Let t be a left integral in H and λ ∈ L and Λ ∈ R such that λ (S−1 (t)) = 1 and Λ(S(t)) = 1. Then v := (μ −1 (g)μ (β ))−1 μ (S(p2 ) f 1 )S(p1 ) f 2 is invertible in H and λ ◦ S = Λ  v. Consequently, L λ → λ ◦ S  v−1 ∈ R is a well-defined bijective map, too. Proof By the “op” version of Proposition 7.20 we have that (φop , S−1 (q2 r2 p2 ) ⊗ q1 r1 p1 ) is a Frobenius system for H, where r is a non-zero right integral in H and φop is the unique element of H ∗ satisfying φop (S−1 (q2 r2 p2 ))q1 r1 p1 = 1 or, equivalently, φop (q1 r1 p1 )q2 r2 p2 = 1.  We set now t = S(r) ∈ lH and take λ ∈ L and Λ ∈ R as in the statement, that is, such that λ (S−1 (t)) = 1 and Λ(S(t)) = 1. We will prove that φop = μ −1 ( p˜1 )S−2 ( p˜2 )  λ ◦ S. We begin by showing that V 1 r1U 1 ⊗V 2 r2U 2 = S−1 (q2t2 p2 ) ⊗ S−1 (q1t1 p1 ).

(7.6.8)

Indeed, by (7.3.4) and (7.3.5) we have q˜11 p1 ⊗ q˜12 p2 S(q˜2 )

= (7.3.2)

=

[q˜1 S( p˜1 )]1U 1 p˜2 ⊗ [q˜1 S( p˜1 )]2U 2 S(q˜2 ) [q˜1 S(q˜21 p˜1 )]1U 1 q˜22 p˜2 ⊗ [q˜1 S(q˜21 p˜1 )]2U 2

(3.2.24)

= U 1 ⊗U 2 .

So we have shown that U = q˜11 p1 ⊗ q˜12 p2 S(q˜2 ).

(7.6.9)

Notice that the “op” version of the above relation is V 1 ⊗V 2 = q1 p˜11 ⊗ S−1 ( p˜2 )q2 p˜12 .

(7.6.10)

7.6 Integrals, Cointegrals and the Fourth Power of the Antipode

295

We next compute V 1 r1U 1 ⊗V 2 r2U 2

V 1 r1 q˜11 p1 ⊗V 2 r2 q˜12 p2 S(q˜2 )

= (7.3.1)

=

S−1 ( f 2 P2 )S−1 (t)1 p1 ⊗ S−1 ( f 1 P1 )S−1 (t)2 p2

(3.2.13),(3.2.19) −1

=

S

(S(x1 ) f 2t2 P2 ) ⊗ S−1 (S(x2 ) f 1t1 P1 )β S(x3 )

(3.1.5),(3.2.17)

S−1 (F 2 x3 g22t2 P2 ) ⊗ S−1 ( f 2 F21 x2 g21t1 P1 )β f 1 F11 x1

(7.5.12),(3.2.1)

S−1 (x3t2 P2 ) ⊗ S−1 (S(x1 )α x2t1 P1 )

= =

(3.2.20)

S−1 (q˜2t2 P2 ) ⊗ S−1 (q˜1t1 P1 )

(7.3.6)

S−1 (q2t2 P2 ) ⊗ S−1 (q1t1 P1 ),

= =

as desired. On the other hand, V 1 r1U 1 ⊗V 2 r2U 2 = q1 p˜11 r1 q˜11 p1 ⊗ S−1 ( p˜2 )q2 p˜12 r2 q˜12 p2 S−1 (q˜2 ) = μ −1 ( p˜1 )q1 r1 p1 ⊗ S−1 ( p˜2 )q2 r2 p2 , from which we conclude that

μ −1 ( p˜1 )q1 r1 p1 ⊗ S−1 ( p˜2 )q2 r2 p2 = S−1 (q2t2 p2 ) ⊗ S−1 (q1t1 p1 ). Note that (3.2.21)

(7.2.7)

hq1 r1 ⊗ q2 r2 = q1 h(1,1) r1 ⊗ S−1 (h2 )q2 h(1,2) r2 = μ −1 (h1 )q1 r1 ⊗ S−1 (h2 )q2 r2 , for all h ∈ H. Since

μ (q˜1 )μ −1 ( p˜1 )q˜2 q1 r1 p1 ⊗ S−1 ( p˜2 )q2 r2 p2 = μ (q˜1 )S−1 (q2t2 p2 S(q˜2 )) ⊗ S−1 (q1t1 p1 ), the latest relation implies q1 r1 p1 ⊗ q2 r2 p2 = μ (q˜1 )q˜2 S−1 (q2t2 p2 ) ⊗ S−1 (q1t1 p1 ),

(7.6.11)

where we also made use of (3.2.24). From the proof of the previous proposition, λ (q2 r2 p2 )q1 r1 p1 = μ (q˜1 )q˜2 . Hence 1H

(3.2.24)

=

μ (S( p˜1 )q˜1 p˜21 )q˜2 p˜22

=

μ −1 ( p˜1 )μ ( p˜21 )λ (q2 r2 p2 )q1 r1 p1 p˜22

(7.6.5)

μ −1 ( p˜1 )μ ( p˜21 )λ (q2 r2 p2 S( p˜22 ))q1 r1 p1

(7.5.11)

=

μ −1 ( p˜1 )λ (S−1 ( p˜2 )q2 r2 p2 )q1 r1 p1

=

λ  μ −1 ( p˜1 )S−1 ( p˜2 ), q2 r2 p2 q1 r1 p1 .

=

From the uniqueness of the map φop it follows that

φop = (λ  μ −1 ( p˜1 )S−1 ( p˜2 )) ◦ S = μ −1 ( p˜1 )S−2 ( p˜2 )  λ ◦ S, as claimed. By the definition of pL it is immediate that d := μ −1 ( p˜1 )S−2 ( p˜2 ) is invertible in H, therefore (λ ◦ S, S−1 (q2 r2 p2 )d ⊗ q1 r1 p1 ) is a Frobenius system for  H (see Proposition 7.9) whenever (λ , r) ∈ L × rH is such that λ (r) = 1. Comparing

296

Finite-Dimensional Quasi-Hopf Algebras

it with (Λ, q˜1t1 p˜1 ⊗ S(q˜2t2 p˜2 )) we conclude that there is an invertible element v ∈ H such that λ ◦ S = Λ  v. According to (7.1.3) we have =

v

(7.3.6),(3.2.13)

λ , S( p˜1 ) f 2 S(t)2 g2 S(q1 )S( p˜2 ) f 1 S(t)1 g1 S(q2 )

(7.3.8),(7.3.1)

λ , S−1 (g2 )q2 (g1 S(t))2U 2 q1 (g1 S(t))1U 1

(7.5.11),(7.2.7)

=

μ (g21 )μ −1 (g1 )λ , q2 S(t)2U 2 S(g22 )q1 S(t)1U 1

=

μ (g21 )μ −1 (g1 )λ , q2 S(t)2U 2 q1 S(t)1U 1 g22

= =

((7.3.2), S(t) ∈

H r

)

(7.3.4),(7.2.7)

((7.5.13), S(t) ∈

H r

λ (S(q˜1t1 p˜1 ))S(q˜2t2 p˜2 )

)

=

μ −1 (g1 )μ (q˜1 g21 )λ ,V 2 S(t)2U 2 q˜2V 1 S(t)1U 1 g22

=

μ (S(g1 )q˜1 g21 )λ (S(t))q˜2 g22 .

By (3.2.20), (3.1.5) and (3.2.17) we compute S(g1 )q˜1 g21 ⊗ q˜2 g22 S(g11 G1 S(x3 ))α g12 G2 S(x2 ) f 1 ⊗ g1 S(x1 ) f 2

= (3.2.1),(7.5.12)

=

S(x2 β S(x3 )) f 1 ⊗ S(x1 ) f 2 = S(p2 ) f 1 ⊗ S(p1 ) f 2 .

(7.6.12)

We also have λ (S(t)) = (S−1 (g−1 )  λ )(t) = μ −1 (g−1 )λ (t) = (μ −1 (g)μ (β ))−1 . Thus v = (μ −1 (g)μ (β ))−1 μ (S(p2 ) f 1 )S(p1 ) f 2 , as claimed. It is easy to see that v−1 = μ −1 (g)μ (β q2 g1 S(q12 ))g2 S(q11 ), and so the proof is finished. Corollary 7.69 Let H be a finite-dimensional unimodular quasi-Hopf algebra and denote by S the antipode of H ∗ . Then S(L ) = R. Proof

In this case we have μ = ε and therefore v = 1H .

We can now prove the formula for the fourth power of the antipode. Theorem 7.70 Let H be a finite-dimensional quasi-Hopf algebra with modular elements g ∈ H and μ ∈ H ∗ . Then, for all h ∈ H, we have S4 (μ −1  (h  μ )) = S3 ( f μ−1 )S(g)hS(g−1 )S3 ( f μ ), where f μ := μ ( f 1 ) f 2 and, for all h∗ ∈ H ∗ and h ∈ H, h∗  h := h∗ (h2 )h1 , h  h∗ = h∗ (h1 )h2 . 

Proof As before, consider λ ∈ L and t ∈ lH such that λ (S−1 (t)) = 1. As we have seen, (λ ,t) defines two Frobenius systems on H, namely (λ ◦ S−1 , q1t1 p1 ⊗ S(q2t2 p2 )) and (λ , q2t2 p2 ⊗ S−1 (q1t1 p1 )). Moreover, they are related through the modular element g of H, in the sense of relations (7.6.1). Denote by χ and η , respectively, the Nakayama automorphisms corresponding to them. On the one hand, by Corollary 7.10, we have χ (h) = g−1 η (h)g, for all h ∈ H. On the other hand, by Lemma 7.57, we know that η = S−1 ◦ χ −1 ◦ S. From these two expressions of η we deduce that χ (h) = g−1 S−1 (χ −1 (S(h)))g, for all h ∈ H, which

7.6 Integrals, Cointegrals and the Fourth Power of the Antipode

297

is clearly equivalent to χ ◦ S−1 ◦ χ ◦ S = Inng−1 , where by Inng−1 we denote the inner automorphism of H produced by g−1 , that is, Inng−1 (h) = g−1 hg, for all h ∈ H. Now, we use (7.2.10) to get (χ ◦ S)(h) = μ (S(h)1 )S2 (S(h)2 ), for all h ∈ H, and so ◦ S = S(S(h)  μ ), for all h ∈ H. If we define Sμ (h) := S(h)  μ , for all h ∈ H, then the last relation reads S−1 ◦ χ ◦ S = S ◦ Sμ . Therefore S−1 ◦ χ

Inng−1 = χ ◦ S−1 ◦ χ ◦ S = χ ◦ S ◦ Sμ = S2 ◦ S2μ . Observe next that (3.2.13)

Sμ (h) = μ (g1 S(h2 ) f 1 )g2 S(h1 ) f 2 = f μ−1 S(μ −1  h) f μ , ∀ h ∈ H. Thus S2μ (h) = f μ−1 S(μ −1  Sμ (h)) f μ = f μ−1 S(μ −1  (S(h)  μ )) f μ , for all h ∈ H, and since S is bijective we conclude that Inng−1 (S−1 (h)) = S2 ( f μ−1 )S3 (μ −1  (h  μ ))S2 ( f μ ), ∀ h ∈ H. By applying S to both sides we get the expression for S4 stated above. Observe that the formula for the fourth power of the antipode in the statement of Theorem 7.70 can be restated, for all h ∈ H, as

μ ( f 1 )S−2 (h)S−1 (g−1 )S( f 2 ) = μ (h1 f 1 )μ −1 (h(2,2) )S−1 (g−1 )S(S(h(2,1) ) f 2 ). (7.6.13) Corollary 7.71 If H is a finite-dimensional unimodular quasi-Hopf algebra then S4 = InnS(g) , where g is the modular element of H. Furthermore, if L = R then S4 is the identity morphism of H. Proof In the unimodular case we have μ = ε and f μ = 1 and if, moreover, L = R then g = 1; see Corollary 7.64. We know how the antipode of H ∗ , or its inverse, carries a left or right cointegral on H. In the result below we will see how the antipode of H itself carries left or right integrals in H. Proposition 7.72 Let t, r be a non-zero left, right integral, respectively, in H. Then S(t) = μ (β )−1 μ (q2t2 p2 )q1t1 p1 , S−1 (t) = μ −1 (g)μ (q2t2 p2 )q1t1 p1 , S(r) = (μ −1 (g)μ (αβ ))−1 μ −1 (q2 r2 p2 )q1 r1 p1 , S−1 (r) = μ (α )−1 μ −1 (q2 r2 p2 )q1 r1 p1 . Proof Consider λ ∈ L obeying λ (S−1 (t)) = 1, so that μ (β )λ (t) = 1. If we define r = μ (t2 p2 )t1 p1 then r h

= (3.2.21)

=

μ (t2 p2 )t1 p1 h μ (t2 h(1,2) p2 S(h2 ))t1 h(1,1) p1

298

Finite-Dimensional Quasi-Hopf Algebras (7.2.6)

=

μ (h1 )μ (t2 p2 )μ −1 (h2 )t1 p1

=

ε (h)r ,

for all h ∈ H. Thus r is a right integral in H. As dimk such that S(t) = cr and S−1 (t) = c r . We have

H r

= 1 there exist c, c ∈ k

(7.6.3)

λ (S(t)) = (S−1 (g−1 )  λ )(t) (7.2.6)

= μ −1 (g−1 )λ (t) = (μ −1 (g)μ (β ))−1 ,

(7.2.3)

(7.6.2)

λ (r ) = μ (t2 p2 )λ (t1 p1 ) = μ (S−1 (β )q2t2 p2 )λ (q1t1 p1 ) = μ −1 (β )μ −1 (g−1 ). Thus c = (μ (β )μ −1 (β ))−1 and c = μ −1 (β g−1 )−1 . These together with (7.2.3) prove the relations stated for S(t) and S−1 (t), respectively. If r is a right integral then denote t = S(r) and consider λ ∈ L such that λ (r) = λ (S−1 (t)) = 1. As in the left-handed case one can show that t  := μ −1 (q2 r2 )q1 r1 is a left integral, hence there exist b, b ∈ k such that S(r) = bt  and S−1 (r) = bt  . The formula for S−1 (r) can be obtained from that for S(t) by replacing H with H op . As the formula for S−1 (t) contains g and we do not have an analogue of g in H op we cannot derive the formula for S(r) from that for S−1 (t). Nevertheless, we can obtain it by computing b as follows. We have λ (S(r)) = (S−1 (g−1 )  λ )(r) = ε (g−1 )λ (r) = λ (t) = μ (β )−1 and

λ (t  ) = λ (q1 r1 )μ −1 (q2 r2 ) λ (q1 r1 p1 )μ −1 (q2 r2 p2 α )

= (7.6.11)

=

μ (q˜1 )λ (q˜2 S−1 (q2t2 p2 ))μ −1 (S−1 (q1t1 p1 )α )

=

μ (q˜1 )φ (q2t2 p2 S(q˜2 ))μ (q1t1 p1 )μ −1 (α ),

where in the second equality we applied the “op” version of (7.2.3), and where φ = λ ◦ S−1 is the notation used for the Frobenius morphism of H. We now make use of the Nakayama automorphism χ of H associated to φ to continue to compute

λ (t  ) = μ (q˜1 )φ (χ (S(q˜2 ))q2t2 p2 )μ (q1t1 p1 )μ −1 (α ) (7.2.2)

= μ (q˜1 )φ (q2t2 p2 )μ (S(χ (S(q˜2 )))q1t1 p1 )μ −1 (α ) = μ (q˜1 )λ (S−1 (q2t2 p2 ))μ −1 (S−1 (q1t1 p1 ))μ −1 (χ (S(q˜2 )))μ −1 (α )

(7.6.2)

= μ (q˜1 )μ −1 (g)μ −1 (Sμ (q˜2 ))μ −1 (α ),

where the last equality follows because, as we have seen in the proof of the last theorem, χ ◦ S = S2 ◦ Sμ . Recall that Sμ (h) = S(h)  μ , so

μ −1 (Sμ (h)) = μ (S(h)1 )μ −1 (S(h)2 ) = ε (S(h)) = ε (h), for all h ∈ H. From here we deduce that λ (t  ) = μ (α )μ −1 (gα ), and thus μ (β )−1 = bμ (α )μ −1 (gα ). Hence S(r) = μ (αβ )−1 μ −1 (gα )−1 μ −1 (q2 r2 )q1 r1 = (μ −1 (g)μ (αβ ))−1 μ −1 (q2 r2 p2 )q1 r1 p1 ,

7.7 A Freeness Theorem for Quasi-Hopf Algebras

299

and the proof is complete. By applying Proposition 7.72 and (7.2.8), we obtain the following result. Corollary 7.73 Let H be a finite-dimensional quasi-Hopf algebra. For all t ∈  and r ∈ rH , we have that S2 (t) = (μ −1 (g)μ (β ))−1t

H l

and S2 (r) = (μ −1 (g)μ (β ))−1 r.

7.7 A Freeness Theorem for Quasi-Hopf Algebras The aim of this section is to show that a finite-dimensional quasi-Hopf algebra K is a free module over any quasi-Hopf subalgebra H. This is by definition a subalgebra of K such that Δ(H) ⊆ H ⊗ H and H ⊗ H ⊗ H contains the reassociator Φ of K. In addition, the distinguished elements α , β that define the antipode S of K belong to H, and S(H) ⊆ H. In other words, the quasi-Hopf algebra structure of K restricts to a quasi-Hopf algebra structure on H. As we shall see, the proof of this freeness theorem has a Frobenius-module theory flavour. This is why we have to start by recalling some concepts and well-known results from this area. The result below is due to Krull and Schmidt and a proof for it can be found in any module theory book; we will indicate several such books in the notes of this chapter. Unless otherwise specified, “module” means either a left or a right module. Theorem 7.74 Every module of finite length over a ring R admits a finite indecom posable decomposition M = ti=1 Mi , with each Mi an indecomposable R-module. It  is unique in the sense that if M = sj=1 N j is another indecomposable decomposition of M then t = s and there exists a bijection σ : {1, . . . ,t} → {1, . . . , s} such that Mi ∼ = Nσ (i) as R-modules, for any 1 ≤ i ≤ t. If R is a finite-dimensional k-algebra then R admits a decomposition of the form (n1 )

R = P1

(nt )

⊕ · · · Pt

,

where ni are non-zero natural numbers and Pi are non-isomorphic indecomposable R-modules; here P(t) denotes the direct sum of t copies of P. In what follows we call P1 , . . ., Pt the principal indecomposable R-modules of R. We say that a left R-module M is faithful if AnnR (M), its annihilator, is the null space. That is, {r ∈ R | r · m = 0, ∀ m ∈ M} = (0). We have a similar definition for the case when M is a right R-module. Lemma 7.75 Let R be a finite-dimensional k-algebra which is left self injective and M a left R-module which is finite dimensional as a k-vector space. Then M is faithful if and only if for any 1 ≤ i ≤ t the left principal indecomposable R-module Pi of R is isomorphic to a direct summand of M. A right-handed version holds as well.

300

Finite-Dimensional Quasi-Hopf Algebras

Proof Consider {m1 , . . . , mn } a basis of M over k. If M is R-faithful then f : R → M n given by f (r) = (r · m1 , . . . , r · mn ) is R-linear and injective. As R is injective as a left R-module there exists a left R-module X such that M n ∼ = R ⊕ X. Writing M, R and X as direct sums of indecomposable R-modules, by the uniqueness of such a decomposition it follows that each Pi is isomorphic (as an R-module) to a direct summand of M. Conversely, assume each Pi is isomorphic to a direct summand of M, say Mi . Then (n1 )

R = P1

(nt )

⊕ · · · ⊕ Pt

(n ) (n ) ∼ = M1 1 ⊕ · · · ⊕ Mt t ,

as left R-modules. So if we take q to be the maximum of n1 , . . ., nt then R embeds into M (q) as a left R-module. Hence, if r ∈ R such that r · m = 0, for all m ∈ M, then rM (q) = 0, and consequently rR = 0. We conclude that r = 0, and therefore M is a faithful left R-module. The right-handed case can be proved in a similar manner, so we are done. Proposition 7.76 Let R be a finite-dimensional k-algebra that is left self injective, and consider M a left R-module which has finite dimension as a k-vector space. Then M is faithful if and only if there exist a positive integer s, a free left R-module F and a non faithful left R-module E such that M (s) ∼ = F ⊕ E as left R-modules. The result is valid for right modules, too. Proof

Assume first that M is R-faithful. By the above lemma we decompose M as (m1 )

M = P1

(mt )

⊕ · · · ⊕ Pt

⊕ Q,

where Q is an R-submodule of M that does not contain submodules isomorphic to one of the Pi s. If we take s to be the smallest common multiple of n1 , . . ., nt with n1 , . . ., nt as in the indecomposable decomposition of R, then M (s) ∼ = P1

(sm1 )

and so if p =

smi ni

is the minimum of

M (s) ∼ = P1

(pn1 )

∼ =R

(p)

sm1 n1 ,

. . .,

(pnt )

⊕ · · · ⊕ Pt

⊕

(smt )

⊕ · · · ⊕ Pt



⊕

smt nt

⊕ Q(s) ,

we have



(sm j −pn j )

Pj

⊕ Q(s)

1≤ j =i≤t (sm −pn j ) Pj j ⊕ Q(s) .

1≤ j =i≤t

 (sm −pn j ) ⊕ Q(s) , so that M (s) ∼ Set F = R(p) and E = 1≤ j =i≤t Pj j = E ⊕ F as left R-modules. Clearly F is a free R-module of rank p. By the way of contradiction, if E is faithful then the previous lemma says that Pi is isomorphic to a (indecomposable) direct summand of Q(s) , and so of Q. This contradicts the choice of Q, and therefore E is not faithful as a left R-module. For the converse, if there is a positive integer s such that M (s) ∼ = E ⊕ F with E, F as in the statement then rM = 0 implies rM (s) = 0, and thus rF = 0. Since F is a free left R-module it follows that r = 0. We conclude that M is R-faithful, as needed.

7.7 A Freeness Theorem for Quasi-Hopf Algebras

301

Corollary 7.77 Let R be a finite-dimensional k-algebra that is left self injective, and M a left R-module of finite dimension over k. Then there exist a positive integer s, a free left R-module F and a non faithful left R-module E such that M (s) ∼ = F ⊕E as left R-modules. We have a similar statement for right modules as well. Proof If M is not faithful we take s = 1, F = 0 and E = M. Otherwise, the last proved result applies. Another important result we need is the following. Recall that if H is a quasi-Hopf algebra and X, Y are left or right H-modules then so is X ⊗Y via the action defined by the comultiplication of H. Proposition 7.78 Let H be a finite-dimensional quasi-Hopf algebra and M a left H-module that is finite dimensional over k. If there exists a finite-dimensional faithful left H-module V such that V ⊗ M ∼ = M dimk (V ) as left H-modules, then M is free as a left H-module. Proof H is a Frobenius algebra and thus left and right self injective; see Proposition 7.13. So Corollary 7.77 applies and we find that V (l) ∼ = F  ⊕ E  and M (s) ∼ = F ⊕E as left H-modules, for some positive integers l, s, free left H-modules F, F  and nonfaithful left H-modules E, E  . We prove first that N := M (s) is a free left H-module. If we denote W := V (l) we have W ⊗N ∼ = M (lsdimkV ) ∼ = N (dimkW ) = (V ⊗ M)(ls) ∼ as left H-modules. Furthermore, W ∼ = F  ⊕ E  and N ∼ = F ⊕ E as left H-modules, so (W ⊗ F) ⊕ (W ⊗ E) ∼ = F (dimkW ) ⊕ E (dimkW ) as left H-modules. We now show that W ⊗ F ∼ = F (dimkW ) as left H-modules and this (dim W ) ∼ k in H M ; see the Krull–Schmidt theorem. Indeed, would imply that W ⊗ E = E if Q is an arbitrary left H-module then · Q ⊗ · H ∼ = Q ⊗ · H as left H-modules, where the dots indicate that · Q ⊗ · H is endowed with the left H-module structure defined by Δ, while Q ⊗ · H has the left H-module structure given by the multiplication of H. An isomorphism is produced by

ς : Q ⊗ · H → · Q ⊗ · H,

ς (x ⊗ h) = h1 p˜1 · x ⊗ h2 p˜2

for all x ∈ Q and h ∈ H. Actually, if ς −1 : Q ⊗ · H → · Q ⊗ · H is given by ς −1 (x ⊗ h) = S−1 (q˜1 h1 ) · x ⊗ q˜2 h2 , for all x ∈ Q and h ∈ H, then

ς (ς −1 (x ⊗ h)) = q˜21 h(2,1) p˜1 S−1 (q˜1 h1 ) · x ⊗ q˜22 h(2,2) p˜2 (3.2.24) (3.2.22) 2 1 −1 1 = q˜1 p˜ S (q˜ ) · x ⊗ q˜22 p˜2 h = x ⊗ h.

In a similar manner it can be checked that ς −1 (ς (x ⊗ h)) = x ⊗ h, for all x ∈ Q and h ∈ H, hence ς is bijective. It is also left H-linear since Δ is multiplicative. This Hlinear isomorphism shows in particular that · Q ⊗ · H ∼ = H (dimk Q) as left H-modules, for any finite-dimensional left H-module Q.

302

Finite-Dimensional Quasi-Hopf Algebras

In our context, since F is H-free, say F ∼ = H (q) for some positive integer q, we get W ⊗F ∼ = (W ⊗ H)(q) ∼ = H (qdimkW ) ∼ = F (dimkW ) = W ⊗ H (q) ∼ as left H-modules, as claimed. We get W ⊗ E ∼ = E (dimkW ) , which implies E (dimkW ) ∼ =W ⊗E ∼ = (F  ⊗ E) ⊕ (E  ⊗ E)

(7.7.1)

as left H-modules. Note that F  is not zero, for otherwise W = E  is not H-faithful, and this cannot happen because V is H-faithful and so is V (l) = W . Now using that F  is H-free, by similar arguments to the ones above we obtain that F  ⊗ E is also free as a left H-module. More precisely, it can be proved that for any left H-module Q the map ς  : · H ⊗ · Q → Q ⊗ · H defined by ς  (h ⊗ x) = S(q2 h2 ) · x ⊗ q1 h1 , for all h ∈ H and x ∈ Q, is left H-linear and an isomorphism with inverse given by ς −1 (x ⊗ h) = h1 p1 ⊗ h2 p2 · x, for all h ∈ H and x ∈ Q, where the dots have the meaning explained a few lines above. Notice that this assertion can be proved by using equations (3.2.21) and (3.2.23); we leave it to the reader to check out all the details. Consequently, since Q ⊗ · H is left H-free, · H ⊗ · Q is also left H-free, and from here we get that U ⊗ Q is H-free for any finite-dimensional left H-free module U. Therefore, if E is not zero then F  ⊗ E is a non-zero free H-module, so a faithful H-module. Then the isomorphism (7.7.1) implies that E (dimkW ) is H-faithful, and thus E is H-faithful, a contradiction. Thus we must have E = 0, and consequently M (s) = N = F, a free left H-module. Finally, we are in the position to prove that M itself is H-free. Toward this end consider a positive integer n such that M (s) ∼ = H (n) as left H-modules. Now view k as a right H-module via the counit ε of H, that is, κ · h = ε (h)κ , for all κ ∈ k, h ∈ H. Then (k ⊗H M)(s) ∼ = k ⊗H M (s) ∼ = k ⊗H H (n) ∼ = (k ⊗H H)(n) ∼ = k(n) . If d is the dimension of k ⊗H M we obtain ds = n, hence M (s) ∼ = H (ds) as left H∼ modules. By the Krull–Schmidt theorem we conclude that M = H (d) , a free left H-module. The proof is finished. We can now prove the announced result. For reasons that will be explained in the notes of this chapter we will refer to it as the quasi-Hopf version of the Nichols– Zoeller theorem. Theorem 7.79 Let K be a finite-dimensional quasi-Hopf algebra and H ⊆ K a quasi-Hopf subalgebra. Then K is a free left and right H-module. Proof In view of the above proposition it suffices to show that for the finite-dimensional left H-module K there exists a finite-dimensional left H-faithful module V such that V ⊗ K ∼ = K (dimk V) as left H-modules. We can take V = K, which clearly has the desired properties. The isomorphisms ς defined in the proof of Proposition 7.78, considered for K in place of H, tells us that Q ⊗ K ∼ = K (dimk Q) as left K-modules, for any left K-module Q. We deduce in particular that K ⊗ K ∼ = K (dimk K) as left K-modules, and therefore as left H-modules as well. Thus K is left H-free.

7.8 Notes

303

Working with K op and H op instead of K and H, respectively, we deduce that K is also right H-free, and this finishes the proof. We present some applications of the freeness theorem. The next result can be seen as an extension of the Lagrange theorem for groups to quasi-Hopf algebras. Corollary 7.80 If K is a finite-dimensional quasi-Hopf algebra and H ⊆ K is a quasi-Hopf subalgebra then dimk H | dimk K. Proof As we have seen, K is left H-free, so there is a positive integer s such that K∼ = H (s) as left H-modules. Thus dimk K = sdimk H and this proves that dimk H is a divisor of dimk K. Corollary 7.81 A quasi-Hopf algebra of prime dimension does not admit proper quasi-Hopf subalgebras. Corollary 7.82 If K is a finite-dimensional semisimple quasi-Hopf algebra then every quasi-Hopf subalgebra H of it is also semisimple. Proof By Theorem 7.28 K contains a left integral t satisfying ε (t) = 1. Since K is a left H-free module there exist h1 , · · · , hn ∈ H such that t = ∑i hi ki , where {ki }i is a fixed basis of K over H. Then for all x ∈ H we have

∑(xhi )ki = x ∑ hi ki = xt = ε (x)t = ε (x) ∑ hi ki , i

i

i

and therefore, for each i, xhi = ε (x)hi , for all x ∈ H. Thus each hi is a left integral in H and since 1 = ε (t) = ∑i ε (hi )ε (ki ) we obtain that at least one of the ε (hi )s is non-zero. By applying Theorem 7.28 again we conclude that H is semisimple, as needed.

7.8 Notes Frobenius algebras started to be studied in the 1930s by Brauer and Nesbitt, who named these algebras after Frobenius; they also introduced the concept of symmetric algebra in [37]. Nakayama discovered the beginnings of the duality property of a Frobenius algebra in [161, 162], and Dieudonn´e used this to characterize Frobenius algebras in [76]. Nakayama also studied symmetric algebras in [160]; the automorphism that carries his name was defined in [162]. Recently, interest in these algebras has been renewed due to connections to monoidal categories and topological quantum field theory; see [208]. Our main sources of inspiration in the presentation of these concepts were the books of Lam [131] and Kadison [123], and the papers [124, 125, 181]. The existence of integrals in finite-dimensional quasi-Hopf algebras was proved in [176], where a short argument of Van Daele [214] that guarantees the existence (and uniqueness) of integrals for finite-dimensional Hopf algebras was generalized to the

304

Finite-Dimensional Quasi-Hopf Algebras

quasi-Hopf algebra setting. For Hopf algebras this result has been known since the late 1960s. Actually, Larson and Sweedler [135] used the fundamental theorem for Hopf modules to prove the existence and uniqueness of integrals in a Hopf algebra, and then the bijectivity of the antipode follows. As we have seen, this fundamental theorem was adapted in [109] for quasi-Hopf bimodules but the proof there relies on the fact that the antipode is bijective. The bijectivity of the antipode has been proved since then in [47], a second proof being given one year later in [199]. The integral theory that was presented here is taken from [47], where it has been proved also that there are no non-zero integrals in the case of an infinite-dimensional quasi-Hopf algebra with bijective antipode. The Maschke theorem is essentially from [172], and the other characterizations for semisimple quasi-Hopf algebras are from [109]. We have used also [109, 124] in the presentation of symmetric algebras and symmetric quasi-Hopf algebras. We reproduced the general theory on cointegrals from [109] but the two sets of characterizations, the examples and the connections between left and right cointegrals are from [47, 49]. As H is a coalgebra in the monoidal category of H-bimodules, and so has a natural left and right H-comodule structure within this category, we get for free that H ∗ is a quasi-Hopf H-bimodule with the structure as in [109]. But we should point out that (apart from the Hopf algebra case) we have multiple choices for the set of coinvariants, and this gives us alternative definitions for cointegrals. The formula for the fourth power of the antipode was proved for unimodular Hopf algebras by Larson [132], and was extended by Radford [183] to any finitedimensional Hopf algebra. Recently, it was generalized for co-Frobenius Hopf algebras in [27]. For quasi-Hopf algebras a first proof appears in [109] but, owing to the Frobenius arguments, we preferred here to present the one given by Kadison in [122]. For module theoretical aspects related to the freeness theorem for quasi-Hopf algebras we refer to the books [131, 10, 70]. It was conjectured by Kaplansky [126] that any Hopf algebra is free over any Hopf subalgebra. In the finite-dimensional case a positive answer was given by Nichols and Zoeller in [169], and this is why the stated freeness theorem is known as the Nichols–Zoeller theorem. In the infinitedimensional case it has been known for a long time that the conjecture is false, examples of Hopf algebras which are not free over a Hopf subalgebra having been given by Oberst and Schneider [170] in 1974. Afterwards, in [184, 185], Radford proved that this conjecture of Kaplansky is also true in some special infinite-dimensional cases. For finite-dimensional quasi-Hopf algebras the freeness theorem has been proved by Schauenburg in [200], and our presentation is taken mostly from there. We also used the books [157, 73], and a simplified argument from [202].

8 Yetter–Drinfeld Module Categories

We introduce the categories of Yetter–Drinfeld modules by computing the left and right centers of a category of modules over a quasi-bialgebra H. We then show that all four categories of Yetter–Drinfeld modules are braided isomorphic. We also introduce the quasi-Hopf algebra structure of the quantum double of a finite-dimensional quasi-Hopf algebra.

8.1 The Left and Right Center Constructions The center construction associates to a monoidal category a braided monoidal one. Definition 8.1 Let C be a monoidal category. Denote by Wr (C ) the category whose objects are pairs (V, c−,V ), with V an object of C and c−,V = (cX,V : X ⊗ V → V ⊗ X)X∈C a natural transformation, satisfying c1,V = rV−1 lV and, for all X,Y ∈ C , X Y V

e

cX⊗Y,V =

X V

where

•

cX,V =

•

Y V

and

V X

V X Y

cY,V =

e

,

(8.1.1)

V Y

and where we assume, for simplicity, that C is strict monoidal. A morphism between (V, c−,V ) and (W, c−,W ) in Wr (C ) is a morphism f : V → X V

W in C obeying

X V

X W h • = fh , where cX,W = ♦ . It is clear that Wr (C ) is a

f ♦

W X

W X

W X

category, called in what follows the right weak center of C . Note that Id(V,c−,V ) = IdV . We will show that Wr (C ) is a pre-braided monoidal category via the following structure: Proposition 8.2

The right weak center Wr (C ) of a monoidal category C is

306

Yetter–Drinfeld Module Categories

monoidal with tensor product X V W •

(V, c−,V ) ⊗ (W, c−,W ) = (V ⊗W, c−,V ⊗W ), where cX,V ⊗W =

, ∀ X ∈ C.

♦

V W X

(8.1.2) The unit object of Wr (C ) is (1, c−,1 := (lX−1 rX : X ⊗ 1 → 1 ⊗ X)X∈C ), and the left and right unit constraints are the same as those of C . The monoidal category Wr (C ) is, moreover, pre-braided with pre-braiding c(V,c−,V ),(W,c−,W ) = cV,W : (V, c−,V ) ⊗ (W, c−,W ) → (W, c−,W ) ⊗ (V, c−,V ),

(8.1.3)

for all (V, c−,V ), (W, c−,W ) in Wr (C ). Proof We first prove that the tensor product of Wr (C ) is well defined. Actually, (V ⊗W, c−,V ⊗W ) is an object of the right weak center of C since, for all X,Y ∈ C , X Y V W

X Y V W

e

•

cX⊗Y,V ⊗W =

!

e

!

=

,

•

♦

♦

V W X Y

V W X Y

where X V

cX,V =

•

Y V

V X

X W

e , cX,W =

, cY,V =

V Y

Y W

and cY,W =

♦

W X

.

!

W Y

According to the definition we have −1 −1 (IdV ⊗ rW lW )aV,1,W (rV−1 lV ⊗ IdW )a−1 c1,V ⊗W = aV,W,1 1,V,W

= rV−1⊗W (IdV ⊗ lW )aV,1,W (rV−1 ⊗ IdW )lV ⊗W = rV−1⊗W lV ⊗W ,

as needed, where in the second equality we used Proposition 1.5 and in the last one (1.1.2). Thus (V ⊗W, c−,V ⊗W ) is indeed an object of Wr (C ), as claimed. f

We prove that the tensor product in C of two morphisms (V, c−,V ) → (W, c−,W ) f

and (V  , c−,V  ) → (W  , c−,W  ) in Wr (C ) is a morphism in Wr (C ). We compute X V W

X V W

 h fh • h f =

f •

♦

V W

 fh =

♦

X

V W

X V W •

,

♦

 h fh

f

X

V W

X

where the diagrammatic notation for cX,V  and cX,W  is the same as the one used for cX,V and cX,W , respectively.

8.1 The Left and Right Center Constructions

307

−1 Clearly (1, l− r− ) together with l and r define a unit object for Wr (C ), and so Wr (C ) is a monoidal category, as desired. It remains to show that c defined in the statement endows Wr (C ) with a prebraided structure. First, cV,W is a morphism in Wr (C ) if and only if

X V W

X V W

•

V W ♦

=

, where cV,W =

♦ •

W V X

, W V

W V X

and this is true since c−,W is natural and satisfies (8.1.1) (of course, with V replaced by W ), and since cX,V : X ⊗V → V ⊗ X is a morphism in C . Now, c obeys (1.5.2) because of its definition and (8.1.1). (1.5.1) in our situation becomes cU,V ⊗W = cU,V ⊗W = (IdV ⊗ cW ,U )(cU,V ⊗ IdW ), where V = (V, c−,V ), etc. and follows from the definition of the tensor product in Wr (C ). In the definition of the objects of the right weak center one can require c−,V to be a natural isomorphism. The new category obtained in this way is denoted by Zr (C ), and called the right center of C . It follows that Zr (C ) is a braided category. Let Π : Zr (C ) → C be the forgetful functor, that is, Π acts as identity on morphisms and Π((V, c−,V )) = V , for all (V, c−,V ) ∈ Zr (C ). We have that Π is a strict monoidal functor, and is bijective on objects. These observations provide a Universal Property for the center construction that we have just described. Proposition 8.3 Let C  be a monoidal category with right center Zr (C  ) and canonical monoidal functor Π : Zr (C  ) → C  . Then for any braided category C and any monoidal functor F : C → C  which is bijective on objects and surjective on morphisms, there exists a unique braided monoidal functor Zr (F) : C → Zr (C  ) such that Π ◦ Zr (F) = F. Proof As usual, assume that C , C  are strict monoidal categories, and that F is a strict monoidal functor. If V is an object of C we define Zr (F)(V ) := (F(V ), c−,F(V ) ), with cX  ,F(V ) = F(cF −1 (X  ),V ), for all X  ∈ C  , where F −1 (X  ) is the unique object of C that is mapped by F to X  . Note that, if (F, ϕ0 , ϕ2 ) is an arbitrary monoidal functor, we have cX  ,F(V ) : X  ⊗ F(V ) = FF −1 (X  ) ⊗ F(V ) F(cF −1 (X  ),V )

−→

F(V ⊗ F −1 (X  ))

ϕ2,F −1 (X  ),V

ϕ −1

−→

2,V,F −1 (X  )

−→

F(F −1 (X  ) ⊗V )

F(V ) ⊗ FF −1 (X  ) = F(V ) ⊗ X  .

Clearly c−,1 is defined by identity morphisms of C  when C  is strict monoidal, otherwise it is defined by the left and right unit constraints of C  . We next compute cX  ⊗Y  ,F(V ) = F(cF −1 (X  ⊗Y  ),V ) = F(cF −1 (X  )⊗F −1 (Y  ),V )

308

Yetter–Drinfeld Module Categories = F(cF −1 (X  ),V ⊗ IdF −1 (Y  ) )F(IdF −1 (X  ) ⊗ cF −1 (Y  ),V ) = (F(cF −1 (X  ),V ) ⊗ IdY  )(IdX  ⊗ F(cF −1 (Y  ),V )) = cX  ,F(V ) ⊗ cY  ,F(V ) ,

and so Zr (F) is well defined on objects. It is well defined on morphisms as well f - W in C one has Zr (F)( f ) := F( f ) : F(V ) → F(W ) and it since for all V satisfies the required property to be a morphism in Zr (C  ) since (F( f ) ⊗ IdX  )cX  ,F(V ) = (F( f ) ⊗ IdX  )F(cF −1 (X  ),V ) = F(( f ⊗ IdF −1 (X  ) )cF −1 (X  ),V ) = F(cF −1 (X  ),W (IdF −1 (X  ) ⊗ f )) = F(cF −1 (X  ),W )(IdX  ⊗ F( f )) = cX  ,F(W ) (IdX  ⊗ F( f )). We also have Π ◦ Zr (F)(V ) = Π ((F(V ), c−,F(V ) )) = F(V ) and Π ◦ Zr (F)( f ) = Π (F( f )) = F( f ). Thus Zr (F) is a well-defined functor that satisfies Π ◦ Zr (F) = F. To complete the proof of the existence we have to check that Zr (F) is a braided monoidal functor. In general, the monoidal structure of F induces a monoidal structure on Zr (F) since, for instance, ϕ2,V,W : F(V ) ⊗ F(W ) → F(V ⊗W ) can be viewed also as a morphism in Zr (C ). Namely,

ϕ2,V,W : (F(V ) ⊗ F(W ), c−,F(V )⊗F(W ) ) → (F(V ⊗W ), c−,F(V ⊗W ) ), or, equivalently,

ϕ2,V,W : Zr (F)(V ) ⊗ Zr (F)(W ) → Zr (F)(V ⊗W ). When F is strict monoidal this fact follows from F(cF −1 (X  ),V ⊗W ) = F((IdV ⊗ cF −1 (X  ),W )(cF −1 (X  ),V ⊗ IdW )) = (IdF(V ) ⊗ F(cF −1 (X  ),W ))(F(cF −1 (X  ),V ) ⊗ IdW ) = (IdF(V ) ⊗ cX  ,F(W ) )(cX  ,F(V ) ⊗ IdW ) = cX  ,F(V )⊗F(W ) . The required property for Zr (F) to be braided monoidal reduces to the definition of this functor: Zr (F)(cV,W ) = F(cV,W ), and so the first part of the proof is done. Suppose now that F  : C → Zr (C  ) is a braided monoidal functor satisfying Π ◦  F = F. If we denote F  (X) = (X  , c−,X  ), then we have: • on objects: Π ◦ F  (X) = F(X), which is equivalent to X  = F(X), and so F  (X) = (F(X), c−,F(X) ), for any object X of C ; • on morphisms: Π ◦ F  ( f ) = F( f ), thus F  ( f ) = F( f ), for any morphism f in C . Since F  is braided monoidal, we have ϕ2,Y,X cF(X),F(Y ) = F(cX,Y )ϕ2,X,Y . Thus −1 cX  ,F(Y ) = ϕ2,Y,F −1 (X  ) F(cF −1 (X  ),Y )ϕ2,F −1 (X  ),Y ,

for all X  ∈ C  and Y ∈ C , implying that F  = Zr (F) and finishing the proof.

8.1 The Left and Right Center Constructions

309

Corollary 8.4 If C is a braided category, there exists a unique braided monoidal functor Zr : C → Zr (C ) such that Π ◦ Zr = IdC . Proof

In Proposition 8.3 take C  = C and F = IdC . Then Zr = Zr (IdC ).

The above result says that, in general, the center of a braided category C is not isomorphic to C itself. All we can say is that C can be identified with a braided subcategory of its right center. This fact will be illustrated when we compute the center of the monoidal category of representations of a quasi-Hopf algebra; see the next sections. In a similar manner we can introduce the left (weak) center of a monoidal category. More precisely, let C be a monoidal category. By Wl (C ) we denote the category whose objects are pairs (V, dV,− ) consisting of an object V of C and a natural transformation dV,− = (dV,X : V ⊗ X → X ⊗ V )X∈C , such that, for all X,Y ∈ C , V X Y

dV,X⊗Y =

•

e

X Y V

V X

,

where

dV,X =

•

V Y

and

e

dV,Y =

X V

,

(8.1.4)

Y V

and where, once more, we assumed that C is strict monoidal. A morphism f : (V, dV,− ) → (W, dW,− ) in Wl (C ) is a morphism f : V → W in C such that dX,W ( f ⊗ IdX ) = (IdX ⊗ f )dV,X , for all X ∈ C . We call Wl (C ) the left weak center of C . It is a monoidal category with tensor product V W X

(V, dV,− ) ⊗ (W, dW,− ) = (V ⊗W, dV ⊗W,− ) , with dV ⊗W,X =

♦ •

,

(8.1.5)

X V W

where the notation is similar to the one used in the right-handed case. The unit object of Wl (C ) is (1, d1,− := (rX−1 lX )X∈C ), and the left and right unit constraints coincide with those of C . Wl (C ) is, moreover, pre-braided via the pre-braiding given by dV,W : (V, dV,− ) ⊗ (W, dW,− ) → (W, dW,− ) ⊗ (V, dV,− ),

(8.1.6)

for all (V, dV,− ), (W, dW,− ) objects of Wl (C ). If we require dV,− to be a natural isomorphism we obtain the notion of left center, denoted by Zl (C ). As in the right-handed case, Zl (C ) is a braided category. The connection with Zr (C ) is presented in the following result. Proposition 8.5 Let C be a monoidal category, and Zl (C ) and Zr (C ) the left and right centers associated to it. Then F : Zr (C ) → Zl (C )in defined by F((V, c−,V )) = (V, c−1 −,V ), for all (V, c−,V ) ∈ Zr (C ), provides a braided monoidal isomorphism. F acts as identity on morphisms.

310

Yetter–Drinfeld Module Categories X Y V

Proof

If cX⊗Y,V =

e

•

V X Y

then

c−1 X⊗Y,V

Y V

e then

e , where, if cY,V =

X Y V

V X Y −1 cY,V =

•

=

V Y

V Y

e , etc. This shows that (V, c−1 ) ∈ Zl (C ), and so F is well defined −,V

Y V

on objects. It can be easily checked that F maps a morphism in Zr (C ) to a morphism in Zl (C ), thus F is well defined on morphisms, too. By the definition of −1 (V, c−,V ) ⊗(W, c−,W ) = (V ⊗W, c−,V ⊗W ) it follows that (V, c−1 −,V ) ⊗(W, c−,W ) = (V ⊗ −1 W, c−,V ⊗W ), so F is a strict monoidal functor. Viewed as a functor from Zr (C ) to Zl (C )in it is, moreover, braided monoidal since the commutativity of the diagram F(V ) ⊗ F(W )

F(V ⊗W )

cVin,W =cV,W

F(cV ,W )=cV ,W

F(W ) ⊗ F(V )

F(W ⊗V )

follows from the definition of the braiding c of Zr (C ). Here we have denoted V := (V, c−,V ) and W := (W, c−,W ). Hence the proof is finished. The following result is completely obvious; we leave the details to the reader. Proposition 8.6

Let C be a monoidal category. Then Wl (C ) " Wr (C ) and Wr (C ) " Wl (C )

as pre-braided categories, and Zl (C ) " Zr (C ) and Zr (C ) " Zl (C ) as braided categories, where C is the reverse monoidal category associated to C .

8.2 Yetter–Drinfeld Modules over Quasi-bialgebras Throughout this section H is a quasi-bialgebra or a quasi-Hopf algebra (sometimes with bijective antipode) over a field k. We introduce several categories of Yetter– Drinfeld modules over H and show that they are isomorphic to the left or right weak centers of H M or MH . Definition 8.7 Let H be a quasi-bialgebra. (1) A left Yetter–Drinfeld module over H is a left H-module M together with a k-linear map (called the left H-coaction)

λM : M → H ⊗ M,

λM (m) = m(−1) ⊗ m(0)

8.2 Yetter–Drinfeld Modules over Quasi-bialgebras

311

such that the following conditions hold, for all h ∈ H and m ∈ M: X 1 m(−1) ⊗ (X 2 · m(0) )(−1) X 3 ⊗ (X 2 · m(0) )(0) = X 1 (Y 1 · m)(−1)1 Y 2 ⊗ X 2 (Y 1 · m)(−1)2 Y 3 ⊗ X 3 · (Y 1 · m)(0) , (8.2.1)

ε (m(−1) )m(0) = m,

(8.2.2)

h1 m(−1) ⊗ h2 · m(0) = (h1 · m)(−1) h2 ⊗ (h1 · m)(0) .

(8.2.3)

The category of left Yetter–Drinfeld modules and k-linear maps that preserve the H-action and H-coaction is denoted by H H YD. (2) A left–right Yetter–Drinfeld module over H is a left H-module M together with a k-linear map (called the right H-coaction)

ρM : M → M ⊗ H,

ρM (m) = m(0) ⊗ m(1)

such that the following conditions hold, for all h ∈ H and m ∈ M: (x2 · m(0) )(0) ⊗ (x2 · m(0) )(1) x1 ⊗ x3 m(1) = x1 · (y3 · m)(0) ⊗ x2 (y3 · m)(1)1 y1 ⊗ x3 (y3 · m)(1)2 y2 ,

(8.2.4)

ε (m(1) )m(0) = m,

(8.2.5)

h1 · m(0) ⊗ h2 m(1) = (h2 · m)(0) ⊗ (h2 · m)(1) h1 .

(8.2.6)

The category of left–right Yetter–Drinfeld modules and k-linear maps that preserve the H-action and H-coaction is denoted by H YD H . (3) A right–left Yetter–Drinfeld module over H is a right H-module M together with a k-linear map (called the left H-coaction)

λM : M → H ⊗ M,

λM (m) = m(−1) ⊗ m(0)

such that the following conditions hold, for all h ∈ H and m ∈ M: m(−1) x1 ⊗ x3 (m(0) · x2 )(−1) ⊗ (m(0) · x2 )(0) = y2 (m · y1 )(−1)1 x1 ⊗ y3 (m · y1 )(−1)2 x2 ⊗ (m · y1 )(0) · x3 ,

(8.2.7)

ε (m(−1) )m(0) = m,

(8.2.8)

m(−1) h1 ⊗ m(0) · h2 = h2 (m · h1 )(−1) ⊗ (m · h1 )(0) .

(8.2.9)

The category of right–left Yetter–Drinfeld modules and k-linear maps that preserve the H-action and H-coaction is denoted by H YD H . (4) A right Yetter–Drinfeld module over H is a right H-module M together with a k-linear map (called the right H-coaction)

ρM : M → M ⊗ H,

ρM (m) = m(0) ⊗ m(1)

such that the following conditions hold, for all h ∈ H and m ∈ M: (m(0) · X 2 )(0) ⊗ X 1 (m(0) · X 2 )(1) ⊗ m(1) X 3 = (m ·Y 3 )(0) · X 1 ⊗Y 1 (m ·Y 3 )(1)1 X 2 ⊗Y 2 (m ·Y 3 )(1)2 X 3 , (8.2.10)

312

Yetter–Drinfeld Module Categories

ε (m(1) )m(0) = m,

(8.2.11)

m(0) · h1 ⊗ m(1) h2 = (m · h2 )(0) ⊗ h1 (m · h2 )(1) .

(8.2.12)

The category of right Yetter–Drinfeld modules and k-linear maps that preserve the H-action and H-coaction is denoted by YD H H. The relation between Yetter–Drinfeld modules and the center construction is the following. Theorem 8.8 Let H be a quasi-bialgebra. Then we have the following isomorphisms of categories: Wl (H M ) ∼ =H H YD ;

Wr (H M ) ∼ = H YD H ;

Wr (MH ) ∼ = YD H H ;

Wl (MH ) ∼ = H YD H .

Proof Take (M, sM,− ) ∈ Wl (H M ), and consider λM = sM,H ◦ (IdM ⊗ ηH ) : M → H ⊗ M, where we denote by ηH : k → H, ηH (1) = 1H . We use the notation λM (m) = m(−1) ⊗ m(0) = sM,H (m ⊗ 1H ). The map λM determines s completely; indeed, if x ∈ X ∈ H M and we consider f : H → X, f (h) = h·x, a left H-linear map, the naturality of s entails that ( f ⊗ IdM ) ◦ sM,H = sM,X (IdM ⊗ f ). Hence, for all m ∈ M, we have sM,X (m ⊗ x) = sM,X ((IdM ⊗ f )(m ⊗ 1H )) = f (m(−1) ) ⊗ m(0) = m(−1) · x ⊗ m(0) .

(8.2.13)

In particular, m = sM,k (m ⊗ 1) = m(−1) · 1 ⊗ m(0) = ε (m(−1) )m(0) , so (8.2.2) holds. If we evaluate (8.1.4), with X = Y = H, at m ⊗ 1H ⊗ 1H , we find (8.2.1). Finally, h · sM,H (m ⊗ 1H ) = sM,H (h1 · m ⊗ h2 ) = (h1 · m)(−1) h2 ⊗ (h1 · m)(0) , h · sM,H (m ⊗ 1H ) = h · (m(−1) ⊗ m(0) ) = h1 m(−1) ⊗ h2 · m(0) , proving (8.2.3). So we have shown that (M, λM ) is a left Yetter–Drinfeld module. Conversely, if (M, λM ) is a left Yetter–Drinfeld module, then (M, sM,− ), with s given by (8.2.13), is an object of Wl (H M ); the details are left to the reader. The proof of the other three isomorphisms can be done in a similar way. Corollary 8.9

H YD, YD H , H YD H H H

and YD H H are pre-braided categories.

Proof The pre-braided structure of Wl (H M ) induces a pre-braided structure on H YD. This structure is such that the forgetful functor H YD → M is strict monoidal. H H H

By using (3.1.2) and (8.1.5), we find that the H-coaction on the tensor product M ⊗ N of two left Yetter–Drinfeld modules M and N is given by

λM⊗N (m ⊗ n) = X 1 (x1Y 1 · m)(−1) x2 (Y 2 · n)(−1)Y 3 ⊗ X 2 · (x1Y 1 · m)(0) ⊗ X 3 x3 · (Y 2 · n)(0) .

(8.2.14)

By using (8.1.6), we also find the pre-braiding: cM,N (m ⊗ n) = m(−1) · n ⊗ m(0) .

(8.2.15)

8.2 Yetter–Drinfeld Modules over Quasi-bialgebras

313

By applying the same procedure, we can make H YD H , H YD H and YD H H into pre-braided categories. On H YD H , we find the following structure. The right H-coaction on the tensor product M ⊗ N of M, N ∈ H YD H can be computed by using (8.1.2). Explicitly, it is given, for all m ∈ M, n ∈ N, by

ρM⊗N (m ⊗ n) = x1 X 1 · (y2 · m)(0) ⊗ x2 · (X 3 y3 · n)(0) ⊗ x3 (X 3 y3 · n)(1) X 2 (y2 · m)(1) y1 .

(8.2.16)

We have seen that the functor forgetting the coaction is strict monoidal, so h · (m ⊗ n) = h1 · m ⊗ h2 · n.

(8.2.17)

The pre-braiding c can be deduced from (8.1.3): cM,N : M ⊗ N → N ⊗ M is given by cM,N (m ⊗ n) = n(0) ⊗ n(1) · m,

(8.2.18)

for m ∈ M and n ∈ N. For completeness’ sake, let us also describe the pre-braided structure of YD H H and H YD . For M, N ∈ YD H , the coaction on M ⊗ N is given by the formula H H

ρ (m ⊗ n) = (m · X 2 )(0) · x1Y 1 ⊗ (n · X 3 x3 )(0) ·Y 2 ⊗ X 1 (m · X 2 )(1) x2 (n · X 3 x3 )(0)Y 3 . The pre-braiding d is the following: dM,N (m ⊗ n) = n(0) ⊗ m · n(1) .

(8.2.19)

Now take M, N ∈ H YD H . The coaction on M ⊗ N is the following:

λ (m ⊗ n) = x3 (n · x2 )(−1) X 2 (m · x1 X 1 )(−1) y1 ⊗ (m · x1 X 1 )(0) · y2 ⊗ (n · x2 )(0) · X 3 y3 . The pre-braiding d is given by dM,N (m ⊗ n) = n · m(−1) ⊗ m(0) .

(8.2.20)

The verification of all these facts is left to the reader. There exist connections between the four Yetter–Drinfeld categories defined above. Recall that if H is a quasi-bialgebra then by H op,cop we have denoted the k-vector space H endowed with the opposite multiplication and comultiplication of H. It is also a quasi-bialgebra with reassociator Φ321 , where Φ is the reassociator of H. Proposition 8.10

We have an isomorphism of monoidal categories F:

H op,cop M

→ MH .

F(M) = M as a k-vector space, with right H-action mh = h · m.

314

Yetter–Drinfeld Module Categories

Proof It is obvious that F is an isomorphism of categories. So we only need to show that it preserves the monoidal structure. Let us first describe the monoidal structure of H op,cop M . The left H op,cop -action on N ⊗ M is h · (n ⊗ m) = h2 · n ⊗ h1 · m. The associativity constraint aP,N,M : (P ⊗ N) ⊗ M → P ⊗ (N ⊗ M) is aP,N,M ((p ⊗ n) ⊗ m) = X 3 · p ⊗ (X 2 · n ⊗ X 1 · m). Now we describe the monoidal structure of H op,cop M . We have M⊗N = N ⊗ M. For m ∈ M, n ∈ N, we write m⊗n = n ⊗ m ∈ M⊗N = N ⊗ M. Then h · (m⊗n) = h2 · m⊗h1 · n.

(8.2.21)

The associativity constraint aM,N,P = a−1 P,N,M : (M⊗N)⊗P = P ⊗ (N ⊗ M) → M⊗(N⊗P) = (P ⊗ N) ⊗ M is given by aM,N,P ((m⊗n)⊗p) = x1 · m⊗(x2 · n⊗x3 · p).

(8.2.22)

It is then clear from (8.2.21–8.2.22) that F behaves well with respect to the monoidal structures of H op,cop M and MH . More precisely, F is strong monoidal with the structure given by

ϕ2 = (ϕ2,M,N : F(M) ⊗ F(N) m ⊗ n → n ⊗ m ∈ F(M⊗N))M,N∈

H op,cop M

and ϕ0 = Idk . So the proof is finished. An immediate consequence of Proposition 8.6, Theorem 8.8 and Proposition 8.10 is then the following. Proposition 8.11 Let H be a quasi-bialgebra. Then we have the following isomorphisms of pre-braided categories, induced by the functor F from Proposition 8.10: ∼ H op,cop YD H H = H op,cop YD and Proof

H

op,cop YD H ∼ . = H op,cop YD H

We have the following isomorphisms of pre-braided categories: ∼ ∼ ∼ H op,cop ∼ YD H H = Wr (MH ) = Wr (H op,cop M ) = Wl (H op,cop M ) = H op,cop YD.

Similarly, H

op,cop YD H ∼ , = Wl (MH ) ∼ = Wl (H op,cop M ) ∼ = Wr (H op,cop M ) ∼ = H op,cop YD H

as pre-braided categories. This finishes the proof. If H is a quasi-Hopf algebra with bijective antipode then the four weak centers in the statement of Theorem 8.8 are equal to the centers.

8.2 Yetter–Drinfeld Modules over Quasi-bialgebras

315

Proposition 8.12 If the antipode of H is bijective then Wr (H M ) = Zr (H M ), Wl (MH ) = Zl (MH ), Wl (H M ) = Zl (H M ) and Wr (MH ) = Zr (MH ). H Consequently, H YD H , H YD H , H H YD and YD H are braided categories. Proof If H is a quasi-Hopf algebra with bijective antipode then the pre-braiding c defined in (8.2.15) is a natural isomorphism. For any M, N ∈ H H YD, the inverse of cM,N is given by 2 2 1 −1 1 1 1 2 2 3 c−1 M,N (n ⊗ m) = q˜1 X · (p · m)(0) ⊗ S (q˜ X (p · m)(−1) p S(q˜2 X )) · n, (8.2.23)

where pR = p1 ⊗ p2 and qL = q˜1 ⊗ q˜2 are the elements defined by (3.2.19) and (3.2.20), respectively. Indeed, on the one hand we have cM,N ◦ c−1 M,N (n ⊗ m) 



2 2 = q˜1 X · (p1 · m)(0) (−1) q˜22 X 3 S−1 q˜1 X 1 (p1 · m)(−1) p2 · n

 ⊗ q˜21 X 2 · (p1 · m)(0) (0) (8.2.3)

=

q˜21 (X 2 · (p1 · m)(0) )(−1) X 3 S−1 (q˜1 X 1 (p1 · m)(−1) p2 ) · n

⊗ q˜22 · (X 2 · (p1 · m)(0) )(0) (8.2.1),(3.2.19) 2 2 = q˜1 X m(−1)2 S−1 (q˜1 X 1 m(−1)1 β ) · n ⊗ q˜22 X 3 · m(0) (3.2.24) = q˜21 p˜1 S−1 (q˜1 ) · n ⊗ q˜22 p˜2 · m = n ⊗ m, for all n ∈ N and m ∈ M. On the other hand, we have: c−1 M,N ◦ cM,N (m ⊗ n) =

q˜21 X 2 · (p1 · m(0) )(0) ⊗ S−1 (q˜1 X 1 (p1 · m(0) )(−1) p2 S(q˜22 X 3 ))m(−1) · n

(5.2.7)

z3 y22 · (p1 · m(0) )(0) ⊗ S−1 (α z2 y21 (p1 · m(0) )(−1) p2 S(y3 ))z1 y1 m(−1) · n

(8.2.3)

z3 · (y21 p1 · m(0) )(0) ⊗ S−1 (α z2 (y21 p1 · m(0) )(−1) y22 p2 S(y3 ))z1 y1 m(−1) · n

= =

(5.5.16) 3 = z · (Y 2 · (p11 · m)(0) )(0) (8.2.3) ⊗ S−1 (α z2 (Y 2 · (p11 · m)(0) )(−1)Y 3 p2 )z1Y 1 (p11 · m)(−1) p12 · n (8.2.1) = (Y 1 p11 · m)(0) ⊗ S−1 (α (Y 1 p11 · m)(−1)2 Y 3 p2 )(Y 1 p11 · m)(−1)1 Y 2 p12 · n (3.2.23) = q1 p11 · m ⊗ S−1 (p2 )q2 p12 · n = m ⊗ n,

for all m ∈ M and n ∈ N, as needed. H H op,cop Since YD H H and H op,cop YD identify as ordinary categories, it follows that YD H is a braided category, too. We have that the inverse of the (pre-)braiding d defined in (8.2.19) is given by −1 dM,N (n ⊗ m) = (m · q˜2 )(0) · X 2 p12 ⊗ n · S(S−1 (X 1 p11 )q˜1 (m · q˜2 )(1) X 3 p2 ), H for all n ∈ N ∈ YD H H and m ∈ M ∈ YD H . Similarly, for H a quasi-bialgebra we have that

H YD

H

(8.2.24)

can be identified as an

316

Yetter–Drinfeld Module Categories cop

H ordinary category with H H cop YD. Thus if H is a quasi-Hopf algebra then H YD is braided; the inverse of c defined in (8.2.18) is given by 1 1 2 3 2 1 2 1 2 c−1 M,N (n ⊗ m) = q1 x S(q x ( p˜ · n)(1) p˜ ) · m ⊗ q2 x · ( p˜ · n)(0)

= X11 x1 S(X 2 x3 ( p˜2 · n)(1) p˜1 )α X 3 · m ⊗ X21 x2 · ( p˜2 · n)(0) , op,cop

for all n ∈ N ∈ H YD H and m ∈ M ∈ H YD H . Finally, H YD H ≡ H op,cop YD H as ordinary categories, hence when H is a quasi-Hopf algebra H YD H is a braided category. The inverse of the (pre-)braiding d in (8.2.20) is given by 2 1 1 1 3 2 1 2 2 d−1 M,N (n ⊗ m) = m · S(q (n · q )(−1) x p˜ )x p˜2 ⊗ (n · q )(0) · x p˜1

= m · S(X 2 (n · X 1 )(−1) x1 p˜1 )α X 3 x3 p˜22 ⊗ (n · X 1 )(0) · x2 p˜21 , for all n ∈ N ∈ H YD H and m ∈ M ∈ H YD H . Remark 8.13 By using the definitions of qR and pL one can rewrite the formulas for c−1 and d−1 in such a way that the bijectivity of the antipode becomes unnecessary for defining them. Therefore, the categories of left–right and right–left Yetter– Drinfeld modules are braided over an arbitrary quasi-Hopf algebra. By combining Proposition 8.5 and Theorem 8.8, we find the following result. Theorem 8.14 Let H be a quasi-Hopf algebra with bijective antipode. Then we in have an isomorphism of braided categories F : H YD H → H H YD, defined as follows. H For M ∈ H YD , F(M) = M as a left H-module; the left H-coaction is

λM (m) = m(−1) ⊗ m(0) = q11 x1 S(q2 x3 ( p˜2 · m)(1) p˜1 ) ⊗ q12 x2 · ( p˜2 · m)(0) ,

(8.2.25)

for all m ∈ M, where qR = q1 ⊗ q2 and pL = p˜1 ⊗ p˜2 are the elements defined by (3.2.19) and (3.2.20), and ρM (m) = m(0) ⊗ m(1) is the right coaction of H on M. The functor F sends a morphism to itself. Proof

F is nothing else than the composition of the isomorphisms H YD

H in

→ Zr (H M )in → Zl (H M ) → H H YD.

For M ∈ H YD H , we compute that the corresponding left Yetter–Drinfeld module structure λM on M is the following

λM (m) = c−1 H,M (m ⊗ 1H ) = m(−1) ⊗ m(0) = q11 x1 S(q2 x3 ( p˜2 · m)(1) p˜1 ) ⊗ q12 x2 · ( p˜2 · m)(0) , in

H as needed. Note that the inverse of F is the functor G : H defined as H YD → H YD YD then G(M) = M as a left H-module, with right H-coaction follows. If M ∈ H H

ρM (m) = m(0) ⊗ m(1) = q˜21 X 2 · (p1 · m)(0) ⊗ q˜22 X 3 S−1 (q˜1 X 1 (p1 · m)(−1) p2 ),

(8.2.26)

for all m ∈ M, where qL = q˜1 ⊗ q˜2 and pR = p1 ⊗ p2 are the elements defined by (3.2.20) and (3.2.19), and λM (m) = m(−1) ⊗ m(0) is the left coaction of H on M.

8.2 Yetter–Drinfeld Modules over Quasi-bialgebras

317

Similarly, we have the following result. Theorem 8.15 Let H be a quasi-Hopf algebra with bijective antipode. Then the in H categories YD H H and YD H are isomorphic as braided categories. Let H be a quasi-bialgebra and F = F1 ⊗ F2 ∈ H ⊗ H a twist with inverse F−1 = Then, by Proposition 3.5 we have an isomorphism of monoidal categories Π : H M → HF M , where Π(M) = M, with the same left H-action. If H is a quasiHopf algebra with bijective antipode, then we can consider the Drinfeld twist f defined in (3.2.15). The antipode S : H op,cop → H f is a quasi-Hopf algebra isomorphism (see Proposition 3.23) and therefore the monoidal categories H op,cop M and H f M are isomorphic. We have seen in Proposition 8.10 that H op,cop M is isomorphic to MH as a monoidal category. Thus we conclude that the monoidal categories H M and MH are isomorphic. By using Proposition 8.6 and Theorem 8.8, we find braided monoidal isomorphisms G1 ⊗ G2 .

∼ = Zl ( H M ) ∼ = Zl (MH ) ∼ = Zr (MH ) ∼ = YD H H, H ∼ ∼ ∼ ∼ H YD = Zr (H M ) = Zr (MH ) = Zl (MH ) = YD H . H H YD H

We summarize our results as follows: Theorem 8.16 Let H be a quasi-Hopf algebra with bijective antipode. Then we have the following isomorphisms of braided categories: H H YD

in in ∼ = H YD H ∼ = H YD H ∼ = YD H H.

H H ∼H ∼ The braided monoidal isomorphisms H H YD = YD H and H YD = YD H can be H described explicitly. Let us compute the functor H H YD → YD H . We have a monoidal isomorphism Π : H M → H f M , where Π(M) = M with the same left H-action. We will denote the tensor product on H f M by ⊗ f . For M, N ∈ f H M , the isomorphism ψ : Π(M ⊗ N) → Π(M) ⊗ Π(N) is given by the formula 1 f 2 ψ (m ⊗ n) = f · m ⊗ f · n. This isomorphism induces an isomorphism between the Hf two left centers, hence between the categories H H YD and H f YD. Take (M, sM,− ) ∈ f ) the corresponding object in Zl (H f M ), and let λ and λ f be Zl (H M ) and (M, sM,− f ψ , and so we compute the associated coactions. Then we have ψ sM,H = sM,H f λ f (m) = sM,H (m ⊗ 1H ) = ψ ((sM,H (g1 · m ⊗ g2 )) = f 1 (g1 · m)(−1) g2 ⊗ f 2 (g1 · m)(0) .

The quasi-Hopf algebra isomorphism S−1 : H f → H op,cop induces an isomorphism of monoidal categories H f M → H op,cop M ∼ = MH . The image of M is M as a k-vector f space, with right H-action given by m · h = S(h) · m. Take (M, sM,− ) ∈ Zl (H f M ) and

318

Yetter–Drinfeld Module Categories

the corresponding object (M,t−,M ) ∈ Zr (MH ). Then the diagram f

M⊗ H f

sM,H

IdM ⊗S−1

H⊗M = M ⊗ H

H ⊗f M S−1 ⊗IdM

tH,M

M⊗H = H ⊗ M

is commutative, and so we compute the right H-coaction ρM on M as follows:

ρM (m) = tH,M (1H ⊗m) = f 2 · (g1 · m)(0) ⊗ S−1 ( f 1 (g1 · m)(−1) )g2 ). We conclude that the braided monoidal isomorphism K : as follows: K(M) = M, with

H YD H

→ YD H H is defined

m · h = S(h) · m,

ρM (m) = f 2 · (g1 · m)(0) ⊗ S−1 ( f 1 (g1 · m)(−1) g2 ). The inverse functor K −1 can be computed in a similar way: K −1 (M) = M with h · m = m · S−1 (h),

λM (m) = g1 S((m · S−1 ( f 1 ))(1) ) f 2 ⊗ (m · S−1 ( f 1 ))(0) · S−1 (g2 ). For completeness’ sake, we also give the formulas for the braided monoidal isomorphism G : H YD H → H YD H . We have G(M) = M with h · m = m · S−1 (h),

ρM (m) = g1 · ( f 2 · m)(0) ⊗ g2 S(( f 2 · m)(−1) ) f 1 . Conversely, G−1 (M) = M with m · h = S(h) · m,

λM (m) = S−1 ( f 2 (g2 · m)(1) g1 ) ⊗ f 1 · (g2 · m)(0) .

fd 8.3 The Rigid Braided Category H H YD

In this section we investigate when the center of a monoidal category is left or right rigid. We then apply the results to the category of left Yetter–Drinfeld modules over a quasi-Hopf algebra with bijective antipode. Recall that the “mate” notion was defined in the last part of Section 1.8. Theorem 8.17 Let C be a monoidal category and Zl (C ) its left center. An object (V, cV,− ) of the left center admits a left dual if and only V has a left dual in C and −1 is an isomorphism, for any object X ∈ C . If this is the case then the the mate of cV,X −1  left dual of (V, cV,− ) is (V ∗ , sV ∗ ,− := (cV,− ) ). Similarly, an object (V, cV,− ) of Zl (C ) admits a right dual if and only if V has a right dual in C and  cV,X is an isomorphism, for any object X ∈ C . If this is the case then (∗V,t∗V,− := ( cV,− )−1 ) is a right dual for (V, cV,− ).

fd 8.3 The Rigid Braided Category H H YD

319

Proof We prove only the first assertion; the second follows in a similar manner. Assume first that (V, cV,− ) has a left dual (V ∗ , sV ∗ ,− ) in Zl (C ), and let ε , η be the associated evaluation and coevaluation morphisms in Zl (C ). Since the forgetful functor Π : Zl (C ) → C , defined by Π(W, cW,− ) = W and Π( f ) = f , is a strict monoidal functor it follows from Proposition 1.67 that V ∗ is a left dual object of V in C via the same morphisms ε , η , viewed now as morphisms in C . Furthermore, V∗ V

the fact that ε = is a morphism in Zl (C ) means 1 V∗ V X V∗

•

V X

=



X

V∗ X

V X

, ∀ X ∈ C , where cV,X = • and sV ∗ ,X =  .

X V X V∗

X

From here we deduce that V∗ X V 

V∗ X V •

V∗ X

= , so sV ∗ ,X =



X



V∗ X

X

=



•



X V∗

−1  = (cV,X ) ; see (1.6.6).

X V∗

−1  We conclude that (cV,X ) is an isomorphism, for all X in C , so the direct implication is proved. −1  ) is an isomorphism for any X ∈ C and V admits a left dual Conversely, if (cV,X −1  ∗ V we show that (V ∗ , sV ∗ ,− := (cV,− ) ) is a left dual of (V, cV,− ) in Zl (C ). To this −1  ∗ end we first need to show that (V , sV ∗ ,− := (cV,− ) ) is an object in Zl (C ). Indeed, since (V, cV,− ) belongs to the left center of C we have that

V∗ X Y X Y V

V X Y

cV,X⊗Y =

•

−1 e , so cV,X⊗Y =

X Y V

e

•

and sV ∗ ,X⊗Y =

V X Y

•

 e ,

X Y V∗

for any objects X, Y of C . The latest equality is equivalent to sV ∗ ,X⊗Y = (IdV ∗ ⊗ sV ∗ ,Y )(sV ∗ ,X ⊗ IdY ); see (1.6.6). It is clear that sV ∗ ,1 identifies with IdV ∗ , therefore (V ∗ , sV ∗ ,− ) is an object in the left center of C , as claimed. Consider now evV and coevV the evaluation and coevaluation morphisms attached to the left dual V ∗ of V . We prove that they are morphisms in Zl (C ) and this would finish the proof, since the equations required for evV and coevV for an adjunction are

320

Yetter–Drinfeld Module Categories

the same in Zl (C ) as in C . So we compute V∗ V X V∗ V X

•



X

=



V∗ V X



•

(1.6.6)

•



=

V∗ V X

•

=

•



. X

X

X

This shows that evV is a morphism in Zl (C ). In a similar manner it can be proved that coevV is a morphism in Zl (C ), so the proof is finished. We next supply, without proof, the right-handed version of Theorem 8.17. Theorem 8.18 Let C be a monoidal category and Zr (C ) its right center. An object (V, c−,V ) of the right center admits a left dual if and only V has a left dual in C and the mate of cX,V is an isomorphism, for any object X ∈ C . If this is the case then the left dual of (V, c−,V ) is (V ∗ , s−,V ∗ := (c−,V )−1 ). Likewise, an object (V, c−,V ) of Zr (C ) admits a right dual if and only if V has a right dual in C and  (c−1 X,V ) is an isomorphism, for any object X ∈ C . If this is the ∗  ∗ case then ( V,t−, V := (c−1 −,V )) is a right dual for (V, c−,V ). Remark 8.19 Observe that, by the first part of Theorem 8.17, when C is a left rigid −1  monoidal category then any object (V, cV,− ) of Zl (C ) has (V ∗ , sV ∗ ,− = (cV,− ) ) as a left dual object in Wl (C ). Similarly, by the second part of Theorem 8.18 it follows that when C is a right rigid monoidal category then any object (V, c−,V ) of Zr (C ) admits (∗V,t−,∗V :=  (c−1 −,V )) as a right dual object in Wr (C ). We next see that the weak center and the center coincide if C is a rigid monoidal category. Theorem 8.20 Let C be a monoidal category. (i) If C is left rigid then Wr (C ) = Zr (C ) and, similarly, if C is right rigid then Wl (C ) = Zl (C ). (ii) If C is a rigid monoidal category then Wl (C ) = Zl (C ) and Wr (C ) = Zr (C ) are rigid monoidal categories. Proof (i) We prove only the first assertion, the second one can be proved in a similar manner. Suppose that C is a left rigid monoidal category and let (V, c−,V ) be an object of Wr (C ). We state that, for any object X of C , cX,V is an isomorphism in C with c−1 X,V = (cX ∗ ,V ) (see the proof of Proposition 1.81 for the concrete definition of (cX ∗ ,V ) ).

fd 8.3 The Rigid Braided Category H H YD

321

Indeed, since c−,V is natural and coevX : 1 → X ⊗ X ∗ is a morphism in C we get V X



V X



cX ∗ ,V

(8.1.1)

cX,V ◦ (cX ∗ ,V ) =

=

cX⊗X ∗ ,V

cX,V

V

=



V X

X



(1.6.6)



=

IdV ⊗X ,

V X

V X

where in the last but one equality, apart from the naturality of c−,V , we also used the fact that c1,V = IdV . Similarly, by the naturality of c−,V and the fact that evX : X ∗ ⊗ X → 1 is a morphism in C , we have that X V X V

c−1 X,V

◦ cX,V =





cX,V

(8.1.1)

=

cX ∗ ⊗X,V



cX ∗ ,V



X V

X V

 =

(1.6.6)

X

=

IdX⊗V ,

V

X V

and this finishes the proof of (i). (ii) Let C be a rigid monoidal category. By part (i) we have Wr (C ) = Zr (C ) and Wl (C ) = Zl (C ). Then by Remark 8.19 it follows that Wl (C ) = Zl (C ) is left rigid, while Wr (C ) = Zr (C ) is right rigid. Since Wl (C ) = Zl (C ) and Wr (C ) = Zr (C ) are braided categories, by Proposition 1.74 we get that Wl (C ) = Zl (C ) and Wr (C ) = Zr (C ) are braided rigid categories, as stated. From now on, throughout this section H is a quasi-Hopf algebra with bijective antipode, so H M fd is rigid monoidal and H H YD is a braided category. We will see fd that the category of finite-dimensional left Yetter–Drinfeld modules H H YD is rigid and supply the explicit structures for the left and right dual objects. We first need a lemma. Lemma 8.21 Let H be a quasi-Hopf algebra and pR = p1 ⊗ p2 , qL = q˜1 ⊗ q˜2 = Q˜ 1 ⊗ Q˜ 2 and f = f 1 ⊗ f 2 the elements defined by (3.2.19), (3.2.20) and (3.2.15), respectively. Then the following relation holds: S(p1 )q˜1 p21 S(Q˜ 2 )1 ⊗ Q˜ 1 q˜2 p22 S(Q˜ 2 )2 = f .

(8.3.1)

Proof In order to prove (8.3.1), we denote by δ = δ 1 ⊗ δ 2 the element defined in (3.2.6), and then compute that S(p1 )q˜1 p21 S(Q˜ 2 )1 ⊗ Q˜ 1 q˜2 p22 S(Q˜ 2 )2 (3.2.19)

=

(3.2.13),(3.2.14)

=

(3.2.6),(3.1.9),(3.2.1)

=

S(x1 )q˜1 x12 β1 S(Q˜ 2 x3 )1 ⊗ Q˜ 1 q˜2 x22 β2 S(Q˜ 2 x3 )2 S(x1 )q˜1 x12 δ 1 S(Q˜ 22 x23 ) f 1 ⊗ Q˜ 1 q˜2 x22 δ 2 S(Q˜ 21 x13 ) f 2 S(z1 x1 )α z2 x12 y1 β S(Q˜ 22 x23 y32 X 3 ) f 1

322

Yetter–Drinfeld Module Categories (3.1.9),(3.2.1)

=

(3.2.2),(3.2.20)

=

⊗ Q˜ 1 z3 x22 y2 X 1 β S(Q˜ 21 x13 y31 X 2 ) f 2 S(x1 )α x2 β S(Q˜ 22 X 3 x3 ) f 1 ⊗ Q˜ 1 X 1 β S(Q˜ 21 X 2 ) f 2 S(Q˜ 22 p˜2 ) f 1 ⊗ Q˜ 1 S(Q˜ 21 p˜1 ) f 1

(3.2.24)

f1 ⊗ f2 = f,

=

as needed, and this finishes the proof. fd Theorem 8.22 Let H be a quasi-Hopf algebra with bijective antipode. Then H H YD is a rigid braided category. For a finite-dimensional left Yetter–Drinfeld module M with basis (i m)i=1,n and corresponding dual basis (i m)i=1,n , the left and right duals M ∗ and ∗ M are equal to Hom(M, k) as a vector space, with the following H-action and H-coaction: For M ∗ :

(h · m∗ )(m) = m∗ (S(h) · m);

(8.3.2)

λM∗ (m∗ ) = m∗(−1) ⊗ m∗(0) n

= ∑ m∗ , f 2 · (g1 · i m)(0) S−1 ( f 1 (g1 · i m)(−1) g2 ) ⊗ i m.

(8.3.3)

i=1

For ∗ M: (h · ∗ m)(m) := ∗ m(S−1 (h) · m);

(8.3.4)

λ∗ M (∗ m) = ∗ m(−1) ⊗ ∗ m(0) n

= ∑ ∗ m, S−1 ( f 1 ) · (S−1 (g2 ) · i m)(0) g1 S((S−1 (g2 ) · i m)(−1) ) f 2 ⊗ i m, (8.3.5) i=1

for all h ∈ H, m∗ ∈ M ∗ , ∗ m ∈ ∗ M and m ∈ M. Here f = f 1 ⊗ f 2 is the twist defined by (3.2.15), with inverse f −1 = g1 ⊗ g2 . Proof The left H-action on M ∗ viewed as an object of H H YD is the same as the left H-action on M ∗ viewed as an object of H M . We compute the left H-coaction, by using Theorem 8.17. By (8.2.23) in Zl (H M ) ∼ =H H YD we have −1 sV,X (v ⊗ x) = q˜21 X 2 · (p1 · x)(0) ⊗ S−1 (q˜1 X 1 (p1 · x)(−1) p2 S(q˜22 X 3 )) · v

for all V, X ∈ H M , x ∈ X and v ∈ V . Now, if we denote by P1 ⊗ P2 another copy of pR and by Q˜ 1 ⊗ Q˜ 2 another copy of qL , we can compute:

λ (m∗ ) = sM∗ ,H (m∗ ⊗ 1H )

 x1 Z 1 · m∗ , α x2Y 1 q˜21 X 2 · p1 y2 Z13 β · i m (0)   

 x13Y 2 q˜22 X 3 S−1 q˜1 X 1 p1 y2 Z13 β · i m (−1) p2 y1 Z 2 ⊗ x23Y 3 y3 Z23 · i m

 (3.2.1) = m∗ , Q˜ 1Y 1 q˜21 X 2 · p1 P2 S(Q˜ 22Y 3 ) · i m (0)  (3.2.19)  

 Q˜ 21Y 2 q˜22 X 3 S−1 q˜1 X 1 p1 P2 S(Q˜ 22Y 3 ) · i m (−1) p2 P1 ⊗ i m =

fd 8.3 The Rigid Braided Category H 323 H YD

 (5.2.7) = m∗ , Q˜ 1Y 1 q˜2 · y21 p1 P2 S(Q˜ 22Y 3 ) · i m (0)  (8.2.3) 

  Q˜ 21Y 2 y3 S−1 q˜1 y21 p1 P2 S(Q˜ 22Y 3 ) · i m (−1) y22 p2 y1 P1 ⊗ i m

 (3.2.25) 1 = m∗ , Q˜ 1Y 1 q˜2 x(2,2) p22 · g1 S(Q˜ 22Y 3 x3 ) · i m (0)  (8.2.3)  

 1 Q˜ 21Y 2 S−1 q˜1 x(2,1) p21 · g1 S(Q˜ 22Y 3 x3 ) · i m (−1) g2 S(x2 ) x11 p1 ⊗ i m

 (3.2.22) = m∗ , Q˜ 1 q˜2 p22 · S(Q˜ 2 )1 g1 · i m (0)  (3.2.13)  

 S−1 q˜1 p21 · S(Q˜ 2 )1 g1 · i m (−1) S(Q˜ 2 )2 g2 p1 ⊗ i m  



 (8.2.3) = m∗ , f 2 · g1 · i m (0) S−1 f 1 · g1 · i m (−1) g2 ⊗ i m, (8.3.1)

as claimed. The structure on ∗ M can be computed in a similar way, we leave the details to the reader. Since all four categories of Yetter–Drinfeld modules over a quasi-Hopf algebra with bijective antipode are somehow braided isomorphic it follows that the rigidity fd property of H H YD transfers to the other three categories. For instance, for later use we record without further details the following: Proposition 8.23 If H is a quasi-Hopf algebra with bijective antipode then the fd braided category H YD H is rigid. For a finite-dimensional left–right Yetter–Drinfeld module M with basis (i m)i=1,n and corresponding dual basis (i m)i=1,n , the left and right duals M ∗ and ∗ M are equal to Hom(M, k) as a vector space, with the following H-action and H-coaction: For M ∗ : (h · m∗ )(m) = m∗ (S(h) · m);

(8.3.6)

ρM∗ (m∗ ) = m∗(0) ⊗ m∗(1) n

= ∑ m∗ , f 1 · (g2 · i m)(0) i m ⊗ S−1 ( f 2 (g2 · i m)(1) g1 ).

(8.3.7)

i=1

For ∗ M: (h · ∗ m)(m) := ∗ m(S−1 (h) · m);

(8.3.8)

ρ∗ M (∗ m) = ∗ m(0) ⊗ ∗ m(1) n

= ∑ ∗ m, S−1 ( f 2 ) · (S−1 (g1 ) · i m)(0) i m ⊗ g2 S((S−1 (g1 ) · i m)(1) ) f 1 ,

(8.3.9)

i=1

for all h ∈ H, m∗ ∈ M ∗ , ∗ m ∈ ∗ M and m ∈ M. Here, as everywhere else, f = f 1 ⊗ f 2 is the twist defined by (3.2.15), with inverse f −1 = g1 ⊗ g2 . fd

Remark 8.24 The left rigid monoidal structure of H YD H presented in Proposition fd 8.23 is designed in such a way that the forgetful functor F : H YD H → k M fd is a left rigid quasi-monoidal functor, in the sense of Definition 3.36. This follows

324

Yetter–Drinfeld Module Categories fd

mostly from the fact that the left dual of an object M in H YD H is built on the left dual object of M regarded now in k M fd , of course with a different evaluation and coevaluation morphisms, and the fact that the left transpose of a morphism f fd in H YD H coincides with the usual left transpose morphism of f in k M fd . The latter follows by Proposition 3.34 since the evaluation and coevaluation morphisms fd of a left dual of an object M in H YD H coincide with those of the left dual of M fd regarded in H M fd , and since H YD H has the associativity constraint defined in the same manner as that of H M fd . fd

We present a sufficient condition for H YD H to be a sovereign category. fd

Theorem 8.25 If H is a sovereign quasi-Hopf algebra then H YD H is a sovereign category. fd

Proof A left dual of V in C := H YD H is nothing but the left dual of V in H M fd , the evaluation and coevaluations morphisms being the same in C and H M fd ; the difference is made only by the H-coaction that turns V ∗ into a left–right Yetter– Drinfeld module, and for this we need V ∈ C . As the H-coactions do not contribute to the definition of the isomorphisms λ in (1.7.1), it follows that they are defined by the same formula in both C and H M fd . For the last assertion, note also that the fd monoidal structure of H YD H was defined in such a way that the forgetful functor to H M fd is a strict monoidal functor. Assume that H is sovereign. By Theorem 3.40, we have a natural monoidal isomorphism i : IdH M fd → (−)∗∗ . In other words, for any V ∈ H M fd we have an isomorphism iV : V → V ∗∗ in H M fd , natural in V , such that the conditions in (3.6.1) are satisfied. From the above comments, to prove that C is sovereign it suffices to show that, for any object W of C , a subcategory of H M fd , the map iW : W w → g−1 · w ∈ W defined in the proof of Proposition 3.41 is an isomorphism in C rather than in H M fd . Here, to avoid any confusion, we denoted by W the k-vector space W endowed with the structure of a left-right Yetter–Drinfeld module obtained from the natural identification W ∼ = W ∗∗ . Let us compute this structure explicitly. It is clear that the left H-module structure on W given by the identification W ∼ = ∗∗ W is h→w = S2 (h)·w, for all h ∈ H, w ∈ W . We next compute the right H-coaction on W , and to this end we need dual bases {wi , wi }i in W and W ∗ , and {wi , w∗i }i in W ∗ and W ∗∗ , respectively. We have (summations implicitly understood): w → wi (w)w∗i → wi (w)w∗i , f 1 · (g2 · w j )(0) w∗ j ⊗ S−1 ( f 2 (g2 · w j )(1) g1 ) = wi (w)w∗i , f 1 · ws g2 · w j , F 1 · (G2 · ws )(0)  w∗ j ⊗ S−1 ( f 2 S−1 (F 2 (G2 · ws )(1) G1 )g1 ) →  f 1 · ws , wg2 · v j , F 1 · (G2 · ws )(0) w j ⊗ S−1 ( f 2 S−1 (F 2 (G2 · ws )(1) G1 )g1 ) = S(g2 )F 1 · (G2 S( f 1 ) · w)(0) ⊗ S−1 ( f 2 S−1 (F 2 (G2 S( f 1 ) · w)(1) G1 )g1 ).

8.4 Yetter–Drinfeld Modules as Modules over an Algebra

325

Here we used the right H-coactions on W ∗ and W ∗∗ = (W ∗ )∗ defined by (8.3.7) and the canonical isomorphism W ∼ = W ∗∗ and its inverse, respectively. We can see now that iW is right H-colinear:

λW iW (w) = S(g2 )F 1 · (G2 S( f 1 )g−1 · w)(0) ⊗ S−1 ( f 2 S−1 (F 2 (G2 S( f 1 )g−1 · w)(1) G1 )g1 ) (3.6.4)

−2 1 2 −1 −1 = S(g2 )F 1 · (g−1 2 · w)(0) ⊗ S (S(g )F (g2 · w)(1) g1 g)

(8.2.6)

−2 1 2 −1 = S(g2 )F 1 g−1 1 · w(0) ⊗ S (S(g )F g2 w(1) g)

(3.6.4) −1

· w(0) ⊗ S−2 (g−1 w(1) g)

(3.6.3) −1

· w(0) ⊗ w(1) = (iW ⊗ IdH )λW (w),

= g = g

for all w ∈ W . This finishes the proof.

8.4 Yetter–Drinfeld Modules as Modules over an Algebra The goal of this section is to prove that for a finite-dimensional quasi-Hopf algebra H the category of Yetter–Drinfeld modules over H is isomorphic to a certain category of representations. As we shall see, this can be done in a more general framework. Our next definition extends the definition of Yetter–Drinfeld modules from the previous sections. Definition 8.26 Let H be a quasi-bialgebra, C an H-bimodule coalgebra and A an H-bicomodule algebra. A left–right Yetter–Drinfeld module is a k-vector space M with the following additional structure: • M is a left A-module; we write · for the left A-action; • we have a k-linear map ρM : M → M ⊗ C, ρM (m) = m(0) ⊗ m(1) , called the right C-coaction on M, such that for all m ∈ M, ε (m(1) )m(0) = m and (θ 2 · m(0) )(0) ⊗ (θ 2 · m(0) )(1) · θ 1 ⊗ θ 3 · m(1) = x˜1ρ · (x˜3λ · m)(0) ⊗ x˜2ρ · (x˜3λ · m)(1)1 · x˜1λ ⊗ x˜3ρ · (x˜3λ · m)(1)2 · x˜2λ ; (8.4.1) • the following compatibility relation holds: u0 · m(0) ⊗ u1 · m(1) = (u[0] · m)(0) ⊗ (u[0] · m)(1) · u[−1] ,

(8.4.2)

for all u ∈ A, m ∈ M. A YD(H)C will be the category of left–right Yetter–Drinfeld modules and maps preserving the actions by A and the coactions by C. Let H be a quasi-bialgebra, A an H-bicomodule algebra and C an H-bimodule coalgebra. Let us call the 3-tuple (H, A,C) a Yetter–Drinfeld datum. We note that, for an arbitrary H-bimodule coalgebra C, the linear dual space of C, C∗ , is an Hbimodule algebra. The multiplication of C∗ is the convolution, that is, (c∗ d ∗ )(c) = c∗ (c1 )d ∗ (c2 ), the unit is ε and the left and right H-module structures are given by (h  c∗  h )(c) = c∗ (h · c · h), for all h, h ∈ H, c∗ , d ∗ ∈ C∗ , c ∈ C.

326

Yetter–Drinfeld Module Categories

In the rest of this section we establish that if H is a quasi-Hopf algebra and C is finite dimensional then the category A YD(H)C is isomorphic to the category of left C∗  A-modules, C∗  A M , where C∗  A is the diagonal crossed product algebra between the H-bimodule algebra C∗ and the H-bicomodule algebra A as in Section 5.6. First we need some lemmas. Lemma 8.27 Let H be a quasi-Hopf algebra with bijective antipode and (H, A,C) a Yetter–Drinfeld datum. We have a functor F : A YD(H)C → C∗  A M , given by F(M) = M as k-vector space, with the C∗  A-module structure defined by (c∗  u)m := c∗ , q˜2ρ · (u · m)(1) q˜1ρ · (u · m)(0) ,

(8.4.3)

for all c∗ ∈ C∗ , u ∈ A and m ∈ M, where q˜ρ = q˜1ρ ⊗ q˜2ρ is the element defined in (4.3.9). F maps a morphism to itself. Proof Let Q˜ 1ρ ⊗ Q˜ 2ρ be another copy of q˜ρ . For all c∗ , d ∗ ∈ C∗ , u, u ∈ A and m ∈ M we compute: [(c∗  u)(d ∗  u )]m (5.6.15)

=

[(Ω1  c∗  Ω5 )(Ω2 u0[−1]  d ∗  S−1 (u1 )Ω4 )  Ω3 u0[0] u ]m

=

d ∗ , S−1 (u1 )Ω4 (q˜2ρ )2 · (Ω3 u0[0] u · m)(1)2 · Ω2 u0[−1]  c∗ , Ω5 (q˜2ρ )1 · (Ω3 u0[0] u · m)(1)1 · Ω1 q˜1ρ · (Ω3 u0[0] u · m)(0)

(5.6.9)

=

2 1 1 2 d ∗ , S−1 ( f 1 X˜ ρ θ 3 u1 )(q˜2ρ )2 · ((X˜ ρ )[0] x˜3λ θ[0] u0[0] u · m)(1)2 · (X˜ ρ )[−1]2 3 1 2 2 x˜2λ θ[−1] u0[−1] c∗ , S−1 ( f 2 X˜ ρ )(q˜ρ )1 · ((X˜ ρ )[0] x˜3λ θ[0] u0[0] u · m)(1)1 1 2 ·x˜1λ θ 1 q˜1ρ · ((X˜ ρ )[0] x˜3λ θ[0] u0[0] u · m)(0)

(8.4.2)

=

(4.3.15)

2 2 d ∗ , S−1 (θ 3 u1 )Q˜ 2ρ x˜3ρ · (x˜3λ θ[0] u0[0] u · m)(1)2 · x˜2λ θ[−1] u0[−1]  2 c∗ , q˜2ρ (Q˜ 1ρ )1 x˜2ρ · (x˜3λ θ[0] u0[0] u · m)(1)1 · x˜1λ θ 1  2 q˜1ρ (Q˜ 1ρ )0 x˜1ρ · (x˜3λ θ[0] u0[0] u · m)(0)

(8.4.1)

=

3 2 2 d ∗ , S−1 (θ 3 u1 )Q˜ 2ρ θ · (θ[0] u0[0] u · m)(1) · θ[−1] u0[−1]  2 1 2 u0[0] u · m)(0) ](1) · θ θ 1  c∗ , q˜2ρ (Q˜ 1ρ )1 · [θ · (θ[0] 2 2 u0[0] u · m)(0) ](0) q˜1ρ (Q˜ 1ρ )0 · [θ · (θ[0]

(8.4.2)

=

(4.4.3)

3

2 d ∗ , S−1 (α X˜ ρ θ 3 u1 )X˜ ρ θ θ1 u0,1 · (u · m)(1)  3

2

2 2 1 1 1 c∗ , q˜2ρ · [(X˜ ρ )[0] θ θ0 u0,0 · (u · m)(0) ](1) · (X˜ ρ )[−1] θ θ 1  2 2 1 q˜1ρ · [(X˜ ρ )[0] θ θ0 u0,0 · (u · m)(0) ](0) (4.4.3)

=

(4.3.1) (3.2.1)

=

(4.3.9)

3 2 d ∗ , S−1 (αθ23 u12 X˜ ρ )θ13 u11 X˜ ρ · (u · m)(1) 

c∗ , q˜2ρ · [θ 2 u0 X˜ ρ · (u · m)(0) ](1) · θ 1 q˜1ρ · [θ 2 u0 X˜ ρ · (u · m)(0) ](0) c∗ , q˜2ρ · [uQ˜ 1ρ · (u · m)(0) ](1) d ∗ , Q˜ 2ρ · (u · m)(1)  1

1

8.4 Yetter–Drinfeld Modules as Modules over an Algebra (8.4.3)

=

327

q˜1ρ · [uQ˜ 1ρ · (u · m)(0) ](0) d ∗ , Q˜ 2ρ · (u · m)(1) (c∗  u)[Q˜ 1ρ · (u · m)(0) ] = (c∗  u)[(d ∗  u )m],

as needed. It is not hard to see that (ε  1A )m = m for all m ∈ M, so M is a left C∗  A-module. The fact that a morphism in A YD(H)C becomes a morphism in C∗  A M can be proved more easily; we leave the details to the reader. We construct now a functor in the opposite direction. Lemma 8.28 Let H be a quasi-Hopf algebra with bijective antipode, (H, A,C) a Yetter–Drinfeld datum, and assume that C is finite dimensional. We have a functor G : C∗  A M → A YD(H)C , given by G(M) = M as k-vector space, with structure defined, for m ∈ M and u ∈ A, by u · m = (ε  u)m,

(8.4.4) n

ρM : M → M ⊗C, ρM (m) = ∑ (ci  ( p˜1ρ )[0] )m ⊗ S−1 ( p˜2ρ ) · ci · ( p˜1ρ )[−1] . (8.4.5) i=1

Here p˜ρ = p˜1ρ ⊗ p˜2ρ is the element defined in (4.3.9), {ci }i=1,n is a basis of C and {ci }i=1,n is the corresponding dual basis of C∗ . G maps a morphism to itself. Proof The most difficult part of the proof is to show that G(M) satisfies the relations (8.4.1) and (8.4.2). It is then straightforward to show that a morphism in C∗  A M is also a morphism in A YD(H)C , and that G is a functor. It is not hard to see that (4.4.3), (3.2.1) and (4.4.4) imply 1

2

3

2 1 2 2 θ θ 1 ⊗ θ θ0 p˜ρ ⊗ θ θ1 p˜ρ S(θ 3 ) = ( p˜1ρ )[−1] ⊗ ( p˜1ρ )[0] ⊗ p˜2ρ .

(8.4.6)

Write p˜ρ = p˜1ρ ⊗ p˜2ρ = P˜ρ1 ⊗ P˜ρ2 . For all m ∈ M we compute: (θ 2 · m(0) )(0) ⊗ (θ 2 · m(0) )(1) · θ 1 ⊗ θ 3 · m(1) n

=

∑ ((ε  θ 2 )(ci  ( p˜1ρ )[0] )m)(0) ⊗ ((ε  θ 2 )(ci  ( p˜1ρ )[0] )m)(1) · θ 1

i=1

⊗ θ 3 S−1 ( p˜2ρ ) · ci · ( p˜1ρ )[−1] (5.6.15)

=

(8.4.5)

n

2 1 p˜ρ )[0] )m ⊗ S−1 (P˜ρ2 ) · c j · (P˜ρ1 )[−1] θ 1 ∑ (c j  (P˜ρ1 )[0] )(ci  (θ0

i, j=1

2 2 2 1 ⊗ θ 3 S−1 (θ1 p˜ρ ) · ci · (θ0 p˜ρ )[−1] (5.6.15)

=

(5.6.9)

n

∑ [c j ci  (X˜ ρ )[0] x˜3λ (θ 1

2

i, j=1

2 1 (P˜ρ1 )[0]0 θ0 p˜ρ )[0] ]m 1

⊗ S−1 ( f 2 X˜ ρ P˜ρ2 ) · c j · (X˜ ρ )[−1]1 x˜1λ θ (P˜ρ1 )[−1] θ 1 3

1

3

2 2 ⊗ θ 3 S−1 ( f 1 X˜ ρ θ (P˜ρ1 )[0]1 θ1 p˜ρ ) · ci 2

2 1 2 1 · (X˜ ρ )[−1]2 x˜2λ (θ (P˜ρ1 )[0]0 θ0 p˜ρ )[−1]

328

Yetter–Drinfeld Module Categories n

∑ [c j ci  (X˜ ρ (P˜ρ1 )0 p˜1ρ )[0] x˜3λ ]m

(4.4.1),(8.4.6)

=

(4.3.5)

1

i, j=1

⊗ S−1 ( f 2 X˜ ρ P˜ρ2 ) · c j · (X˜ ρ (P˜ρ1 )0 p˜1ρ )[−1]1 x˜1λ 3

1

2 1 ⊗ S−1 ( f 1 X˜ ρ (P˜ρ1 )1 p˜2ρ ) · ci · (X˜ ρ (P˜ρ1 )0 p˜1ρ )[−1]2 x˜2λ n

∑ [c j ci  ((x˜1ρ )0 p˜1ρ )[0] x˜3λ ]m

(4.3.14)

=

i, j=1

⊗ x˜2ρ S−1 ( f 2 ((x˜1ρ )1 p˜2ρ )2 g2 ) · c j · ((x˜1ρ )0 p˜1ρ )[−1]1 x˜1λ ⊗ x˜3ρ S−1 ( f 1 ((x˜1ρ )1 p˜2ρ )1 g1 ) · ci · ((x˜1ρ )0 p˜1ρ )[−1]2 x˜2λ (3.2.13)

=

(4.2.10)

n

∑ [ci  ((x˜1ρ )0 p˜1ρ )[0] x˜3λ ]m

i=1

⊗ x˜2ρ · (S−1 ((x˜1ρ )1 p˜2ρ ) · ci · ((x˜1ρ )0 p˜1ρ )[−1] )1 · x˜1λ ⊗ x˜3ρ · (S−1 ((x˜1ρ )1 p˜2ρ ) · ci · ((x˜1ρ )0 p˜1ρ )[−1] )2 · x˜2λ n

∑ [(x˜1ρ )0[−1]  ci  S−1 ((x˜1ρ )1 )  ((x˜1ρ )0 p˜1ρ )[0] x˜3λ ]m

=

i=1

⊗ x˜2ρ · (S−1 ( p˜2ρ ) · ci · ( p˜1ρ )[−1] )1 · x˜1λ ⊗ x˜3ρ · (S−1 ( p˜2ρ ) · ci · ( p˜1ρ )[−1] )2 · x˜2λ (5.6.15)

=

n

∑ [(ε  x˜1ρ )(ci  ( p˜1ρ )[0] )(ε  x˜3λ )]m

i=1

⊗ x˜2ρ · (S−1 ( p˜2ρ ) · ci · ( p˜1ρ )[−1] )1 · x˜1λ ⊗ x˜3ρ · (S−1 ( p˜2ρ ) · ci · ( p˜1ρ )[−1] )2 · x˜2λ (8.4.4)

=

(8.4.5)

x˜1ρ · (x˜3λ · m)(0) ⊗ x˜2ρ · (x˜3λ · m)(1)1 · x˜1λ ⊗ x˜3ρ · (x˜3λ · m)(1)2 · x˜2λ .

Similarly, we compute: u0 · m(0) ⊗ u1 · m(1) n

= (5.6.15)

=

∑ (ε  u0 )(ci  ( p˜1ρ )[0] )m ⊗ u1 S−1 ( p˜2ρ ) · ci · ( p˜1ρ )[−1]

i=1 n

∑ (u0,0[−1]  ci  S−1 (u0,1 )  u0,0[0] ( p˜1ρ )[0] )m

i=1

⊗ u1 S−1 ( p˜2ρ ) · ci · ( p˜1ρ )[−1] n

= (4.3.10)

=

(5.6.15)

=

∑ (ci  (u0,0 p˜1ρ )[0] )m ⊗ u1 S−1 (u0,1 p˜2ρ ) · ci · (u0,0 p˜1ρ )[−1]

i=1 n

∑ (ci  ( p˜1ρ u)[0] )m ⊗ S−1 ( p˜2ρ ) · ci · ( p˜1ρ u)[−1]

i=1 n

∑ (ci  ( p˜1ρ )[0] )(ε  u[0] )m ⊗ S−1 ( p˜2ρ ) · ci · ( p˜1ρ )[−1] u[−1]

i=1

8.4 Yetter–Drinfeld Modules as Modules over an Algebra (8.4.5)

=

329

(u[0] · m)(0) ⊗ (u[0] · m)(1) · u[−1] ,

for all u ∈ A and m ∈ M, and this finishes the proof. One can now prove the desired isomorphism of categories. Theorem 8.29 Let H be a quasi-Hopf algebra with bijective antipode and (H, A,C) a Yetter–Drinfeld datum, assuming C to be finite dimensional. Then the categories C A YD(H) and C∗  A M are isomorphic. Proof We have to verify that the functors F and G defined in Lemmas 8.27 and 8.28 are inverse to each other. Let M ∈ A YD(H)C . The structures on G(F(M)) (using first Lemma 8.27 and then Lemma 8.28) are denoted by · and ρ  M . For any u ∈ A and m ∈ M we have that u ·  m = (ε  u)m = ε , q˜2ρ · (u · m)(1) q˜1ρ · (u · m)(0) = u · m because ε (h · c) = ε (h)ε (c) and ε (m(1) )m(0) = m for all h ∈ H, c ∈ C, m ∈ M. We now compute for m ∈ M that  ρM (m) = (8.4.3)

=

n

∑ (ci  ( p˜1ρ )[0] )m ⊗ S−1 ( p˜2ρ ) · ci · ( p˜1ρ )[−1]

i=1 n

∑ ci , q˜2ρ · (( p˜1ρ )[0] · m)(1) q˜1ρ · (( p˜1ρ )[0] · m)(0) ⊗ S−1 ( p˜2ρ ) · ci · ( p˜1ρ )[−1]

i=1 (8.4.2)

=

(4.3.13)

=

q˜1ρ ( p˜1ρ )0 · m(0) ⊗ S−1 ( p˜2ρ )q˜2ρ ( p˜1ρ )1 · m(1) m(0) ⊗ m(1) = ρM (m).

Conversely, take M ∈ C∗  A M . We want to show that F(G(M)) = M. If we denote the left C∗  A-action on F(G(M)) by → then by using Lemmas 8.27 and 8.28 we find, for all c∗ ∈ C∗ , u ∈ A and m ∈ M: (c∗  u) → m =

c∗ , q˜2ρ · (u · m)(1) q˜1ρ · (u · m)(0)

=

∑ c∗ , q˜2ρ S−1 ( p˜2ρ ) · ci · ( p˜1ρ )[−1] (ε  q˜1ρ )(ci  ( p˜1ρ )[0] )(ε  u)m

n

(5.6.15)

=

i=1 n

∑ c∗ , q˜2ρ S−1 ((q˜1ρ )1 p˜2ρ ) · ci · ((q˜1ρ )0 p˜1ρ )[−1] 

i=1 i (4.3.12),(5.6.15)

=

(c  ((q˜1ρ )0 p˜1ρ )[0] )(ε  u)m

(c∗  1A )(ε  u)m = (c∗  u)m,

and this finishes our proof. There is a relation between the functor F from Lemma 8.27 and the map Γ as in Proposition 5.51. Proposition 8.30 Let H be a quasi-Hopf algebra with bijective antipode, (H, A,C) a Yetter–Drinfeld datum and M an object in A YD(H)C ; consider the map Γ : C∗ →

330

Yetter–Drinfeld Module Categories

C∗  A as in Proposition 5.51. Then the left C∗  A-module structure on M given in Lemma 8.27 and the map Γ are related by the formula: Γ(c∗ )m = c∗ , m(1) m(0) , ∀ c∗ ∈ C∗ , m ∈ M. Proof

We compute:

Γ(c∗ )m

=

(( p˜1ρ )[−1]  c∗  S−1 ( p˜2ρ )  ( p˜1ρ )[0] )m

=

( p˜1ρ )[−1]  c∗  S−1 ( p˜2ρ ), q˜2ρ · (( p˜1ρ )[0] · m)(1) q˜1ρ · (( p˜1ρ )[0] · m)(0)

=

c∗ , S−1 ( p˜2ρ )q˜2ρ · (( p˜1ρ )[0] · m)(1) · ( p˜1ρ )[−1] q˜1ρ · (( p˜1ρ )[0] · m)(0)

(8.4.2)

=

(4.3.13)

=

c∗ , S−1 ( p˜2ρ )q˜2ρ ( p˜1ρ )1 · m(1) q˜1ρ ( p˜1ρ )0 · m(0) c∗ , m(1) m(0) ,

finishing the proof. By taking C = A = H in Theorem 8.29 we get the following: Corollary 8.31 Let H be a finite-dimensional quasi-Hopf algebra. Then the categories H YD H and H ∗  H M are isomorphic, where H ∗  H is the diagonal crossed product algebra between the H-bimodule algebra H ∗ and the H-bicomodule algebra H as in Examples 5.49 (i), specialized for A = H.

8.5 The Quantum Double of a Quasi-Hopf Algebra Throughout this section, H is a finite-dimensional quasi-Hopf algebra, so has bijective antipode (see Theorem 7.18). The category of finite-dimensional left–right fd Yetter–Drinfeld modules, H YD H , is a left rigid monoidal category; see Proposition 8.23. It is, moreover, a braided category, but to define the quantum double of H as fd a quasi-Hopf algebra we will not need this extra property of H YD H . Actually, we will exploit it in Section 10.4, where we will show that the quantum double of H is, moreover, a quasitriangular quasi-Hopf algebra. fd By Remark 8.24, the forgetful functor F : H YD H → k M fd is a left rigid quasifd monoidal functor. By Corollary 8.31 it follows that the categories H YD H and fd are isomorphic, where H ∗  H is the diagonal crossed product algebra H ∗  H M between the H-bimodule algebra H ∗ and the H-bicomodule algebra H. From now on we will denote H ∗  H by D(H) and call it the quantum double of H. Recall from Section 5.6 that if Ω ∈ H ⊗5 is given by Ω = Ω1 ⊗ Ω2 ⊗ Ω3 ⊗ Ω4 ⊗ Ω5 1 1 = X(1,1) y1 x1 ⊗ X(1,2) y2 x12 ⊗ X21 y3 x22 ⊗ S−1 ( f 1 X 2 x3 ) ⊗ S−1 ( f 2 X 3 ), (8.5.1)

where f ∈ H ⊗ H is the twist defined in (3.2.15), then the quantum double D(H) = H ∗  H is the k-vector space H ∗ ⊗ H endowed with the associative unital k-algebra

8.5 The Quantum Double of a Quasi-Hopf Algebra

331

structure given by (ϕ  h)(ψ  h ) = [(Ω1  ϕ  Ω5 )(Ω2 h(1,1)  ψ  S−1 (h2 )Ω4 )]  Ω3 h(1,2) h = [(Ω1  ϕ  Ω5 )(Ω2  ψ2  Ω4 )]  Ω3 [(S

−1

(8.5.2)

(ψ1 )  h)  ψ3 ]h

and 1D(H) = ε  1H . Note that in the above relations we used the dual quasi-Hopf algebra structure of H ∗ and the H-bimodule structure of H ∗ from Example 4.15. It is easy to see that (ε  h)(ϕ  h ) = h(1,1)  ϕ  S−1 (h2 )  h(1,2) h

(8.5.3)

and (ϕ  h)(ε  h ) = ϕ  hh for all ϕ ∈ H ∗ and h, h ∈ H. Thus D(H) contains H as a k-subalgebra, modulo the identification h ≡ ε  h, h ∈ H. Summing up, we have a finite-dimensional k-algebra, namely D(H), with the fd property that D(H) M fd is a left rigid monoidal category (as it identifies with H YD H as an ordinary category) such that the forgetful functor U : D(H) M → k M fd is a left rigid quasi-monoidal functor. By Theorem 3.38 it follows that D(H) admits a quasiHopf algebra structure. The purpose of this section is to describe this structure of D(H) explicitly. Since we will very often use the categorical isomorphism between H fd fd , for the convenience of the reader, note that, by Lemma 8.28, D(H) M and H YD D(H) is a left–right Yetter–Drinfeld module over H via the structure given by: h · (ϕ  h ) = (ε  h)(ϕ  h ) = h(1,1)  ϕ  S−1 (h2 )  h(1,2) h ,

(8.5.4)

ϕ  h → ∑(ei  p12 )(ϕ  h) ⊗ S−1 (p2 )ei p11 i

= ∑ ei (Ω2 (p12 )(1,1)  ϕ  S−1 ((p12 )2 )Ω4 ) i

 Ω3 (p12 )(1,2) h ⊗ S−1 (p2 )Ω5 ei Ω1 p11 ,

(8.5.5)

for all ϕ ∈ H ∗ and h, h ∈ H, where {ei , ei }i are dual bases in H and H ∗ . Similarly, by Lemma 8.27, any left–right Yetter–Drinfeld module M is a left D(H)module via the left D(H)-action given by (ϕ  h)m = ϕ , q2 (h · m)(1) q1 · (h · m)(0) , ∀ ϕ ∈ H ∗ , h ∈ H, m ∈ M.

(8.5.6)

Consequently, if M, N ∈ H YD H then M ⊗ N ∈ H YD H with structure as in (8.2.17) and (8.2.16), and therefore M ⊗ N is a left D(H)-module with structure given by (ϕ  h)(m ⊗ n) = ϕ , q2 (h · (m ⊗ n))(1) q1 · (h · (m ⊗ n))(0) = ϕ , q2 (h1 · m ⊗ h2 · n)(1) q1 · (h1 · m ⊗ h2 · n)(0) = ϕ , q2 x3 (X 3 y3 h2 · n)(1) X 2 (y2 h1 · m)(1) y1  q11 x1 X 1 · (y2 h1 · m)(0) ⊗ q12 x2 · (X 3 y3 h2 · n)(0) . (8.5.7) By using the reconstruction theorem for quasi-bialgebras (Proposition 3.3) we find the quasi-bialgebra structure of D(H).

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Yetter–Drinfeld Module Categories

Proposition 8.32 Let H be a finite-dimensional quasi-Hopf algebra H. Then D(H) has a unique quasi-bialgebra structure with respect to which the isomorphism of categories in Corollary 8.31 becomes a monoidal isomorphism. Proof By the proof of Proposition 3.3, the comultiplication of D(H), denoted in what follows by ΔD , is given by ΔD (ϕ  h) = (ϕ  h)(1D(H) ⊗ 1D(H) ) = (ϕ  h)((ε  1H ) ⊗ (ε  1H )) (8.5.7)

= ϕ , q2 x3 (X 3 y3 h2 · (ε  1H ))(1) X 2 (y2 h1 · (ε  1H ))(1) y1  q11 x1 X 1 · (y2 h1 · (ε  1H ))(0) ⊗ q12 x2 · (X 3 y3 h2 · (ε  1H ))(0)

(8.5.4)

= ϕ , q2 x3 (ε  X 3 y3 h2 )(1) X 2 (ε  y2 h1 )(1) y1  q11 x1 X 1 · (ε  y2 h1 )(0) ⊗ q12 x2 · (ε  X 3 y3 h2 )(0) ,

for all ϕ ∈ H ∗ and h ∈ H. If we take ϕ = ε in (8.5.5) we get that (ε  h)(0) ⊗ (ε  h)(1) = ∑ ei  p12 h ⊗ S−1 (p2 )ei p11 ,

(8.5.8)

i

for all h ∈ H. Thus, if we denote by P1 ⊗ P2 a second copy of pR , we can compute: ΔD (ϕ  h)

∑ϕ , q2 x3 S−1 (p2 )ei p11 X 2 S−1 (P2 )e j P11 y1 

=

i, j

q11 x1 X 1 · (e j  P21 y2 h1 ) ⊗ q12 x2 · (ei  p12 X 3 y3 h2 ) (8.5.4)

=

∑ϕ , q2 x3 S−1 (p2 )ei p11 X 2 S−1 (P2 )e j P11 y1  i, j

(q11 x1 X 1 )(1,1)  e j  S−1 ((q11 x1 X 1 )2 )  (q11 x1 X 1 )(1,2) P21 y2 h1 ⊗ (q12 x2 )(1,1)  ei  S−1 ((q12 x2 )2 )  (q12 x2 )(1,2) p12 X 3 y3 h2 (5.5.17)

=

∑ϕ , S−1 (Y 3 )q2Y22 S−1 ((q1Y12 )2 p2 )ei (q1Y12 )(1,1) p11 X 2 i, j

S−1 ((Y 1 X 1 )2 P2 )e j (Y 1 X 1 )(1,1) P11 y1 e j  (Y 1 X 1 )(1,2) P21 y2 h1 ⊗ ei  (q1Y12 )(1,2) p12 X 3 y3 h2 (3.2.21)

=

∑ϕ , S−1 (Y 3 )q2 S−1 (q12 p2 )ei (q11 p1 )1Y12 X 2 S−1 ((Y 1 X 1 )2 P2 )e j i, j

(Y 1 X 1 )(1,1) P11 y1 e j  (Y 1 X 1 )(1,2) P21 y2 h1 ⊗ ei  (q11 p1 )2Y22 X 3 y3 h2 (3.2.23)

=

(Y 1 X 1 )(1,1) P11 y1  ϕ2  X 2 S−1 ((Y 1 X 1 )2 P2 )  (Y 1 X 1 )(1,2) P21 y2 h1 ⊗Y12  ϕ1  S−1 (Y 3 )  Y22 X 3 y3 h2 .

Combined with (8.5.3), this computation shows that, for all ϕ ∈ H ∗ , h ∈ H: ΔD (ϕ  h) = (ε  X 1Y 1 )(p11 x1  ϕ2  Y 2 S−1 (p2 )  p12 x2 h1 ) ⊗(X12  ϕ1  S−1 (X 3 )  X22Y 3 x3 h2 ).

(8.5.9)

8.5 The Quantum Double of a Quasi-Hopf Algebra

333

Similar computations yield explicit formulas for the counit εD and the reassociator ΦD of D(H). Namely, by the proof of Proposition 3.3 we have

εD (ϕ  h) = (ϕ  h) · 1k (8.5.6)

= ϕ , q2 (h · 1k )(1) q1 · (h · 1k )(0) = ε (h)ϕ , q2 q1 · 1k = ε (h)ε (q1 )ϕ (q2 ) = ε (h)ϕ (S−1 (α )),

for all ϕ ∈ H ∗ and h ∈ H, while ΦD = aD(H),D(H),D(H) (1D(H) ⊗ 1D(H) ⊗ 1D(H) ) = X 1 · (ε  1H ) ⊗ X 2 · (ε  1H ) ⊗ X 3 · (ε  1H ) (8.5.4)

= (ε  X 1 ) ⊗ (ε  X 2 ) ⊗ (ε  X 3 ).

Concluding, D(H) is a quasi-bialgebra with multiplication given in (8.5.2), unit 1D = ε  1H , comultiplication ΔD as in (8.5.9), and counit and reassociator given by

εD (ϕ  h) = ε (h)ϕ (S−1 (α )), ∀ ϕ  h ∈ D(H),

(8.5.10)

ΦD = (ε  X ) ⊗ (ε  X ) ⊗ (ε  X ),

(8.5.11)

1

2

3

and this is the unique quasi-bialgebra structure on the diagonal crossed product algebra D(H) that turns the isomorphism in Corollary 8.31 into an isomorphism of monoidal categories. We end this section by computing the quasi-Hopf algebra structure of D(H). For this, we will make use of the reconstruction theorem for quasi-Hopf algebras as it was stated in Theorem 3.38. Theorem 8.33 Let H be a finite-dimensional quasi-Hopf algebra. Then D(H), the quantum double of H, is a quasi-Hopf algebra that contains H as a quasi-Hopf subalgebra. Proof Let {ei , ei }i be dual bases in H and H ∗ and denote by {θi j }i, j the basis of D(H)∗ dual to the basis {ei  e j }i, j of D(H), that is, θi j (es  et ) = δi,s δ j,t , where δi, j is the Kronecker delta. By (3.5.1), specialized to our context, the antipode SD of D(H) is given by SD (ϕ  h)

=

∑(ϕ  h) · θi, j , 1D ei  e j i, j

(8.5.6)

=

∑ϕ , q2 (h · θi, j )(1) q1 · (h · θi, j )(0) , ε  1H ei  e j i, j

(8.3.6),(8.5.4)

=

∑ϕ , q2 (h · θi, j )(1) (h · θi, j )(0) , ε  S(q1 )ei  e j , i, j

H∗

for all ϕ ∈ and h ∈ H. By Proposition 8.23 we have that D(H)∗ is a left–right Yetter–Drinfeld module over H with the left H-action defined by h · θi, j =

∑(h · θi, j )(es  et )θs,t s,t

334

Yetter–Drinfeld Module Categories (8.3.6)

=

∑ θi, j (S(h) · (es  et ))θs,t s,t

(8.5.4)

=

∑ θi, j (S(h)(1,1)  es  S−1 (S(h)2 )  S(h)(1,2) et )θs,t s,t

=

∑ es (S−1 (S(h)2 )ei S(h)(1,1) )e j (S(h)(1,2) et )θs,t ,

(8.5.12)

s,t

and the right H-action, see (8.3.7), given by

θi, j → ∑θi, j , f 1 · (g2 · (es  et ))(0) θs,t ⊗ S−1 ( f 2 (g2 · (es  et ))(1) g1 ), s,t

for all i, j, extended by linearity. We conclude that SD (ϕ  h) (8.5.4)

=

∑θs,t , f 1 · (ε  g2 S(q1 ))(0) ϕ , q2 S−1 ( f 2 (ε  g2 S(q1 ))(1) g1 ) s,t

S(h)(1,1)  es  S−1 (S(h)2 )  S(h)(1,2) et (8.5.8)

=

∑ θst , f 1 · (ei  p12 g2 S(q1 ))ϕ , q2 S−1 ( f 2 S−1 (p2 )ei p11 g1 )

(8.5.2) i,s,t

(ε  S(h))(es  et ) (8.5.4)

=

1 1 p11 g1 )(ε  S(h))(es  f(1,2) p12 g2 S(q1 )) ∑ϕ , q2 S−1 ( f 2 S−1 ( f21 p2 )es f(1,1) s

(7.3.1)

= (ε  S(h) f 1 )(p11U 1  ϕ ◦ S−1  f 2 S−1 (p2 )  p12U 2 ),

(8.5.2)

for all ϕ ∈ H ∗ and h ∈ H. Similarly, the general formulas in (3.5.3) give the elements αD and βD that together with SD define the antipode of D(H). More precisely,

αD = ∑ evD (θi, j ⊗ 1D )ei  e j =

(8.5.4)

i, j

∑ θi, j (ε  α )ei  e j = ε  α , i, j

and, similarly, since coevD (1k ) = ∑ β · (ei  e j ) ⊗ θi, j we get that i, j

βD = ∑ θi, j (1D )β · (ei  e j ) = β · (ε  1)=ε  β . i, j

Here evD and coevD are the evaluation and coevaluation morphisms of the left dual fd object of D(H) in H YD H as in Proposition 8.23. Hence, the formulas

αD = ε  α , SD (ϕ  h) = (ε  S(h) f

1

)(p11U 1

S

βD = ε  β , −1

(ϕ ) 

f 2 S−1 (p2 )

(8.5.13) 

p12U 2 ),

(8.5.14)

together with the ones found in the proof of Proposition 8.32 define on D(H) a quasiHopf algebra structure. It is immediate that iD : H h → ε  h ∈ D(H) is an injective quasi-Hopf algebra morphism, so H can be regarded as a quasi-Hopf subalgebra of D(H).

8.6 The Quasi-Hopf Algebras Dω (H) and Dω (G)

335

8.6 The Quasi-Hopf Algebras Dω (H) and Dω (G) Let H be a cocommutative Hopf algebra with antipode S over a base field k (in particular, the cocommutativity implies S2 = IdH ). Since H is cocommutative, we can introduce an even more simplified version of Sweedler’s sigma notation: for h ∈ H, we denote Δ(h) = h ⊗ h, (IdH ⊗ Δ)(Δ(h)) = (Δ ⊗ IdH )(Δ(h)) = h ⊗ h ⊗ h, and so on. With this notation, the antipode and counit axioms read: S(h)h = hS(h) = ε (h)1H , ε (h)h = hε (h) = h. We now recall some facts concerning Hopf crossed products and cohomology. Let H be a cocommutative Hopf algebra and A a commutative left H-module algebra, with H-action denoted by H ⊗ A → A, h ⊗ a → h · a. Assume that we are given a linear map σ : H ⊗ H → A, which is normalized (i.e. σ (1H , h) = σ (h, 1H ) = ε (h)1A for all h ∈ H) and convolution invertible. Suppose that, moreover, σ satisfies the 2-cocycle condition:

σ (x, y)σ (xy, z) = [x · σ (y, z)]σ (x, yz), ∀ x, y, z ∈ H. Then, if we define a multiplication on A ⊗ H by (a # h)(b # g) = a(h · b)σ (h, g) # hg, (we denoted a#h := a ⊗ h, for a ∈ A, h ∈ H), this multiplication is associative and 1A ⊗ 1H is a unit, hence A ⊗ H becomes an algebra, which will be denoted by A#σ H and will be called the Hopf crossed product of A and H. Suppose again that H is a cocommutative Hopf algebra and A is a commutative left H-module algebra, and denote by Ψ : H ⊗ A → A the H-module structure of A. We denote by Regq+ (H, A) the set of k-linear maps g : H ⊗q → A which are normalized (i.e. g(h1 ⊗ · · · ⊗ hq ) = ε (h1 ) · · · ε (hq )1A whenever at least one of the hi s equals 1H ) and convolution invertible. We denote by Z q (H, A), Bq (H, A) and H q (H, A) the qcocycles, q-coboundaries and q-cohomology group of the complex determined by Regq+ (H, A) and the maps Dq : Regq+ (H, A) → Regq+1 + (H, A), given by Dq (u) = [Ψ(Id ⊗ u)] ∗ [u−1 (m ⊗ Id ⊗ · · · ⊗ Id)] ∗ [u(Id ⊗ m ⊗ Id ⊗ · · · ⊗ Id)] ∗ · · · ∗ [u±1 (Id ⊗ · · · ⊗ m)] ∗ [u∓1 ⊗ ε ]. Here m denotes multiplication on H and u−1 the convolution inverse of u. This cohomology is called the Sweedler cohomology of H with coefficients in A. From now on, for the remainder of this section, we assume that H is a finitedimensional cocommutative Hopf algebra. Thus, H ∗ is a commutative Hopf algebra, with unit ε , counit ε (ϕ ) = ϕ (1H ), for all ϕ ∈ H ∗ , multiplication (ϕψ )(h) = ϕ (h)ψ (h), for all ϕ , ψ ∈ H ∗ and h ∈ H, comultiplication Δ(ϕ ) = ϕ1 ⊗ ϕ2 if and only if ϕ (hg) = ϕ1 (h)ϕ2 (g), for all h, g ∈ H, and antipode S(ϕ ) = ϕ ◦ S, for all ϕ ∈ H ∗ .

336

Yetter–Drinfeld Module Categories

Assume that we are given a k-linear map ω : H ⊗ H ⊗ H → k that is convolution invertible and satisfies the conditions:

ω (x, y, zt)ω (xy, z,t) = ω (y, z,t)ω (x, yz,t)ω (x, y, z), ∀ x, y, z,t ∈ H, (8.6.1) ω (1H , x, y) = ω (x, 1H , y) = ω (x, y, 1H ) = ε (x)ε (y), ∀ x, y ∈ H.

(8.6.2)

Such a map ω is exactly a 3-cocycle in the Sweedler cohomology defined above. Since H is finite dimensional, we can identify (H ⊗H ⊗H)∗ with H ∗ ⊗H ∗ ⊗H ∗ , so we can consider ω ∈ H ∗ ⊗H ∗ ⊗H ∗ ; we denote ω = ω1 ⊗ ω2 ⊗ ω3 and its convolution inverse ω −1 = ω 1 ⊗ ω 2 ⊗ ω 3 . We define the element Φ ∈ H ∗ ⊗ H ∗ ⊗ H ∗ by Φ := ω −1 = ω 1 ⊗ ω 2 ⊗ ω 3 . Since H ∗ is a commutative algebra, obviously (H ∗ , Δ, ε , Φ) is a quasi-bialgebra, where Δ and ε are the ones that give the usual coalgebra structure of H ∗ (dual to the algebra structure of H). Moreover, if we define β ∈ H ∗ by the formula β (h) = ω (h, S(h), h), then it is easy to see that (H ∗ , Δ, ε , Φ, S, α = ε , β ) is a quasi-Hopf algebra, which will be denoted by Hω∗ . We can consider the diagonal crossed product (Hω∗ )∗  Hω∗ as in Section 5.6. On the other hand, we will construct a certain Hopf crossed product H ∗ #σ H, as follows. We introduce first the following notation: g x = S(x)gx, for all g, x ∈ H. Next, we define the linear map θ : H ⊗ H ⊗ H → k, by

θ (g; x, y) = ω (g, x, y)ω (x, y, g (xy))ω −1 (x, g x, y),

(8.6.3)

for all g, x, y ∈ H, where ω −1 is the convolution inverse of ω . It is easy to see that θ is also normalized and convolution invertible. By using the 3-cocycle condition for ω several times, one can get the following relation:

θ (g; x, y)θ (g; xy, z) = θ (g x; y, z)θ (g; x, yz),

(8.6.4)

for all g, x, y, z ∈ H. Since H is cocommutative, H ∗ becomes a commutative left H-module algebra, with action H ⊗ H ∗ → H ∗ , h ⊗ ϕ → h • ϕ , where h • ϕ = h  ϕ  S(h), and as before  and  denote the left and right regular actions of H on H ∗ given by (h  ϕ )(a) = ϕ (ah) and (ϕ  h)(a) = ϕ (ha) for all h, a ∈ H and ϕ ∈ H ∗ . Hence, (h • ϕ )(a) = ϕ (a h) for all h, a ∈ H and ϕ ∈ H ∗ . Now define the linear map σ : H ⊗ H → H ∗ by σ (x, y)(g) = θ (g; x, y). Since θ is normalized and convolution invertible, σ is also normalized and convolution invertible; one can easily see that the relation (8.6.4) is equivalent to the fact that σ is a 2-cocycle, that is:

σ (x, y)σ (xy, z) = [x • σ (y, z)]σ (x, yz),

(8.6.5)

for all x, y, z ∈ H. Hence, we can consider the Hopf crossed product H ∗ #σ H, which will be denoted by Dω (H), and which is an associative algebra with unit ε #1H . Its multiplication is given, for all ϕ , ϕ  ∈ H ∗ , h, h ∈ H, by (ϕ ⊗ h)(ϕ  ⊗ h ) = ϕ (h  ϕ   S(h))σ (h, h ) ⊗ hh .

(8.6.6)

8.6 The Quasi-Hopf Algebras Dω (H) and Dω (G) Theorem 8.34

337

The linear map w : (Hω∗ )∗  Hω∗ → H ∗ #σ H defined by

w(h  ϕ ) = ω 2 (h)ω 3 (S(h))ω 1 (h  ϕ  S(h))#h, ∀ h ∈ H, ϕ ∈ H ∗ ,

(8.6.7)

is an algebra isomorphism, with inverse W : H ∗ #σ H → (Hω∗ )∗  Hω∗ given by W (ϕ #h) = p11 (h)p2 (S(h))(ϕ1  h  S(ϕ3 ))  p12 ϕ2 , ∀ ϕ ∈ H ∗ , h ∈ H,

(8.6.8)

where we denote by p1 ⊗ p2 = x1 ⊗ x2 β S(x3 ) the element for Hω∗ given by (3.2.19) and by  and  the regular actions of H on H ∗ and of H ∗ on H. Proof We will construct the map w by using the Universal Property of the diagonal crossed product (Proposition 5.61). We define the linear maps

γ : Hω∗ → H ∗ #σ H, γ (ϕ ) = ϕ #1H ,

v : H = (Hω∗ )∗ → H ∗ #σ H, v(h) = ε #h.

One can easily see that γ is an algebra map and the relations (5.6.27) and (5.6.29) are satisfied, that is, we have

γ (ϕ1 )v(h  ϕ2 ) = v(ϕ1  h)γ (ϕ2 ), v(1H ) = ε #1H , for all ϕ ∈ H ∗ and h ∈ H. So the only thing left to prove is the relation (5.6.28), that is,

ε #hh = (ω 1 #1H )(ε #ω1 ω 1  h  ω 2 )(ω2 #1H )(ε #ω 2  h  ω 3 ω3 )(ω 3 #1H ), where we denote by ω −1 = ω 1 ⊗ ω 2 ⊗ ω 3 another copy of ω −1 . We compute: (ω 1 #1H )(ε #ω1 ω 1  h  ω 2 )(ω2 #1H )(ε #ω 2  h  ω 3 ω3 )(ω 3 #1H ) = (ω 1 #ω1 ω 1  h  ω 2 )(ω2 #ω 2  h  ω 3 ω3 )(ω 3 #1H ) = ω1 (h)ω 1 (h)ω 2 (h)ω 2 (h )ω 3 (h )ω3 (h )(ω 1 #h)(ω2 (h  ω 3  S(h ))#h ) = ω1 (h)ω 1 (h)ω 2 (h)ω 2 (h )ω 3 (h )ω3 (h ) (ω 1 (h  ω2  S(h))(hh  ω 3  S(hh ))σ (h, h )#hh ). When we evaluate this in g ⊗ ϕ ∈ H ⊗ H ∗ we obtain:

ω (h, S(h)gh, h )ω −1 (h, h , S(hh )ghh )ω −1 (g, h, h )θ (g; h, h )ϕ (hh ) = ω (h, S(h)gh, h )ω −1 (h, h , S(hh )ghh )ω −1 (g, h, h )

ω (g, h, h )ω (h, h , S(hh )ghh )ω −1 (h, S(h)gh, h )ϕ (hh ) = ε (g)ϕ (hh ) = (ε #hh )(g ⊗ ϕ ). Thus, Proposition 5.61 yields an algebra map w : (Hω∗ )∗  Hω∗ → H ∗ #σ H, defined by the formula w(h  ϕ ) = γ (q1 )v(h  q2 )γ (ϕ ), where q1 ⊗ q2 = X 1 ⊗ S−1 (α X 3 )X 2 is the element for Hω∗ given by (3.2.19). An easy computation shows that this map is identical to the one given by (8.6.7).

338

Yetter–Drinfeld Module Categories

To prove that w is bijective with inverse W , since the underlying vector spaces have the same (finite) dimension, it is enough to prove that w ◦W = Id. We compute: w(W (ϕ #h)) = w(p11 (h)p2 (S(h))ϕ1 (h)ϕ3 (S(h))h  p12 ϕ2 ) = p11 (h)p2 (S(h))ϕ1 (h)ϕ3 (S(h))ω 2 (h)ω 3 (S(h))

ω 1 (h  p12 ϕ2  S(h))#h. When we evaluate this in g ⊗ ψ ∈ H ⊗ H ∗ we obtain: p11 (h)p2 (S(h))ϕ1 (h)ϕ3 (S(h))ω 2 (h)ω 3 (S(h))p12 (S(h)gh)ϕ2 (S(h)gh)ω 1 (g)ψ (h) = p1 (gh)p2 (S(h))ϕ (g)ω −1 (g, h, S(h))ψ (h) = ω1 (gh)ω2 (S(h))β (S(h))ω3 (h)ω −1 (g, h, S(h))ϕ (g)ψ (h) = ω (gh, S(h), h)ω (S(h), h, S(h))ω −1 (g, h, S(h))ϕ (g)ψ (h). To finish the proof it will be enough to prove that

ω (gh, S(h), h)ω (S(h), h, S(h))ω −1 (g, h, S(h)) = ε (g)ε (h). Note first that the 3-cocycle condition for ω applied to the elements x = h, y = S(h), z = h, t = S(h) yields

ω (S(h), h, S(h)) = ω −1 (h, S(h), h).

(8.6.9)

So it is enough to prove that

ω (gh, S(h), h)ω −1 (h, S(h), h)ω −1 (g, h, S(h)) = ε (g)ε (h). But this relation follows immediately by applying the 3-cocycle condition for ω to the elements x = g, y = h, z = S(h), t = h. The quantum double D(Hω∗ ) of the quasi-Hopf algebra Hω∗ has as underlying algebra structure the diagonal crossed product (Hω∗ )∗  Hω∗ , so Theorem 8.34 implies: Theorem 8.35 Dω (H) = H ∗ #σ H is a quasi-Hopf algebra. Proof Most of the structure of Dω (H) may be obtained, by a straightforward computation, by transferring the structure from D(Hω∗ ) via the isomorphism (8.6.7). We write down the structures obtained in this way: • the counit:

ε : Dω (H) → k, ε (ϕ #h) = ϕ (1H )ε (h), ∀ ϕ ∈ H ∗ , h ∈ H; • the reassociator: Φ = (ω 1 #1H ) ⊗ (ω 2 #1H ) ⊗ (ω 3 #1H ) ∈ Dω (H) ⊗ Dω (H) ⊗ Dω (H); • αDω (H) = ε #1H , βDω (H) = β #1H ; • the comultiplication: define the linear map γ : H ⊗ H ⊗ H → k by

γ (g, h; x) = ω (g, h, x)ω (x, g x, h x)ω −1 (g, x, h x),

(8.6.10)

8.6 The Quasi-Hopf Algebras Dω (H) and Dω (G)

339

for all g, h, x ∈ H. Then define the linear map ν : H → (H ⊗ H)∗ , ν (h)(x ⊗ y) = γ (x, y; h). Identifying (H ⊗ H)∗ with H ∗ ⊗ H ∗ , we will write, for any h ∈ H, ν (h) = ν1 (h) ⊗ ν2 (h) ∈ H ∗ ⊗ H ∗ . Then the comultiplication of Dω (H) is defined, for all ϕ ∈ H ∗ , h ∈ H, by Δ : Dω (H) → Dω (H) ⊗ Dω (H), Δ(ϕ #h) = (ν1 (h)ϕ1 #h) ⊗ (ν2 (h)ϕ2 #h). (8.6.11) The only part of the structure of Dω (H) that is difficult to obtain by transferring from D(Hω∗ ) (the computations become very unpleasant) is the antipode, so we need to give a direct proof of the fact that the linear map s : Dω (H) → Dω (H), s(ϕ #h) = [ε #S(h)][σ −1 (h, S(h))S(ϕν1−1 (h))ν2−1 (h)#1H ],

(8.6.12)

for all ϕ ∈ H ∗ , h ∈ H, is the antipode, where we denote by ν −1 the convolution inverse of ν , with notation ν −1 (h) = ν1−1 (h) ⊗ ν2−1 (h) ∈ H ∗ ⊗ H ∗ . By using the relation (8.6.4), it follows that

θ (x; S(h), h) = θ (hxS(h); h, S(h)),

(8.6.13)

for all x, h ∈ H, and by using this relation we obtain s((ϕ #h)1 )(ϕ #h)2 = ε (ϕ #h)(ε #1H ),

(8.6.14)

for all ϕ ∈ H ∗ , h ∈ H, where we denote, as usual, Δ(ϕ #h) = (ϕ #h)1 ⊗ (ϕ #h)2 . By using the 3-cocycle relation for ω , one obtains the following identity:

γ (x, y; h)γ (xy, z; h)ω (x h, y h, z h) = γ (x, yz; h)γ (y, z; h)ω (x, y, z),

(8.6.15)

for all h, x, y, z ∈ H. By using this relation we obtain

γ (x, S(x); h)β (x h) = γ (S(x), x; h)β (x),

(8.6.16)

from which we get that (ϕ #h)1 β s((ϕ #h)2 ) = ε (ϕ #h)β ,

(8.6.17)

for all ϕ ∈ H ∗ , h ∈ H. The relation (ω 1 #1H )β s(ω 2 #1H )(ω 3 #1H ) = ε #1H ,

(8.6.18)

is immediate, while the relation s(ω1 #1H )(ω2 #1H )β s(ω3 #1H ) = ε #1H

(8.6.19)

follows immediately by using (8.6.9). By applying the 3-cocycle relation for ω repeatedly, we get:

γ (x, y; h)γ (x h, y h; h )θ (x; h, h )θ (y; h, h ) = γ (x, y; hh )θ (xy; h, h ),

(8.6.20)

for all h, h , x, y ∈ H, and by using this relation and (8.6.4) we obtain s((ϕ #h)(ϕ  #h )) = s(ϕ  #h )s(ϕ #h),

(8.6.21)

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for all ϕ , ϕ  ∈ H ∗ and h, h ∈ H, and since obviously s(ε #1H ) = ε #1H , it follows that s is an anti-algebra homomorphism. Thus, s is indeed the antipode of Dω (H). Moreover, by (8.6.9) we get that β is convolution invertible with inverse β −1 = β ◦ S. By using (8.6.20), (8.6.13) and (8.6.16), s2 (ϕ #h) = (β −1 #1H )(ϕ #h)(β #1H ), for all ϕ ∈ H ∗ , h ∈ H. We will see how Dω (H) depends on the cohomology class of ω . Suppose that ω  is another normalized 3-cocycle on H, which lies in the same cohomology class as ω ; that is, there exists a k-linear map t : H ⊗ H → k, normalized and convolution invertible, such that ω  = ω d(t), where d(t) : H ⊗ H ⊗ H → k is given by d(t)(a ⊗ b ⊗ c) = t(b, c)t −1 (ab, c)t(a, bc)t −1 (a, b), for all a, b, c ∈ H, and t −1 is the convolution inverse of t. We will denote by θ , θ  , θd(t) and σ , σ  , σd(t) the maps associated (as above) to ω , ω  , d(t), respectively. Now define the map τ = τt : H → H ∗ by τ (x)(g) = t(x, g x)t −1 (g, x), for all x, g ∈ H. Obviously τ (1H ) = ε and τ is convolution invertible. Since ω  = ω d(t), it follows that θ  = θ θd(t) , so we have

θ  (g; x, y)θ −1 (g; x, y) = θd(t) (g; x, y), ∀ g, x, y ∈ H. A straightforward computation yields:

θd(t) (g; x, y) = τ (x)(g)τ (y)(g x)τ −1 (xy)(g), ∀ g, x, y ∈ H. Now, for the cohomology H n (H, H ∗ ), since τ ∈ Reg1+ (H, H ∗ ), we can consider B2 (H, H ∗ ), and it is easy to see that

D1 (τ ) ∈

D1 (τ )(x, y)(g) = τ (x)(g)τ (y)(g x)τ −1 (xy)(g), ∀ g, x, y ∈ H, hence we have θd(t) (g; x, y) = D1 (τ )(x, y)(g), that is σd(t) = D1 (τ ). By identifying (H ⊗ H)∗ = H ∗ ⊗ H ∗ , we will write t = t1 ⊗ t2 ∈ H ∗ ⊗ H ∗ . Now define F ∈ Dω (H) ⊗ Dω (H) by F = (t1 ⊗ 1H ) ⊗ (t2 ⊗ 1H ). Since t is normalized and convolution invertible (let t −1 = l1 ⊗ l2 be the convolution inverse of t), it follows immediately that F is invertible with inverse F −1 = (l1 ⊗ 1H ) ⊗ (l2 ⊗ 1H ), and satisfies the relation (ε ⊗ Id)(F) = (Id ⊗ ε )(F) = 1, that is, F is a gauge transformation on Dω (H).  Define the linear map T : Dω (H) → Dω (H) by T (ϕ ⊗ h) = ϕτ (h) ⊗ h, for all ϕ ∈ H ∗ and h ∈ H. By a direct computation, using the facts that σ  = σ σd(t) and σd(t) = D1 (τ ), it follows that T is an algebra isomorphism. Consider now Dω (H)F −1 , that is, Dω (H) with its algebra structure, but with comultiplication given by ΔF −1 (a) = F −1 Δ(a)F, for all a ∈ Dω (H). By using the fact that γ  = γγd(t) , it follows that (T ⊗ T ) ◦ Δ = ΔF −1 ◦ T . By using ω  = ω d(t), −1 (Id ⊗ Δ)(F −1 )Φ(Δ ⊗ it follows that (T ⊗ T ⊗ T )(Φ ) = ΦF −1 , where ΦF −1 = F23 Id)(F)F12 . Obviously, we have ε ◦ T = ε . Summing up, we obtain that T is an isomorphism of quasi-bialgebras.

8.6 The Quasi-Hopf Algebras Dω (H) and Dω (G)

341

In conclusion, we have obtained the following result: Proposition 8.36 The above map T is an isomorphism of quasi-bialgebras be  tween Dω (H) and Dω (H)F −1 . Consequently, Dω (H) and Dω (H) are twist equivalent quasi-bialgebras. Now let G be a finite group, with multiplication denoted by juxtaposition and unit denoted by e. Let ω be a normalized 3-cocycle on G, that is, ω : G × G × G → k∗ is a map such that ω (x, y, z)ω (tx, y, z)−1 ω (t, xy, z)ω (t, x, yz)−1 ω (t, x, y) = 1 for all t, x, y, z ∈ G, and ω (x, y, z) = 1 whenever x, y or z is equal to 1. We can take H = k[G], the group algebra of G, which is a finite-dimensional cocommutative Hopf algebra, and extend ω by linearity to a map ω : H ⊗ H ⊗ H → k, which turns out to be a Sweedler 3-cocycle on H. So, we can consider the quasi-Hopf algebra Dω (H), which will be denoted by Dω (G). As a linear space, Dω (G) = k[G]∗ ⊗ k[G], which has the basis {pg ⊗ x}, with g, x ∈ G, where pg (h) = δg,h for any g, h ∈ G (δ is the Kronecker delta). In view of the formulas presented above, Dω (G) becomes a quasiHopf algebra with the following structure: • the multiplication: (pg ⊗ x)(ph ⊗ y) = δg,xhx−1 θ (g, x, y)(pg ⊗ xy), where

θ (g, x, y) = ω (g, x, y)ω (x, y, (xy)−1 gxy)ω (x, x−1 gx, y)−1 ; • the unit: 1 = ∑g∈G pg ⊗ e; • the comultiplication: Δ : Dω (G) → Dω (G) ⊗ Dω (G), Δ(pg ⊗ x) =

∑ γ (x, u, v)(pu ⊗ x) ⊗ (pv ⊗ x), where

uv=g

γ (x, u, v) = ω (u, v, x)ω (x, x−1 ux, x−1 vx)ω (u, x, x−1 vx)−1 ; • • • •

the counit: ε : Dω (G) → k, ε (pg ⊗ x) = δg,e ; the reassociator: Φ = ∑x,y,z∈G ω (x, y, z)−1 (px ⊗ e) ⊗ (py ⊗ e) ⊗ (pz ⊗ e); α = 1, β = ∑g∈G ω (g, g−1 , g)(pg ⊗ e); the antipode: s : Dω (G) → Dω (G), s(pg ⊗ x) = θ (g−1 , x, x−1 )−1 γ (x, g, g−1 )−1 (px−1 g−1 x ⊗ x−1 ).

Now, consider the element λ = ∑y∈G pe ⊗ y ∈ Dω (G). We check that λ is a left integral in Dω (G). If g, x ∈ G, we have: (pg ⊗ x)λ =

∑ δg,e θ (g, x, y)(pg ⊗ xy).

y∈G

If g = e, then (pe ⊗ x)λ = ∑y∈G pe ⊗ xy = λ = ε (pe ⊗ x)λ . If g = e, then (pg ⊗ x)λ = 0 = ε (pg ⊗ x)λ , since ε (pg ⊗ x) = δg,e . This shows that λ is indeed a left integral in Dω (G). Similarly, one can prove that λ is a right integral as well, so Dω (G) is unimodular.

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Yetter–Drinfeld Module Categories

Since ε (λ ) = ∑y∈G δe,e = |G|, by applying Theorem 7.28 we obtain that Dω (G) is semisimple if and only if the characteristic of k does not divide |G|, the order of G.

8.7 Algebras within Categories of Yetter–Drinfeld Modules We will show that the algebra H0 within the monoidal category H M considered in Definition 4.4 actually has a commutative algebra structure within H H YD. We start by presenting the Yetter–Drinfeld module structure of H0 . Recall that H0 = H as k-vector spaces. Lemma 8.37 Let H be a quasi-Hopf algebra with bijective antipode. Then H is a left Yetter–Drinfeld module with the following structure: h  h = h1 h S(h2 ),

λH (h) = h(−1) ⊗ h(0) := X 1Y11 h1 g1 S(q2Y22 )Y 3 ⊗ X 2Y21 h2 g2 S(X 3 q1Y12 ),

(8.7.1) (8.7.2)

for all h, h ∈ H, where f −1 = g1 ⊗ g2 and qR = q1 ⊗ q2 are the elements defined by (3.2.16) and (3.2.19), respectively. Proof We know that H is a left H-module via . Now we will prove that the relations (8.2.1)–(8.2.3) hold for our structure. For this, observe first that (3.2.13) implies

λH (h) = (X 1 ⊗ X 2 )Δ(Y 1 hS(Y 2 ))U[Y 3 ⊗ S(X 3 )], ∀ h ∈ H,

(8.7.3)

where U = U 1 ⊗U 2 = U1 ⊗ U2 is the element defined in (7.3.1). Therefore, for any h ∈ H we have: Z 1 (T 1  h)(−1)1 T 2 ⊗ Z 2 (T 1  h)(−1)2 T 3 ⊗ Z 3  (T 1  h)(0) (8.7.3)

=

(Z 1 X11 ⊗ Z 2 X21 ⊗ Z13 X 2 )(Δ ⊗ Id)(Δ(Y 1 (T 1  h)S(Y 2 ))U) [Y13 T 2 ⊗Y23 T 3 ⊗ S(Z23 X 3 )]

(3.1.9),(3.1.7)

=

(Z 1 X11 ⊗ Z 2 X21 ⊗ Z13 X 2 )Φ−1 (Id ⊗ Δ)(Δ(Y 1 T 1 hS(Y12 T 2 )))Φ (Δ ⊗ Id)(Δ(S(W 1 )U))(W 2 ⊗W 3 ⊗ 1H )[Y22 T 3 ⊗Y 3 ⊗ S(Z23 X 3 )]

(7.5.1),(3.1.9)

=

(Z 1 ⊗ X 1 Z12 ⊗ X 2 Z22 )(Id ⊗ Δ)(Δ(Y 1 T 1 hS(Y12 T 2 ))U)(1 ⊗U) [Y22 T 3 ⊗Y 3 ⊗ S(X 3 Z 3 )]

(7.3.2)×2

=

Z 1Y11 (T 1 hS(T 2 ))1 U1 T 3 ⊗ {[(X 1 ⊗ X 2 )Δ(Z 2Y21 (T 1 hS(T 2 ))2 U2 S(Z13Y 2 )] U[Z23Y 3 ⊗ S(X 3 )]}

(3.1.9),(8.7.3)

=

(8.7.3)

=

Z 1 h(−1) ⊗ {(X 1 ⊗ X 2 )Δ(Y 1 Z12 h(0) S(Y 2 Z22 ))U[Y 3 Z 3 ⊗ S(X 3 )]} Z 1 h(−1) ⊗ (Z 2  h(0) )(−1) Z 3 ⊗ (Z 2  h(0) )(0) .

By ε (α ) = ε (β ) = 1, ε (g1 )g2 = 1, ε (q2 )q1 = 1, (3.1.8) and (3.1.11) we deduce that ε (h(−1) )h(0) = h, for all h ∈ H, thus we only have to show the compatibility relation (8.2.3). For all h, h ∈ H we calculate: (h1  h )(−1) h2 ⊗ (h1  h )(0)

8.7 Algebras within Categories of Yetter–Drinfeld Modules

343

X 1 (Y 1 h(1,1) h S(Y 2 h(1,2) ))1 U1Y 3 h2

=

⊗ X 2 (Y 1 h(1,1) h S(Y 2 h(1,2) ))2 U2 S(X 3 ) (3.1.7)

X 1 (h1Y 1 h S(Y 2 ))1 S(h(2,1) )1 U1 h(2,2)Y 3

=

⊗ X 2 (h1Y 1 h S(Y 2 ))2 S(h(2,1) )2 U2 S(X 3 ) (7.3.2),(8.7.3)

X 1 h(1,1) (Y 1 h S(Y 2 ))1 U1Y 3 ⊗ X 2 h(1,2) (Y 1 h S(Y 2 ))2 U2 S(X 3 h2 )

(3.1.7),(8.7.3)

h1 h(−1) ⊗ h2  h(0) ,

= =

as needed. This finishes the proof. Let H be a quasi-Hopf algebra with bijective antipode and H0 the algebra within associated to H, as in Definition 4.4. By the above lemma, it follows that H0 is also an object in H H YD with the same action and coaction as H. Furthermore, we next show that H0 is an algebra within H H YD. H In general, an object A ∈ H YD has an algebra structure within H H YD if and only YD such that mA is if there exist morphisms mA : A ⊗ A → A and η A : k → A in H H YD and η is a unit for mA . In other associative up to the associativity constraint of H H A words, A admits an algebra structure (A, mA , η A ) within H M such that HM

λA (aa ) = X 1 (x1Y 1 · a)(−1) x2 (Y 2 · a )(−1)Y 3 ⊗ [X 2 · (x1Y 1 · a)(0) ][X 3 x3 · (Y 2 · a )(0) ],

(8.7.4)

λA (1A ) = 1H ⊗ 1A ,

(8.7.5)

for all a, a ∈ A, where we denote by · the left action of H on A, by λA the left Hcoaction on A and by 1A the unit of A. Note that the two equalities above express the fact that mA and η A are left H-colinear morphisms. Proposition 8.38 Let H be a quasi-Hopf algebra with bijective antipode and H0 the H-module algebra from Definition 4.4. Then H0 is an algebra in the monoidal category H H YD. Proof We only have to show that the relations (8.7.4) and (8.7.5) hold. To prove (8.7.4) we set qR = q1 ⊗ q2 = Q1 ⊗ Q2 , f −1 = g1 ⊗ g2 = G1 ⊗ G2 and γ = γ 1 ⊗ γ 2 , where qR , f −1 and γ are the elements defined by (3.2.19), (3.2.16) and (3.2.5), respectively, and for all h, h ∈ H we calculate: X 1 (x1Y 1  h)(−1) x2 (Y 2  h )(−1)Y 3 ⊗ [X 2  (x1Y 1  h)(0) ] ◦ [X 3 x3  (Y 2  h )(0) ] (8.7.1),(8.7.2)

=

(3.2.13)



X 1 Z 1 (T 1 x11Y11 hS(T 2 x21Y21 ))1 g1 S(q2 )T 3 x2U 1 (V 1Y12 h S(V 2Y22 ))1 G1 S(Q2 )V 3Y 3 ⊗ [X12 Z 2 (T 1 x11Y11 hS(T 2 x21Y21 ))2 g2 S(X22 Z 3 q1 )] ◦ [X13 x13U 2 (V 1Y12 h S(V 2Y22 ))2 G2 S(X23 x23U 3 Q1 )]

(3.1.9),(3.2.13)

=

(3.2.21),(3.1.7)

X 1 Z 1 y11 (T 1Y11 hS(x1 T 2Y21 ))1 g1 S(q2 )x2U 1 (T13V 1Y12 h S(V 2Y22 ))1 G1 S(Q2 )V 3Y 3 ⊗ [X12 Z 2 y12 (T 1Y11 hS(x1 T 2Y21 ))2 g2 S(X22 Z 3 y2 q1 )] ◦ [(X 3 y3 x3 )1U 2 (T13V 1Y12 h S(V 2Y22 ))2 G2 S((X 3 y3 x3 )2U 3 T23 Q1 )]

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Yetter–Drinfeld Module Categories

(3.1.9),(4.1.9)

X 1 (T 1Y11 hS(x1 T 2Y21 ))1 g1 S(q2 )x2U 1 (T13V 1Y12 h S(V 2Y22 ))1 G1 S(Q2 )

=

(3.1.7),(3.2.1)

V 3Y 3 ⊗ X 2 (T 1Y11 hS(x1 T 2Y21 ))2 g2 S(z1 q1 )α z2 x13U 2 (T13V 1Y12 h S(V 2Y22 ))2 G2 S(X 3 z3 x23U 3 T23 Q1 )

(3.2.21),(3.2.13)

=

(3.1.9)

X 1 (T 1Y11 hS(z1 x1 T 2Y21 ))1 g1 S(q2 )z2 (x2 T13V 1Y12 h S(V 2Y22 ))1 G1 S(Q2 )V 3Y 3 ⊗ X 2 (T 1Y11 hS(z1 x1 T 2Y21 ))2 g2 S(q1 )α z3 (x2 T13V 1Y12 h S(V 2Y22 ))2 G2 S(X 3 x3 T23 Q1 )

(3.1.9),(3.2.13)

2 X 1 (T 1V 1 (Y11 y1 )1 hS(z1 T12V 2 (Y11 y1 )2 ))1 g1 S(q2 )z2 T(2,1) (V 3Y21 y2 h )1

=

(3.2.21),(3.1.9)

S(Y 2 y3 )1 G1 S(Q2 )Y 3 ⊗ X 2 (T 1V 1 (Y11 y1 )1 hS(z1 T12V 2 (Y11 y1 )2 ))2 g2 2 S(q1 )α z3 T(2,2) (V 3Y21 y2 h )2 S(Y 2 y3 )2 G2 S(X 3 T 3 Q1 ) (3.1.7),(3.2.13)

1 1 X 1 (V 1Y(1,1) y11 hS(z1V 2Y(1,2) y12 ))1 g1 S(q2 )z2 (V 3Y21 y2 h S(Y 2 y3 ))1

=

(3.2.21),(3.2.1)

1 1 G1 S(Q2 )Y 3 ⊗ X 2 (V 1Y(1,1) y11 hS(z1V 2Y(1,2) y12 ))2 g2

S(q1 )α z3 (V 3Y21 y2 h S(Y 2 y3 ))2 G2 S(X 3 Q1 ) (3.1.7)×2,(3.2.21)

=

(3.2.1),(3.2.13)×2

X 1Y11 (V 1 y11 hS(z1V 2 y12 ))1 g1 S(q2 )z2 (V 3 y2 h S(y3 ))1 G1 S(Q2Y22 )Y 3 ⊗ X 2Y21 (V 1 y11 hS(z1V 2 y12 ))2 g2 S(q1 )α z3 (V 3 y2 h S(y3 ))2 G2 S(X 3 Q1Y12 )

(3.2.19),(3.2.13)

=

(3.2.5) (3.2.14)

=

X 1Y11 (V 1 y11 hS(V 2 y12 ))1 g1 γ 1 (V 3 y2 h S(y3 ))1 G1 S(Q2Y22 )Y 3 ⊗ X 2Y21 (V 1 y11 hS(V 2 y12 ))2 g2 γ 2 (V 3 y2 h S(y3 ))2 G2 S(X 3 Q1Y12 )

λH0 (V 1 y11 hS(V 2 y12 )α V 3 y2 h S(y3 ))

(3.1.9),(3.2.1)

=



λH0 (h ◦ h ).

Now we prove the relation in (8.7.5). For this, recall that the unit for H0 is β , and therefore, by (8.7.3), we have:

λH0 (1H0 ) = (X 1 ⊗ X 2 )Δ(Y 1 β S(Y 2 ))U[Y 3 ⊗ S(X 3 )] (7.3.5)

(3.2.19)

= (X 1 ⊗ X 2 )pR [1 ⊗ S(X 3 )] = 1H ⊗ 1H0 ,

as desired. We end this section by showing that H0 is commutative as an algebra within H H YD. We first prove some formulas; some of them are of independent interest. Lemma 8.39 Let H be a quasi-Hopf algebra with bijective antipode. Then we have Δ(S(p1 ))U(p2 ⊗ 1H ) = f −1 , S(g )α g = S(β ), 1

2

f β S( f ) = S(α ), 1

S(q22 X 3 ) f 1 ⊗ S(q1 X 1 β S(q21 X 2 ) f 2 )

2

= (IdH ⊗ S)(qL ),

(8.7.6) (8.7.7) (8.7.8)

where U, qR = q1 ⊗ q2 and pR = p1 ⊗ p2 , qL , f = f 1 ⊗ f 2 and f −1 = g1 ⊗ g2 are the elements defined by (7.3.1), (3.2.19), (3.2.20), (3.2.15) and (3.2.16), respectively. Proof The relation (8.7.6) is an immediate consequence of (3.2.13) and (3.2.23), and the equalities in (8.7.7) were checked in the proof of Proposition 3.23; for the convenience of the reader we have recorded them one more time here.

8.7 Algebras within Categories of Yetter–Drinfeld Modules

345

It remains to prove that the relation (8.7.8) holds. By (3.2.26) we obtain: (IdH ⊗ Δ)(qR ) = (1H ⊗ S−1 (x3 g2 ) ⊗ S−1 (x2 g1 ))(qR ⊗ 1H ) (Δ ⊗ IdH )(qR )Φ−1 (IdH ⊗ Δ)(Δ(x1 )). By using the formula in (5.5.17) we get that 1 1 (IdH ⊗ Δ)(qR ) = Q1Y 1 x11 ⊗ S−1 (x3 g2 )Q2 q1Y12 x(2,1) ⊗ S−1 (Y 3 x2 g1 )q2Y22 x(2,2) , (8.7.9) where qR = q1 ⊗ q2 = Q1 ⊗ Q2 , so we can now compute:

S(q22 X 3 ) f 1 ⊗ S(q1 X 1 β S(q21 X 2 ) f 2 ) (8.7.9)

=

1 1 S(q2Y22 x(2,2) X 3 )Y 3 x2 ⊗ S(Q1Y 1 x11 X 1 β S(Q2 q1Y12 x(2,1) X 2 )x3 )

(3.1.7)

1 1 S(q2Y22 X 3 x21 )Y 3 x2 ⊗ S(Q1Y 1 X 1 x(1,1) β S(Q2 q1Y12 X 2 x(1,2) )x3 )

(3.2.1)

S(q2Y22 X 3 x1 )Y 3 x2 ⊗ S(Q1Y 1 X 1 β S(Q2 q1Y12 X 2 )x3 )

(3.1.9)

S(q2 y2 X13Y 2 x1 )y3 X23Y 3 x2 ⊗ S(Q1 X 1Y11 β S(Q2 q1 y1 X 2Y21 )x3 )

= = =

(3.2.1),(3.2.19)

S(X13 x1 )α X23 x2 ⊗ S(Q1 X 1 β S(Q2 X 2 )x3 )

(3.2.1),(3.1.11)

S(x1 )α x2 ⊗ S(Q1 β S(Q2 )x3 )

(3.2.19),(3.2.2)

S(x1 )α x2 ⊗ S(x3 ),

= = =

as needed. We next show the commutativity of the algebra H0 within the braided category H YD. H

Proposition 8.40 Let H be a quasi-Hopf algebra with bijective antipode. Then H0  is commutative as an algebra in H H YD, that is, for all h, h ∈ H: h ◦ h = (h(−1)  h ) ◦ h(0) . Proof

For all h, h ∈ H we have:

(h(−1)  h ) ◦ h(0) (8.7.2)

=

(8.7.1),(4.1.9)

=

(X 1Y11 h1 g1 S(q2Y22 )Y 3  h ) ◦ X 2Y21 h2 g2 S(X 3 q1Y12 ) 1 1 Z 1 X11Y(1,1) h(1,1) g11 S(q2Y22 )1Y13 h S(x1 Z 2 X21Y(1,2) h(1,2) g12 S(q2Y22 )2Y23 )

α x2 Z13 X 2Y21 h2 g2 S(x3 Z23 X 3 q1Y12 ) (3.1.9),(3.2.1)

=

1 1 Z 1Y(1,1) h(1,1) g11 S(q2Y22 )1Y13 h S(Z 2Y(1,2) h(1,2) g12 S(q2Y22 )2Y23 )

α Z 3Y21 h2 g2 S(q1Y12 ) (3.2.13)

=

Z 1 [Y 1 hS(Y 2 )](1,1) g11 S(q2 )1Y13 h S(Z 2 [Y 1 hS(Y 2 )](1,2) g12 S(q2 )2Y23 )

α Z 3 [Y 1 hS(Y 2 )]2 g2 S(q1 ) (3.1.7),(3.2.1)

=

(3.2.17)

=

Y 1 hS(Y 2 )Z 1 g11 S(q2 )1Y13 h S(Z 2 g12 S(q2 )2Y23 )α Z 3 g2 S(q1 ) Y 1 hS(Y 2 )g1 S(X 3 ) f 1 S(q2 )1Y13 h S(g21 G1 S(X 2 ) f 2 S(q2 )2Y23 )α g22 G2 S(q1 X 1 )

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Yetter–Drinfeld Module Categories

(3.2.1),(8.7.7)

=

Y 1 hS(X 3Y 2 ) f 1 S(q2 )1Y13 h S(q1 X 1 β S(X 2 ) f 2 S(q2 )2Y23 )

(3.2.13)

Y 1 hS(q22 X 3Y 2 ) f 1Y13 h S(q1 X 1 β S(q21 X 2 ) f 2Y23 )

(8.7.8)

Y 1 hS(x1Y 2 )α x2Y13 h S(x3Y23 ) = h ◦ h ,

= =

(4.1.9)

as required. Remark 8.41 In general, H0 does not have a coalgebra, bialgebra or Hopf algebra structure within H H YD. Later on we shall see that it is possible to endow H0 with such structures in the case when H is a quasitriangular quasi-Hopf algebra. If H is a finite-dimensional quasi-Hopf algebra then H0 becomes a left D(H)module algebra, where D(H) is the quantum double of H. This follows from the following three results, which we have already proved: • H0 is an algebra within the monoidal category of left Yetter–Drinfeld modules H YD (see Proposition 8.38); H in

H • there is a braided monoidal isomorphism between H (see TheoH YD and H YD rem 8.16); • the category H YD H is monoidally isomorphic to D(H) M (see Proposition 8.32).

By using these isomorphisms we can transfer the algebra structure of H0 from to D(H) M . In this way we associate to any finite-dimensional quasi-Hopf algebra H a left D(H)-module algebra, called the Schr¨odinger representation of H. Our next goal is to find the explicit structure of H0 as a left D(H)-module algebra. For this, first we compute the algebra structure of H0 in H YD H , and second the left D(H)-module algebra structure of H0 , as desired. H YD H

Proposition 8.42 Let H be a quasi-Hopf algebra with bijective antipode. Then H0 is an algebra in the monoidal category H YD H with the left H-module structure defined in (8.7.1) and with the right H-coaction ρH0 : H0 → H0 ⊗ H given for all h ∈ H by

ρH0 (h) = h(0) ⊗ h(1) = x1 q˜2 y22 h2 g2 S(x2 y31 ) ⊗ x3 y32 S−1 (q˜1 y21 h1 g1 )y1 ,

(8.7.10)

where qL = q˜1 ⊗ q˜2 and f −1 = g1 ⊗ g2 are the elements defined in (3.2.20) and (3.2.16), respectively. Moreover, if H is finite dimensional, then H0 is a left D(H)module algebra via the action (ϕ  h) → h = ϕ , q2 x3 y32 S−1 (q˜1 y21 (h  h )1 g1 )y1  q11 x1 q˜2 y22 (h  h )2 g2 S(q12 x2 y31 ),

(8.7.11)

for all ϕ ∈ H ∗ and h, h ∈ H, where qR = q1 ⊗ q2 is the element defined in (3.2.19). Proof The functor G described in the proof of Theorem 8.14 is strong monoidal, so it carries algebras to algebras. Moreover, G is a strict monoidal functor, so if A is H an algebra in H H YD then G(A) is an algebra in H Y D with the same multiplication and unit. Now, G acts as identity on objects at the level of actions. Thus G(H0 ) = H0

8.8 Cross Products of Algebras in H M , H MH , H H YD

347

as left H-module algebras, so we only have to show that that corresponding right H-action on H0 through the functor G is the one claimed in (8.7.10). By (3.2.13) and (7.3.1) we get a second formula for the left H-coaction on H0 defined in (8.7.2), namely

λH0 (h) = h(−1) ⊗ h(0) = (X 1 ⊗ X 2 )Δ(Y 1 hS(Y 2 ))U(Y 3 ⊗ S(X 3 )). Now, for any h ∈ H we calculate:

ρH0 (h)

(8.2.26)

=

= (8.7.1),(5.5.16)

=

q˜21 Z 2  (p1  h)0 ⊗ q˜22 Z 3 S−1 (q˜1 Z 1 (p1  h)−1 p2 )

 q˜21 Z 2  [X 2 Y 1 (p1  h)S(Y 2 ) 2 U 2 S(X 3 )]

 ⊗ q˜22 Z 3 S−1 (q˜1 Z 1 X 1 Y 1 (p1  h)S(Y 2 ) 1 U 1Y 3 p2 ) q˜21 Z 2  [X 2 x21 h2 S(x12 p1 )2U 2 S(X 3 )] ⊗ q˜22 Z 3 x3 S−1 (q˜1 Z 1 X 1 x11 h1 S(x12 p1 )1U 1 x22 p2 )

(8.7.1),(7.3.2)

=

q˜21  [Z12 X 2 x21 h2 S(p1 )2U 2 S(Z22 X 3 x2 )] ⊗ q˜22 Z 3 x3 S−1 (q˜1 Z 1 X 1 x11 h1 S(p1 )1U 1 p2 )

(3.1.9),(8.7.6),(8.7.1) 2 = q˜(1,1) x1 X 2 h2 g2 S(q˜2(1,2) x2 X13 ) ⊗ q˜22 x3 X23 S−1 (q˜1 X 1 h1 g1 ) (3.1.7),(5.2.7) = x1 q˜2 y22 h2 g2 S(x2 y31 ) ⊗ x3 y32 S−1 (q˜1 y21 h1 g1 )y1 ,

as needed. The last assertion is a consequence of (8.5.6) and (8.7.10), the details are left to the reader.

8.8 Cross Products of Algebras in H M , H MH , H H YD We recall from Section 2.1 that, if C is a monoidal category with associativity constraint aU,V,W : (U ⊗V ) ⊗W → U ⊗ (V ⊗W ) and unit 1, and (A, mA , η A ), (B, mB , η B ) are two algebras in C , then a morphism R : B ⊗ A → A ⊗ B in C is called a twisting morphism between A and B if the following conditions hold: R ◦ (IdB ⊗ η A ) = η A ⊗ IdB , R ◦ (η B ⊗ IdA ) = IdA ⊗ η B , R ◦ (mB ⊗ IdA ) = R ◦ (IdB ⊗ mA ) =

(8.8.1)

(IdA ⊗ mB ) ◦ aA,B,B ◦ (R ⊗ IdB ) ◦ a−1 B,A,B ◦(IdB ⊗ R) ◦ aB,B,A ,

(8.8.2)

(mA ⊗ IdB ) ◦ a−1 A,A,B ◦ (IdA ⊗ R) ◦ aA,B,A −1 ◦(R ⊗ IdA ) ◦ aB,A,A .

(8.8.3)

Given such a twisting morphism, A ⊗ B becomes an algebra in C , with multiplication

μ = (mA ⊗ mB ) ◦ a−1 A,A,B⊗B ◦ (IdA ⊗ aA,B,B ) ◦ (IdA ⊗ R ⊗ IdB ) ◦(IdA ⊗ a−1 B,A,B ) ◦ aA,B,A⊗B

(8.8.4)

and unit η = η A ⊗ η B . This algebra structure on A ⊗ B is denoted by A#R B and is called the cross product of A and B afforded by the twisting morphism R. It has, moreover, the property that the morphisms iA := IdA ⊗ η B : A → A#R B and iB := η A ⊗ IdB : B → A#R B are morphisms of algebras in C .

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Yetter–Drinfeld Module Categories

Also from Section 2.1 we recall that, if cU,V : U ⊗V → V ⊗U is a braiding on C , then for any two algebras A and B in C , the morphism R = cB,A : B ⊗ A → A ⊗ B is a twisting morphism, and in this case A#R B is denoted by A ⊗+ B and is called the c-tensor product of A and B. Proposition 8.43 Let H be a quasi-bialgebra, A an H-bimodule algebra and A an algebra in H H YD. We regard A as an H-bimodule algebra with trivial right H-action. Define the linear map R : A ⊗ A → A ⊗ A,

R(a ⊗ ϕ ) = a(−1) · ϕ ⊗ a(0) , ∀ a ∈ A, ϕ ∈ A .

(8.8.5)

Then R is a twisting morphism between A and A in the monoidal category H MH . We will denote by A ! A the H-bimodule algebra A #R A, the cross product of A and A in H MH . Proof The fact that R is right H-linear is obvious, and the fact that it is left H-linear follows immediately from (8.2.3), so R is indeed a morphism in H MH . The relations (8.8.1) follow immediately from (8.2.2) and (8.7.5). The relation (8.8.2) reduces to (8.7.4), while the relation (8.8.3) reduces to (8.2.1). Remark 8.44 The explicit structure of A ! A is the following: the unit is 1A ⊗ 1A , the left H-action is h · (ϕ ⊗ a) = h1 · ϕ ⊗ h2 · a, the right H-action is (ϕ ⊗ a) · h = ϕ · h ⊗ a, and the multiplication is (by (8.8.4)): (ϕ ⊗ a)(ϕ  ⊗ a ) = (y1 X 1 · ϕ )(y2Y 1 (x1 X 2 · a)(−1) x2 X13 · ϕ  ) ⊗ (y31Y 2 · (x1 X 2 · a)(0) )(y32Y 3 x3 X23 · a ). We are mainly interested in the following particular case of Proposition 8.43. Corollary 8.45 Let H be a quasi-Hopf algebra with bijective antipode, A a left H-module algebra and H0 the algebra in H H YD as in Section 8.7. Define the map R : H0 ⊗ A → A ⊗ H0 , R(h⊗a) = h(−1) ·a⊗h(0) = X 1Y11 h1 g1 S(q2Y22 )Y 3 ·a⊗X 2Y21 h2 g2 S(X 3 q1Y12 ). (8.8.6) Then R is a twisting morphism between A and H0 in the monoidal category H M . We will denote by A  H0 the left H-module algebra A#R H0 , the cross product of A and H0 in H M . Its unit is 1A ⊗ β , the H-action is h · (a ⊗ h ) = h1 · a ⊗ h2  h , and the multiplication is (a ⊗ h)(a ⊗ h ) = (y1 X 1 · a)(y2Y 1 (x1 X 2  h)(−1) x2 X13 · a ) ⊗ (y31Y 2  (x1 X 2  h)(0) ) ◦ (y32Y 3 x3 X23  h ), where ◦ is the multiplication of H0 . More generally, if C is a left H-module algebra and A is an algebra in H H YD, then C ! A is a left H-module algebra, which will be denoted by C  A. Since the braiding c in H H YD is given by m ⊗ n → m(−1) · n ⊗ m(0) , we obtain the following:

8.8 Cross Products of Algebras in H M , H MH , H H YD

349

Corollary 8.46 Let H be a quasi-Hopf algebra with bijective antipode and A an algebra in H H YD. Then the left H-module algebra A  H0 is actually an algebra in H YD and it coincides with the c-tensor product algebra A ⊗ H in H YD. + 0 H H Lemma 8.47 Let H be a quasi-Hopf algebra and A a left H-module algebra. Consider the map j : H → A#H, j(h) = 1A #h, as well as the map i0 : A → A#H defined by (5.1.5). Then the following relation holds, for all h ∈ H and a ∈ A: j(h) ◦ i0 (a) = i0 (Y 1 X11 h1 g1 S(T 2 X22 )α T 3 X 3 · a) ◦ j(Y 2 X21 h2 g2 S(Y 3 T 1 X12 )), (8.8.7) where f −1 = g1 ⊗ g2 is given by (3.2.16) and ◦ is the multiplication in (A#H) j . Proof

We compute:

j(h) ◦ i0 (a) =

(1A #h) ◦ (x1 · a#x2 β S(x3 ))

=

(1A #X 1 h)(1A #S(y1 X 2 )α y2 X13 )(x1 · a#x2 β S(x3 ))(1A #S(y3 X23 ))

=

3 X11 h1 S(y1 X 2 )1 α1 y21 X(1,1) x1 · a 3 #X21 h2 S(y1 X 2 )2 α2 y22 X(1,2) x2 β S(y3 X23 x3 )

(3.1.7),(3.2.14)

=

(3.2.13) (3.2.1),(3.2.5)

=

X11 h1 g1 S(y12 X22 )γ 1 y21 x1 X13 · a 3 3 #X21 h2 g2 S(y11 X12 )γ 2 y22 x2 X(2,1) β S(X(2,2) )S(y3 x3 )

X11 h1 g1 S(T 2t21 y12 X22 )α T 3t 2 y21 x1 X 3 · a #X21 h2 g2 S(T 1t11 y11 X12 )α t 3 y22 x2 β S(y3 x3 )

(3.1.9),(3.1.7)

=

(3.2.1)

X11 h1 g1 S(T 2 X22 )α T 3 X 3 · a #X21 h2 g2 S(T 1 X12 )S(y1 )α y2 β S(y3 )

(3.2.2)

X11 h1 g1 S(T 2 X22 )α T 3 X 3 · a#X21 h2 g2 S(T 1 X12 )

(5.1.6)

i0 (Y 1 X11 h1 g1 S(T 2 X22 )α T 3 X 3 · a) ◦ j(Y 2 X21 h2 g2 S(Y 3 T 1 X12 )),

= =

finishing the proof. Proposition 8.48 Let H be a quasi-Hopf algebra with bijective antipode and A a left H-module algebra. Then the linear map Π : A  H0 → (A#H) j ,

Π(a ⊗ h) = x1 · a#x2 hS(x3 ), ∀ a ∈ A, h ∈ H,

is an isomorphism of left H-module algebras. Proof Note first that Π is bijective, with inverse given by Π−1 (a#h) = X 1 · a ⊗ X 2 hS(X 3 ). For proving that Π is a morphism of left H-module algebras, we will use the Universal Property of A  H0 as a cross product algebra in H M (see Proposition 2.9). We know that i0 : A → (A#H) j is a morphism of left H-module algebras, and it is easy to see that j : H0 → (A#H) j is also a morphism of left H-module algebras. Moreover, the relation (2.1.5) reduces in this case exactly to (8.8.7). We can thus use

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Yetter–Drinfeld Module Categories

the Universal Property of A  H0 , which provides a morphism w : A  H0 → (A#H) j of left H-module algebras, which is moreover given by w(a⊗h) = i0 (a)◦ j(h). Relation (5.1.6) shows that actually we have w = Π. If we assume, moreover, that A is an algebra in H H YD, in which case A  H0 beYD, then we intend to show that comes the c-tensor product algebra A ⊗+ H0 in H H YD in a natural way and that Π becomes an (A#H) j also becomes an algebra in H H H isomorphism of algebras in H YD. The next result is a generalization of the fact that H0 is an algebra in H H YD; the proof is similar to the one given for H0 and will be omitted. Proposition 8.49 If H is a quasi-Hopf algebra with bijective antipode, (B, λ , Φλ ) a left H-comodule algebra and v : H → B a morphism of left H-comodule algebras, then Bv becomes an algebra in H H YD with coaction Bv → H ⊗ Bv , b → X 1Y11 b[−1] g1 S(q2Y22 )Y 3 ⊗ v(X 2Y21 )b[0] v(g2 S(X 3 q1Y12 )), where f −1 = g1 ⊗ g2 is given by (3.2.16) and qR = q1 ⊗ q2 = Z 1 ⊗ S−1 (α Z 3 )Z 2 . Moreover, the map v : H0 → Bv is a morphism of algebras in H H YD. The next result is a part of Proposition 9.13 from Chapter 9, so its proof will be given there. Proposition 8.50 Let H be a quasi-bialgebra and A an algebra in H H YD, with coaction denoted by A → H ⊗ A, a → a(−1) ⊗ a(0) . Then (A#H, λ , Φλ ) is a left Hcomodule algebra, with structures:

λ : A#H → H ⊗ (A#H),

λ (a#h) = T 1 (t 1 · a)(−1)t 2 h1 ⊗ (T 2 · (t 1 · a)(0) #T 3t 3 h2 ),

Φλ = X 1 ⊗ X 2 ⊗ (1A #X 3 ) ∈ H ⊗ H ⊗ (A#H), for all a ∈ A and h ∈ H. As a consequence of these results, we obtain: Corollary 8.51 Let H be a quasi-Hopf algebra with bijective antipode and A an j H algebra in H H YD. Then (A#H) becomes an algebra in H YD, via the left H-coaction j j λ(A#H) j : (A#H) → H ⊗ (A#H) , given by 1 λ(A#H) j (a#h) = X 1Y11 T 1 (t 1 · a)(−1)t 2 h1 g1 S(q2Y22 )Y 3 ⊗ [X12Y(2,1) T 2 · (t 1 · a)(0) 1 #X22Y(2,2) T 3t 3 h2 g2 S(X 3 q1Y12 )],

and the map j : H0 → (A#H) j is a morphism of algebras in H H YD. Lemma 8.52 Let H be a quasi-Hopf algebra with bijective antipode and A an H j algebra in H H YD. Then i0 is a morphism of algebras in H YD from A to (A#H) . Proof We already know that i0 is a morphism of left H-module algebras from A to (A#H) j , so the only thing left to prove is that λ(A#H) j ◦ i0 = (IdH ⊗ i0 ) ◦ λA .

8.9 Notes

351

Now, by denoting pR = P1 ⊗ P2 another copy of pR , we see that (λ(A#H) j ◦ i0 )(a) =

λ(A#H) j (p1 · a#p2 )

=

X 1Y11 T 1 (t 1 p1 · a)(−1)t 2 p21 g1 S(q2Y22 )Y 3 1 1 ⊗[X12Y(2,1) T 2 · (t 1 p1 · a)(0) #X22Y(2,2) T 3t 3 p22 g2 S(X 3 q1Y12 )]

(3.2.25)

=

1 1 X 1Y11 T 1 (Z(1,1) p11 P1 · a)(−1) Z(1,2) p12 P2 S(q2Y22 Z 3 )Y 3 1 1 1 ⊗[X12Y(2,1) T 2 · (Z(1,1) p11 P1 · a)(0) #X22Y(2,2) T 3 Z21 p2 S(X 3 q1Y12 Z 2 )]

(8.2.3),(3.1.7)

=

1 1 X 1 T 1Y(1,1) Z(1,1) p11 (P1 · a)(−1) P2 S(q2Y22 Z 3 )Y 3 1 1 ⊗[X12 T 2Y(1,2) Z(1,2) p12 · (P1 · a)(0) #X22 T 3Y21 Z21 p2 S(X 3 q1Y12 Z 2 )]

(5.5.17)

=

X 1 T 1 q1(1,1,1) p11 (P1 · a)(−1) P2 S(q2 ) ⊗[X12 T 2 q1(1,1,2) p12 · (P1 · a)(0) #X22 T 3 q1(1,2) p2 S(q12 )S(X 3 )]

(3.2.21)

=

X 1 T 1 p11 q11 (P1 · a)(−1) P2 S(q2 ) ⊗[X12 T 2 p12 q12 · (P1 · a)(0) #X22 T 3 p2 S(X 3 )]

(8.2.3),(5.5.16)

=

X 1 y1 (q11 P1 · a)(−1) q12 P2 S(q2 ) ⊗[X12 y21 p1 · (q11 P1 · a)(0) #X22 y22 p2 S(X 3 y3 )]

(3.2.23)

=

a(−1) ⊗ (p1 · a(0) #p2 ) = ((IdH ⊗ i0 ) ◦ λA )(a),

finishing the proof. Theorem 8.53 Let H be a quasi-Hopf algebra with bijective antipode and A an j 1 2 3 algebra in H H YD. Then the map Π : A ⊗+ H0 → (A#H) , Π(a ⊗ h) = x · a#x hS(x ), H is an isomorphism of algebras in H YD. Proof We proved that j : H0 → (A#H) j and i0 : A → (A#H) j are morphisms of algebras in H H YD, and together with the commutation relation (8.8.7) this allows us to apply the Universal Property of the cross product algebra A ⊗+ H0 = A#R H0 in H the category H H YD, obtaining thus a morphism of algebras in H YD between A ⊗+ H0 and (A#H) j , say ω , which has to be given by ω (a ⊗ h) = i0 (a) ◦ j(h). It turns out that ω coincides with the map Π, finishing the proof.

8.9 Notes The center construction is due to Drinfeld (unpublished), Joyal and Street [120] and Majid [145]. The connection between the left and right center constructions has been observed in [51]. The category of Yetter–Drinfeld modules over a quasi-bialgebra H was introduced by Majid in [147] by computing the center of the monoidal category H M . The aim in [147] was to define the quantum double D(H) of a quasi-Hopf algebra H by an implicit Tannaka–Krein reconstruction procedure, in such a way

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Yetter–Drinfeld Module Categories

that D(H) M identifies as braided monoidal category with the category of Yetter– Drinfeld modules over H. This goal was achieved afterwards in [107, 108, 195, 64]; an explicit construction of the quantum double (as a diagonal crossed product) was given for the first time by Hausser and Nill in [107, 108], an isomorphic copy of it being constructed afterwards by Schauenburg in [195]. That in the finite-dimensional case Yetter–Drinfeld modules are modules over the quantum double algebra D(H) is from [108, 60]. The rigidity of the center is taken from [207, 195]; the rigid monoidal structure of the category of Yetter–Drinfeld modules over a quasi-Hopf algebra is taken from [51] as well as the suitable braided monoidal isomorphisms between the four types of categories of Yetter–Drinfeld modules. The content of Section 8.7 is taken from [54, 50], and that of Section 8.8 is taken from [8]. The quasi-Hopf algebra Dω (G) was introduced in [77] for physical reasons, in relation to the work of Dijkgraaf and Witten on topological field theories. In [9], Altschuler and Coste proved that Dω (G) is, moreover, ribbon and used it to construct invariants for knots and links. The quasi-Hopf algebra Dω (H) was introduced in [56] as a generalization of Dω (G). The quasi-Hopf algebra Hω∗ was introduced in [177], where it was also proved that the center of Hω∗ M is braided equivalent to Dω (H) M , implying that Dω (H) should be the quantum double of Hω∗ . The explicit isomorphism between Dω (H) and D(Hω∗ ) constructed in Theorem 8.34 is from [57].

9 Two-sided Two-cosided Hopf Modules

We introduce the category of two-sided two-cosided Hopf modules over a quasi-bialgebra H and show that it is braided monoidally equivalent to the category of Yetter–Drinfeld modules over H, provided that H is a quasi-Hopf algebra. We use this equivalence to obtain certain structure theorems for bicomodule algebras and bimodule coalgebras over H. Finally, we show that a Hopf algebra within this braided monoidal category identifies with a quasi-Hopf algebra with projection.

9.1 Two-sided Two-cosided Hopf Modules Throughout this section, H is a quasi-bialgebra or a quasi-Hopf algebra, A is an H-bicomodule algebra and C is an H-bimodule coalgebra. Recall that H MH is a monoidal category and that the underlying quasi-coalgebra structure of H provides a monoidal coalgebra structure for H in H MH . In Section 6.1 we defined the category H H MH as the category of right H-corepresentations in H MH . Similarly, we can define H M H as the category of H-bicomodules within M , and this leads to the notion H H H H of two-sided two-cosided Hopf module over H. We can generalize this last concept as follows. Definition 9.1 Let H be a quasi-bialgebra, (A, λ , ρ , Φλ , Φρ , Φλ ,ρ ) an H-bicomodule algebra and C an H-bimodule coalgebra. A two-sided two-cosided (H, A,C)Hopf module is a k-vector space M with the following additional structure: • M is an (H, A)-two-sided Hopf module, that is, M ∈ H MAH ; as usual, we denote H (m) = m the left H-action and the right A-action by ·, and we write ρM (0) ⊗ m(1) for the right H-coaction on m ∈ M; C : M → C ⊗ M, λ C (m) = m • we have a k-linear map λM {−1} ⊗ m{0} , called the left M C-coaction on M, such that, for all m ∈ M, ε (m{−1} )m{0} = m and C C C (m)) = (IdC ⊗ λM )(λM (m)) · Φλ ; Φ · (Δ ⊗ IdM )(λM

(9.1.1)

• M is a (C, H)-“bicomodule,” in the sense that, for all m ∈ M, C H H C Φ · (λ M ⊗ IdH )(ρM (m)) = (IdC ⊗ ρM )(λM (m)) · Φλ ,ρ ;

(9.1.2)

354

Two-sided Two-cosided Hopf Modules

• the following compatibility relations hold: C λM (h · m) = h1 · m{−1} ⊗ h2 · m{0} ,

(9.1.3)

C λM (m · u) =

(9.1.4)

m{−1} · u[−1] ⊗ m{0} · u[0] ,

for all h ∈ H, m ∈ M and u ∈ A. CMH H A

will be the category of two-sided two-cosided Hopf modules and maps preserving the actions by H and A and the coactions by H and C. H When C = A = H we have that H H MH is just the category of H-bicomodules within H MH , as we have already explained. We conclude this section by constructing a functor between the categories H MAH and CH MAH . This will allow us to construct two-sided two-cosided Hopf modules from right quasi-Hopf (H, A,C)-modules.

Proposition 9.2 Let H be a quasi-bialgebra, A an H-bicomodule algebra and C an H-bimodule coalgebra. Then for any right quasi-Hopf (H, A,C)-Hopf module M we can define on the k-vector space C ⊗ M a two-sided two-cosided (H, A,C)-Hopf module structure as follows: • C ⊗ M is an (H, A)-bimodule via the actions given for all h ∈ H, c ∈ C, m ∈ M and u ∈ A by h · (c ⊗ m) · u := h1 · c · u[−1] ⊗ h2 · m · u[0] ; • the right H-coaction on M is defined for all c ∈ C and m ∈ M by H ρC⊗M (c ⊗ m) := (x1 · c · Θ1 ⊗ x2 · m(0) · Θ2 ) ⊗ x3 m(1) Θ3 ;

• the left C-coaction on M is defined for all c ∈ C and m ∈ M by C λC⊗M (c ⊗ m) := X 1 · c1 · x˜1λ ⊗ (X 2 · c2 · x˜2λ ⊗ X 3 · m · x˜3λ ).

In this way we have a well-defined functor C ⊗ − : sends a morphism ϑ to IdC ⊗ ϑ . Proof

H H MA

→ CH MAH . This functor

This is a straightforward computation.

Example 9.3 Let H be a quasi-Hopf algebra, A an H-bicomodule algebra and C an H-bimodule coalgebra. Then C ⊗ V = C ⊗ A ⊗ C and C ⊗ U = C ⊗ C ⊗ A are isomorphic two-sided two-cosided (H, A,C)-Hopf modules, where V and U are the isomorphic quasi-Hopf (H, A)-bimodules considered in Examples 6.5. The monoidal structure of H MHH in Proposition 6.8 gives rise to a monoidal strucH ture on H H MH . The proof of the next result is immediate. H Proposition 9.4 Let H be a quasi-bialgebra and M, N two objects in H H MH . Then M ⊗H N is a two-sided two-cosided Hopf module over H with the structure given by (6.1.14), (6.1.15) and

λM⊗H N : M ⊗H N m ⊗H n → m{−1} n{−1} ⊗ m{0} ⊗H n{0} ∈ H ⊗ M ⊗H N, (9.1.5)

9.2 Two-sided Two-cosided Hopf Modules versus Yetter–Drinfeld Modules 355 H for all m ∈ M, n ∈ N and h, h ∈ H. In this way the category H H MH becomes a strict H monoidal category with unit object H = Hε , considered in H H MH with the structure as in Example 6.7, completed with λH = Δ.

9.2 Two-sided Two-cosided Hopf Modules versus Yetter–Drinfeld Modules Our goal is to prove that the functors defined in Theorem 6.16 restrict to a category equivalence between two-sided two-cosided Hopf modules and Yetter–Drinfeld modules. Roughly speaking, we can say that CH MAH is obtained from H MAH by “adding” a left C-coaction plus compatibilities. Thus the equivalence between the categories H H MA and MA should extend “somehow” to an equivalence between the category C M H and a category whose objects are right A-modules, a sort of left C-comodules H A plus a compatibility relation between these two structures. It turns out that the category we are looking for is just C YD(H)A , the category of generalized right-left Yetter–Drinfeld modules associated to the datum (H, A,C). It can be introduced as op,cop )Ccop . More explicitly: Aop YD(H Definition 9.5 Let H be a quasi-bialgebra, C an H-bimodule coalgebra and A an H-bicomodule algebra. A right-left (H, A,C)-Yetter–Drinfeld module is a right Amodule together with a k-linear map λM : M → C ⊗ M (we denote λM (m) = m{−1} ⊗ m{0} for all m ∈ M) such that the following relations hold: m{−1} · θ 1 ⊗ θ 3 · (m{0} · θ 2 ){−1} ⊗ (m{0} · θ 2 ){0} = x˜ρ2 · (m · x˜ρ1 ){−1}1 · x˜λ1 ⊗ x˜ρ3 · (m · x˜ρ1 ){−1}2 · x˜λ2 ⊗ (m · x˜ρ1 ){0} · x˜λ3 , (9.2.1)

ε (m{−1} )m{0} = m,

(9.2.2)

m{−1} · u[−1] ⊗ m{0} · u[0] = u1 · (m · u0 ){−1} ⊗ (m · u0 ){0} ,

(9.2.3)

for all m ∈ M and u ∈ A. C YD(H)A will be the category of right-left Yetter–Drinfeld modules and maps preserving the actions by A and coactions by C. In order to prove the claimed equivalence of categories we first need some lemmas. Lemma 9.6 Let H be a quasi-bialgebra, C an H-bimodule coalgebra and A an H-bicomodule algebra. We have a functor F : C YD(H)A → CH MAH , defined for any M ∈ C YD(H)A by F(M) = M ⊗ H, where the structure of M ⊗ H in CH MAH is: • M ⊗ H is an (H, A)-bimodule via (for all h, h ∈ H, m ∈ M and u ∈ A) h · (m ⊗ h ) · u = m · u0 ⊗ hh u1 ;

(9.2.4)

• for all m ∈ M and h ∈ H, the left C-coaction and the right H-coaction on M ⊗ H are given by C λM⊗H (m ⊗ h) = h1 X˜ρ2 · (m · X˜ρ1 ){−1} · θ 1 ⊗ (m · X˜ρ1 ){0} · θ 2 ⊗ h2 X˜ρ3 θ 3 , (9.2.5) H ρM⊗H (m ⊗ h) = m · X˜ρ1 ⊗ h1 X˜ρ2 ⊗ h2 X˜ρ3 .

(9.2.6)

356

Two-sided Two-cosided Hopf Modules

If χ : M → N is a morphism in C YD(H)A then F(χ ) = χ ⊗ IdH . Proof The functor F in the statement is the functor F defined in the proof of Theorem 6.16, restricted to the category of right-left Yetter–Drinfeld modules. To see this, compare (6.3.3), (6.3.4) to (9.2.4), (9.2.6). Now, we will check that (9.1.1) holds for our structures. For all m ∈ M and h ∈ H we have C Φ · (Δ ⊗ IdM⊗H )(λM⊗H (m ⊗ h)) · Φ−1 λ (9.2.5)

=

(9.2.4)

2 1 2 X 1 h(1,1) (X˜ ρ )1 · (m · X˜ ρ ){−1}1 · θ11 x˜1λ ⊗ X 2 h(1,2) (X˜ ρ )2 1 1 3 · (m · X˜ ρ ){−1}2 · θ21 x˜2λ ⊗ (m · X˜ ρ ){0} · θ 2 (x˜3λ )0 ⊗ X 3 h2 X˜ ρ θ 3 (x˜3λ )1

(3.1.7),(4.3.2)

=

(4.4.2)

2

1

1

h1 X˜ ρ (Y˜ ρ )1 x˜2ρ · (m · x˜1ρ (Y˜ ρ )0 x˜1ρ ){−1}1 · x˜1λ θ 1 3

2

1

1

2 ⊗ h(2,1) (X˜ ρ )1Y˜ ρ x˜3ρ · (m · X˜ ρ (Y˜ ρ )0 x˜1ρ ){−1}2 · x˜2λ θ[−1] θ 1

2

1

3

1

3

2 ⊗ (m · X˜ ρ (Y˜ ρ )0 x˜1ρ ){0} · x˜3λ θ[0] θ ⊗ h(2,2) X 3 (X˜ ρ )2Y˜ ρ θ 3 θ (9.2.1)

=

(9.2.3)

3

1

2 1 1 h1 X˜ ρ · (m · X˜ ρ ){−1} · (Y˜ ρ )[−1] θ θ 1 3

2

1 3 2 1 1 2 ⊗ h(2,1) (X˜ ρ )1Y˜ ρ θ · ((m · X˜ ρ ){0} · (Y˜ ρ )[0] θ ){−1} · θ[−1] θ 1

1

2

2

3

3

2 ⊗ ((m · X˜ ρ ){0} · (Y˜ ρ )[0] θ ){0} · θ[0] θ ⊗ h(2,2) X 3 (X˜ ρ )2Y˜ ρ θ 3 θ (9.2.3),(4.4.3)

=

(9.2.5)

3

C C (IdC ⊗ λM⊗H )(λM⊗H (m ⊗ h)),

as required. The relation (9.1.2) can be proved in a similar way. The relation (9.1.3) follows directly from the definitions, and the relation (9.1.4) follows from (4.3.1), (9.2.3), (4.4.1) and (9.2.4). The converse of the above result is also true if H is a quasi-Hopf algebra. Lemma 9.7 Let H be a quasi-Hopf algebra with bijective antipode, C an Hbimodule coalgebra and A an H-bicomodule algebra. Then we have a functor G : C M H → C YD(H) . If M is a two-sided two-cosided Hopf module then G(M) = A H A M co(H) , where M co(H) is viewed as a Yetter–Drinfeld module via the structure m ← u = S−1 (u1 ) · m · u0 ,

(9.2.7)

C λG(M) (m) := m{−1} ⊗ m{0} = x˜ρ3 S−1 ( f 2 (x˜ρ2 )2 p2 ) · m{−1} · (x˜ρ1 )[−1] θ 1

⊗S−1 ( f 1 (x˜ρ2 )1 p1 θ 3 ) · m{0} · (x˜ρ1 )[0] θ 2 , (9.2.8) where pR = p1 ⊗ p2 , f = f 1 ⊗ f 2 are the elements defined in (3.2.19), (3.2.15). Proof

H then M is a two-sided (H, A)-Hopf module, so it makes sense If M ∈ CH MA

to consider the set of coinvariants (of the first type) of M, M co(H) . From the proof of Theorem 6.16 we know that M co(H) is a right A-module via the action defined by (9.2.7). The most difficult part of the proof is to show that λ C is well defined, that is, λ C (M co(H) ) ⊆ C ⊗ M co(H) , and that (9.2.1) and (9.2.3) hold for our context.

9.2 Two-sided Two-cosided Hopf Modules versus Yetter–Drinfeld Modules 357 Write f = f 1 ⊗ f 2 = F 1 ⊗ F 2 and f −1 = g1 ⊗ g2 . For m ∈ M co(H) we compute:   x˜ρ3 S−1 ( f 2 (x˜ρ2 )2 p2 ) · m{−1} · (x˜ρ1 )[−1] θ 1 ⊗ S−1 ( f 1 (x˜ρ2 )1 p1 θ 3 ) · m{0} · (x˜ρ1 )[0] θ 2 (0)   ⊗ S−1 ( f 1 (x˜ρ2 )1 p1 θ 3 ) · m{0} · (x˜ρ1 )[0] θ 2 (1)

(6.1.6)

=

(3.2.13)

x˜3ρ S−1 ( f 2 (x˜2ρ )2 p2 ) · m{−1} · (x˜1ρ )[−1] θ 1 2 ⊗ S−1 (F 2 f21 (x˜2ρ )(1,2) p12 θ23 g2 ) · m{0}(0) · (x˜1ρ )[0]0 θ0 2 ⊗ S−1 (F 1 f11 (x˜2ρ )(1,1) p11 θ13 g1 ) · m{0}(1) · (x˜1ρ )[0]1 θ1

(9.1.2),(3.2.17) 3 −1 1 = x˜ρ S (F 2 f22 X 3 (x˜2ρ )2 p2 ) · m(0){−1} · (x˜1ρ )0[−1] θ θ 1 (4.4.1) 2 2 ⊗ S−1 (F 1 f12 X 2 (x˜2ρ )(1,2) p12 θ23 g2 ) · m(0){0} · (x˜1ρ )0[0] θ θ0 3

2 ⊗ S−1 ( f 1 X 1 (x˜2ρ )(1,1) p11 θ13 g1 ) · m(1) · (x˜1ρ )1 θ θ1 (6.3.1),(9.1.3)

=

(9.1.4),(3.2.13)

1

x˜3ρ S−1 ( f 2 q22 (X˜ ρ )(2,2) X 3 (x˜2ρ )2 p2 ) · m{−1} · (X˜ ρ (x˜1ρ )0 )[−1] θ θ 1 3

1

2

2 ⊗ S−1 ( f 1 q21 (X˜ ρ )(2,1) X 2 (x˜2ρ )(1,2) p12 θ23 g2 ) · m{0} · (X˜ ρ (x˜1ρ )0 )[0] θ θ0 3

1

3

2 ⊗ S−1 (q1 (X˜ ρ )1 X 1 (x˜2ρ )(1,1) p11 θ13 g1 )X˜ ρ (x˜1ρ )1 θ θ1 3

2

(4.4.3),(4.3.10) 3 3 −1 2 2 = x˜ρ y S ( f q2 ((x˜2ρ )2 y2 )(2,2) X 3 p2 ) · m{−1} · (x˜1ρ )[−1] θ 1 (3.1.7),(4.3.2) 3 1 3 ⊗ S−1 ( f 1 q21 ((x˜2ρ )2 y2 )(2,1) X 2 p12 θ(2,2) (Y˜ ρ )2 g2 ) · m{0} · (x˜1ρ )[0] θ 2Y˜ ρ 3 ⊗ S−1 (q1 ((x˜2ρ )2 y2 )1 X 1 p11 θ(2,1) (Y˜ ρ )1 g1 )(x˜2ρ )1 y1 θ13Y˜ ρ 3

2

(4.3.11),(3.1.7) 3 −1 2 2 = x˜ρ S ( f (x˜ρ )2Y 3 p2 ) · m{−1} · (x˜1ρ )[−1] θ 1 (5.5.16),(3.1.9) 3 1 ⊗ S−1 ( f 1 (x˜2ρ )1Y 2 y3 (p1 θ 3 )(2,2) (Y˜ ρ )2 g2 ) · m{0} · (x˜1ρ )[0] θ 2Y˜ ρ 3 2 ⊗ S−1 (α Y21 y2 (p1 θ 3 )(2,1) (Y˜ ρ )1 g1 )Y11 y1 (p1 θ 3 )1Y˜ ρ (3.1.7)

=

(3.2.1)

x˜3ρ S−1 ( f 2 (x˜2ρ )2 p2 ) · m{−1} · (x˜1ρ )[−1] θ 1 ⊗ S−1 (q2 (Y˜ ρ )2 g2 )   1 3 2 · S−1 ( f 1 (x˜2ρ )1 p1 θ 3 ) · m{0} · (x˜1ρ )[0] θ 2 · Y˜ ρ ⊗ S−1 (q1 (Y˜ ρ )1 g1 )Y˜ ρ , 3

C which shows that λG(M) is well defined. The relation in (9.2.1) follows from (9.2.8), (9.2.7) and the axioms of a two-sided Hopf module. Note that the desired relation comes out as     x˜2ρ · (m ← x˜1ρ ){−1} · x˜1λ ⊗ x˜3ρ · (m ← x˜1ρ ){−1} · x˜2λ ⊗ (m ← x˜1ρ ){0} ← x˜3λ 1

{−1}

=m

· θ ⊗ θ · (m 1

3

{−1}

2 2 {−1}

←θ )

⊗ (m{−1} ← θ 2 ){0} .

Finally, (9.2.3) follows from similar computations, left to the reader. We can prove now the main result of this section. Theorem 9.8 Let H be a quasi-Hopf algebra with bijective antipode, C an Hbimodule coalgebra and A an H-bicomodule algebra. Then the categories C YD(H)A and CH MAH are equivalent.

358

Two-sided Two-cosided Hopf Modules

Proof As we have already discussed, the functors F and G from Lemma 9.6 and Lemma 9.7 provide the inverse equivalences between the categories of two-sided (H, A)-Hopf modules and right A-modules presented in Theorem 6.16. So it suffices to check that for any M ∈ C YD(H)A the right A-module isomorphism

ξM : G(F(M)) = {m · q˜1ρ ⊗ q˜2ρ | m ∈ M} → M, ξM (m · q˜1ρ ⊗ q˜2ρ ) = m, ∀ m ∈ M, from the proof of Theorem 6.16 is left C-colinear, and that for all M ∈ CH MAH the isomorphism

ζM : M → M co(H) ⊗ H = F(G(M)), ζM (m) = E(m(0) ) ⊗ m(1) , ∀ m ∈ M from the same proof is also left C-colinear. We check first that ξM is left C-colinear. Note that (4.3.2) and (3.2.1) imply x˜1ρ ⊗ (x˜2ρ )1 p1 ⊗ x˜3ρ S−1 ((x˜2ρ )2 p2 ) = X˜ ρ ( p˜1ρ )0 ⊗ X˜ ρ ( p˜1ρ )1 ⊗ S−1 (X˜ ρ p˜2ρ ), (9.2.9) 1

2

3

where pR = p1 ⊗ p2 is as in (3.2.19). Now, if we denote by Q˜ 1ρ ⊗ Q˜ 2ρ another copy of q˜ρ then from (9.2.8), (9.2.5) and (9.2.4) we have C λG(F(M)) (m · q˜1ρ ⊗ q˜2ρ )

=

1

x˜3ρ S−1 ( f 2 (x˜2ρ )2 p2 )(q˜2ρ )1 X˜ ρ · (m · q˜1ρ X˜ ρ ){−1} · θ (x˜1ρ )[−1] θ 1 2

1

2 1 3 3 2 3 ⊗ (m · q˜1ρ X˜ ρ ){0} · θ (x˜1ρ )[0]0 θ0 ⊗ S−1 ( f 1 (x˜2ρ )1 p1 θ 3 )(q˜2ρ )2 X˜ ρ θ (x˜1ρ )[0]1 θ1 (4.4.1),(9.2.3) −1 2 3 2 2 1 = S ( f Y˜ ρ p˜ρ )(q˜2ρ )1 X˜ ρ (Y˜ ρ ( p˜1ρ )0 )0,1 (9.2.9) 1 1 1 1 1 · (m · q˜1ρ X˜ ρ (Y˜ ρ ( p˜1ρ )0 )0,0 ){−1} · θ θ 1 ⊗ (m · q˜1ρ X˜ ρ (Y˜ ρ ( p˜1ρ )0 )0,0 ){0} 2 2 3 3 2 3 1 · θ θ0 ⊗ S−1 ( f 1Y˜ ρ ( p˜1ρ )1 θ 3 )(q˜2ρ )2 X˜ ρ (Y˜ ρ ( p˜1ρ )0 )1 θ θ1 (4.3.1) 1 = S−1 ( p˜2ρ )q˜2ρ (Q˜ 1ρ ( p˜1ρ )0,0 )1 · (m · q˜1ρ (Q˜ 1ρ ( p˜1ρ )0,0 )0 ){−1} · θ θ 1 (4.3.15) 2 2 3 3 ⊗ (m · q˜1ρ (Q˜ 1ρ ( p˜1ρ )0,0 )0 ){0} · θ θ0 ⊗ S−1 (( p˜1ρ )1 θ 3 )Q˜ 2ρ ( p˜1ρ )0,1 θ θ1 (4.3.11) 1 2 2 = (Q˜ 1ρ )1 · (m · (Q˜ 1ρ )0 ){−1} · θ θ 1 ⊗ (m · (Q˜ 1ρ )0 ){0} · θ θ0 (4.3.13) 3 3 ⊗ S−1 (θ 3 )Q˜ 2ρ θ θ1 (9.2.3),(4.3.9) 1 2 2 1 1 3 2 3 3 = m{−1} · (X˜ ρ )[−1] θ θ 1 ⊗ m{0} · (X˜ ρ )[0] θ θ0 ⊗ S−1 (α X˜ ρ θ 3 )X˜ ρ θ θ1 (4.4.3),(3.2.1) = m{−1} ⊗ m{0} · q˜1ρ ⊗ q˜2ρ , (4.3.9) C C◦ for all m ∈ M. Having this explicit formula for λG(F(M)) one can easily see that λM C C ξM = (IdC ⊗ ξM ) ◦ λG(F(M)) , and this means that λG(F(M)) is left C-colinear. C H Now let M be an object of H MA . For simplicity we prove that ζM−1 is left CC is given for all colinear. From (9.2.5), (9.2.8) and (9.2.7) we obtain that λF(G(M)) co(H)

m∈M

and h ∈ H by

C λF(G(M)) (m ⊗ h) 1 2 1 1 = h1 X˜ ρ x˜3ρ S−1 ( f 2 (X˜ ρ )12 (x˜2ρ )2 p2 ) · m{−1} · (X˜ ρ )0[−1] (x˜1ρ )[−1] θ θ 1

9.2 Two-sided Two-cosided Hopf Modules versus Yetter–Drinfeld Modules 359 3 2 2 2 1 1 3 ) · m{0} · (X˜ ρ )0[0] (x˜1ρ )[0] θ θ0 ⊗ h2 X˜ ρ θ 3 , ⊗ S−1 ( f 1 (X˜ ρ )11 (x˜2ρ )1 p1 θ θ1

and since ζM−1 (m ⊗ h) = hS−1 ( p˜2ρ ) · m · p˜1ρ , after a straightforward computation we obtain for all m ∈ M co(H) and h ∈ H: C C (IdC ⊗ ζM−1 ) ◦ λF(G(M)) (m ⊗ h) = λM ◦ ζM−1 (m ⊗ h). C is left C-colinear. The above equality shows that λF(G(M))

Remark 6.17 suggests another equivalence between a category of two-sided twocosided Hopf modules and a category of Yetter–Drinfeld modules. As before, let H be a quasi-Hopf algebra with bijective antipode, A an H-bicomodule algebra and C an H-bimodule coalgebra. As we have already seen, Aop is an H op -bicomodule algebra and C is also an H op -bimodule coalgebra. As in Remark 6.17, we can define the op category of two-sided (A, H)-Hopf modules as being A MHH := H op MAHop , and then op the category CA MHH := CH op MAHop . By Theorem 9.8 we have an equivalence between C M H and C YD(H op ) op . The latter category will be denoted in what follows by A H A C YD(H). A Proposition 9.9

Consider the functors C A YD(H)

F G

C H A MH ,

defined as follows: • For M ∈ CA YD(H) we have F(M) = M ⊗ H ∈ CA MHH with structure u · (m ⊗ h) · h = u0 · m ⊗ u1 hh ,

(9.2.10)   C λM⊗H (m ⊗ h) = Θ1 · (x˜1ρ · m){−1} · x˜2ρ h1 ⊗ Θ2 · (x˜1ρ · m){0} ⊗ Θ3 x˜3ρ h2 , (9.2.11) H ρM⊗H (m ⊗ h) = (x˜1ρ · m ⊗ x˜2ρ h1 ) ⊗ x˜3ρ h2 ,

(9.2.12)

for all u ∈ A, h, h ∈ H and m ∈ M. If f : M → N is a morphisms in CA YD(H) then F( f ) = f ⊗ IdH . • If M ∈ CA MHH then G(M) = M co(H) , the set of alternative coinvariants of M, which is an object of CA YD(H) via the structure u → m = u0 · m · S(u1 ),

(9.2.13)

1 2 3 λ C co(H) (m) = Θ1 (X˜ ρ )[−1] · m{−1} · g1 S(q2 (X˜ ρ )2 )X˜ ρ M 1 2 ⊗Θ2 (X˜ ρ )[0] · m{0} · g2 S(Θ3 q1 (X˜ ρ )1 ),

(9.2.14)

for all u ∈ A and m ∈ M co(H) . On morphisms we have that G( f ) = f |Mco(H) , for any morphism f : M → N in CA MHH . Then F and G are inverse equivalence functors.

360

Two-sided Two-cosided Hopf Modules H H 9.3 The Categories H H MH and H YD

The goal of this section is to show that in the situation when C = A = H the pair H H (F, G) in Proposition 9.9 provides a monoidal equivalence between H H MH and H YD. In other words we want to show that the monoidal equivalence between H MHH and H H H H M in Section 6.4 yields a monoidal equivalence between H MH and H YD. Towards this end we first need some preparatory work. Lemma 9.10

In any quasi-Hopf algebra H with bijective antipode we have:

q1 Q11 z1 y11 ⊗ S(Q2 z3 y1(2,2) )y2 ⊗ S(q2 Q12 z2 y1(2,1) )y3 = X 1 ⊗ S(q2 x21 X22 )x2 X13 ⊗ S(q1 x11 X12 )α x3 X23 ,

(9.3.1)

where q1 ⊗ q2 = Q1 ⊗ Q2 are two copies of the element qR defined in (3.2.19). Proof

The formula (9.3.1) is a consequence of the following computation:

q1 Q11 z1 y11 ⊗ S(Q2 z3 y1(2,2) )y2 ⊗ S(q2 Q12 z2 y1(2,1) )y3 (3.1.7)

q1 (Q1 y11 )1 z1 ⊗ S(Q2 y12 z3 )y2 ⊗ S(q2 (Q1 y11 )2 z2 )y3

(6.5.1)

q1 X11 z1 ⊗ S(X 2 z3 )q˜1 X13 ⊗ S(q2 X21 z2 )q˜2 X23

= =

(3.2.19),(3.2.22)

=

Y 1 X11 z1 ⊗ S(Y13 X 2 z3 )q˜1 (Y23 X 3 )1 ⊗ S(Y 2 X21 z2 )α q˜2 (Y23 X 3 )2

(3.1.9)

Y 1 ⊗ S(X 2Y22 )q˜1 X13Y13 ⊗ S(X 1Y12 )α q˜2 X23Y23

(6.5.1)

Y 1 ⊗ S(q2 x21Y22 )x2Y13 ⊗ S(q1 x11Y12 )α x3Y23 ,

= =

as we stated. Theorem 9.11 If H is a quasi-Hopf algebra with bijective antipode then the cateH H gories H H MH and H YD are strong monoidally equivalent. Proof By Corollary 6.25, we have that the functors F and G that provide the equivH H alence between H H MH and H YD in Proposition 9.9 yield a monoidal equivalence H between H MH and H M . So, according to Proposition 1.31, it is enough to prove that the strong monoidal structure of G (considered as functor from H MHH to H M ) extends to a strong monoidal structure of G when it is considered as functor between H M H and H YD. In other words, it suffices to prove that φ 2,M,N defined in (6.4.16) H H H H is left H-colinear, for all M, N ∈ H H MH . H As before, for M ∈ H H MH we denote by M m → λM (m) = m{−1} ⊗m{0} ∈ H ⊗M its left H-coaction and by M m → ρM (m) = m(0) ⊗ m(1) ∈ M ⊗ H its right Hcoaction. With this notation, we have that, for all m ∈ M co(H) and n ∈ N co(H) ,

λ(M⊗

co(H) φ 2,M,N (m ⊗H n) H N) (6.4.16) = λ(M⊗ N)co(H) (q1 x11 · m · S(q2 x21 )x2 ⊗H n · S(x3 )) H (9.2.14) 1 1 1 1 = X Y1 (q x1 · m · S(q2 x21 )x2 ){−1} (n · S(x3 )){−1} g1 S(Q2Y22 )Y 3 ⊗ X 2Y21 · (q1 x11 · m · S(q2 x21 )x2 ){0} ⊗H (n · S(x3 )){0} · g2 S(X 3 Q1Y12 )

H H 9.3 The Categories H H MH and H YD (3.2.13)

=

(3.2.26)

361

1 X 1Y11 (q1 Q11 z1 y11 x11 )1 m{−1} G1 S(Q2 z3 y1(2,2) x(2,2) )y2 x12 n{−1} g1 S(Q22Y22 x23 )Y 3 1 ⊗ X 2Y21 (q1 Q11 z1 y11 x11 )2 · m{0} · G2 S(q2 Q12 z2 y1(2,1) x(2,1) )y3 x22

⊗ H n{0} · g2 S(X 3 Q1Y12 x13 ) (9.3.1)

=

1 1 X 1Y11 Z11 x(1,1) m{−1} G1 S(q2 y12 Z22 x(2,2) )y2 Z13 x12 n{−1} g1 S(Q2Y22 x23 )Y 3 1 1 ⊗ X 2Y21 Z21 x(1,2) · m{0} · G2 S(q1 y11 Z12 x(2,1) )α y3 Z23 x22

⊗ H n{0} · g2 S(X 3 Q1Y12 x13 ). On the other hand, we compute that (IdH ⊗ φ 2,M,N )λMco(H) ⊗N co(H) (m ⊗ n) (8.2.14)

=

X 1 (x1Y 1 → m)[−1] x2 (Y 2 → n)[−1]Y 3 ⊗ φ 2,M,N (X 2 → (x1Y 1 → m)[0] ⊗ X 3 x3 → (Y 2 → n)[0] )

(9.2.14)

=

X 1U 1 Z11 (x1Y 1 → m){−1} g1 S(q2 Z22 )Z 3 x2 T 1V11 (Y 2 → n){−1} G1 S(Q2V22 )

V 3Y 3 ⊗ φ 2,M,N X 2 → (U 2 Z21 · (x1Y 1 → m){0} · g2 S(U 3 q1 Z12 ))  ⊗ X 3 x3 → (T 2V21 · (Y 2 → n){0} · G2 S(T 3 Q1V12 ))

(9.2.13)

2 X 1U 1 Z11 (x1Y 1 )(1,1) m{−1} g1 S(q2 Z22 (x1Y 1 )(2,2) )Z 3 x2 T 1V11Y(1,1) n{−1} G1

=

(9.3.1)

2 S(Q2V22Y(2,2) )V 3Y 3 ⊗W 1 X12U 2 Z21 (x1Y 1 )(1,2) · m{0} · 2 g2 S(t 1W 2 X22U 3 q1 Z12 (x1Y 1 )(2,1) )α t 2W13 X13 T 2V21Y(1,2) 2 ⊗ H n{0} · G2 S(t 3W23 X23 x23 T 3 Q1V12Y(2,1) ) (3.1.9)

=

2 X 1U11 Z11 (x1Y 1 )(1,1) m{−1} g1 S(q2 Z22 (x1Y 1 )(2,2) )Z 3 x2 T 1V11Y(1,1) n{−1} 2 G1 S(Q2V22Y(2,2) )V 3Y 3 ⊗ X 2U21 Z21 (x1Y 1 )(1,2) · m{0} 2 · g2 S(t 1 X13U 2 q1 Z12 (x1Y 1 )(2,1) )α t 2 (X23U 3 )1 x13 T 2V21Y(1,2) 2 ⊗ H n{0} · G2 S(t 3 (X23U 3 )2 x23 T 3 Q1V12Y(2,1) )

(3.2.21)

=

(3.1.9)

1 X 1 Z11Y(1,1) m{−1} g1 S(q2 (x1 Z 2Y21 )2 )x2 Z13 T 1 (V11Y12 )1 n{−1} G1 1 S(Q2 (V 2Y22 )2 )V 3Y 3 ⊗ X 2 Z21Y(1,2) · m{0} · g2 S(t 1 q1 (x1 Z 2Y21 )1 )α t 2

(x3 Z23 )1 T 2 (V 1Y12 )2 ⊗H n{0} · G2 S(X 3t 3 (x3 Z23 )2 T 3 Q1 (V 2Y22 )1 ) (3.2.21),(3.1.7)

=

(3.1.9)

X 1 (Z 1Y11 )1 m{−1} g1 S(q2 (t 1 x1 Z 2Y21 )2 )t 2 (x2 Z13V 1Y12 )1 n{−1} G1 S(Q2 (V 2Y22 )2 )V 3Y 3 ⊗ X 2 (Z 1Y11 )2 · m{0} · g2 S(q1 (t 1 x1 Z 2Y21 )1 )α t 3 (x2 Z13V 1Y12 )2 ⊗H n{0} · G2 S(X 3 x3 Z23 Q1 (V 2Y22 )1 )

(3.1.9),(3.2.21)

=

(3.1.9)

X 1 (Z 1V 1 (Y11 x1 )1 )1 m{−1} g1 S(q2 (t 1 Z12V 2 (Y11 x1 )2 )2 )t 2 (Z22V 3 )1 (Y21 x2 )1 n{−1} G1 S(Q2 (Y 2 x3 )2 )Y 3 ⊗ X 2 (Z 1V 1 (Y11 x1 )1 )2 · m{0} · g2 S(q1 (t 1 Z12V 2 (Y11 x1 )2 )1 )α t 3 (Z23V 3 )2 (Y21 x2 )2 ⊗ H n{0} · G2 S(X 3 Z 3 Q1 (Y 2 x3 )1 )

(3.1.7)

=

(3.2.21),(3.2.1)

1 X 1Y11V11 x(1,1) m{−1} g1 S(q2 (t 1V 2 x21 )2 )t 2V13 x12 n{−1} G1 S(Q2Y22 x23 )Y 3

362

Two-sided Two-cosided Hopf Modules 1 ⊗ X 2Y21V21 x(1,2) · m{0} · g2 S(q1 (t 1V 2 x21 )1 )α t 3V23 x22

⊗ H n{0} · G2 S(X 3 Q1Y12 x13 ). Hence, by comparing the two computations performed above we get that φ 2,M,N is left H-colinear, as needed.

9.4 A Structure Theorem for Bicomodule Algebras We will make use of the ideas in Section 6.5 in order to give a structure theorem for H algebras within the strict monoidal category (H H MH , ⊗H , H). H We start with the description of an algebra in H H MH . H Lemma 9.12 For H a quasi-bialgebra, an algebra in H H MH is a 4-tuple (A, λ , ρ , i) consisting of a k-algebra A, a k-algebra morphism i : H → A, and k-linear maps λ : A → H ⊗ A and ρ : A → A ⊗ H such that the following conditions hold:

• λ (i(h)) = h1 ⊗ i(h2 ) and ρ (i(h)) = i(h1 ) ⊗ h2 , for all h ∈ H; • (A, λ , ρ , Φλ := X 1 ⊗X 2 ⊗i(X 3 ), Φρ := i(X 1 )⊗X 2 ⊗X 3 , Φλ ,ρ := X 1 ⊗i(X 2 )⊗X 3 ) is an H-bicomodule algebra, where Φ = X 1 ⊗ X 2 ⊗ X 3 is the reassociator of H. H H H H Proof Let A be an algebra in H H MH . Since the forgetful functors H MH → H MH H H H and H MH → H MH are strong monoidal we get that A, with the same H-bimodule structure, is both an algebra in H MHH and H H MH . Hence, by Lemma 6.26 and its lefthanded version, we deduce that A is a k-algebra and that there exist i : H → A an algebra morphism, and λ : A → H ⊗ A and ρ : A → A ⊗ H linear maps, such that

• λ (i(h) = h1 ⊗ i(h2 ) and ρ (i(h)) = i(h1 ) ⊗ h2 , for all h ∈ H; • (A, λ , Φλ := X 1 ⊗ X 2 ⊗ i(X 3 )) is a left H-comodule algebra and (A, ρ , Φρ := i(X 1 ) ⊗ X 2 ⊗ X 3 ) is a right H-comodule algebra. There is only one property of A that we have not yet explored: namely, the compatibility between the left and right H-coactions on A. More precisely, by (9.1.2) we have, for all u ∈ A, that X 1 u0[−1] ⊗ i(X 2 )u0[0] ⊗ X 3 u1 = u[−1] X 1 ⊗ u[0]0 i(X 2 ) ⊗ u[0]1 X 3 , and this means that (A, λ , ρ , Φλ := X 1 ⊗ X 2 ⊗ i(X 3 ), Φρ := i(X 1 ) ⊗ X 2 ⊗ X 3 , Φλ ,ρ := X 1 ⊗ i(X 2 ) ⊗ X 3 ) is an H-bicomodule algebra. The converse follows from Lemma 6.26 and its left-handed version. H Examples of algebras in H H MH are given by some smash product algebras.

Proposition 9.13 Let H be a quasi-bialgebra and A an algebra in H H YD, with coaction denoted by A → H ⊗ A, a → a[−1] ⊗ a[0] . Then (A#H, λ , ρ , j) is an algebra H in H H MH , where j : H → A#H is the canonical inclusion map and

λ : A#H → H ⊗ (A#H), λ (a#h) = T 1 (t 1 · a)[−1]t 2 h1 ⊗ (T 2 · (t 1 · a)[0] #T 3t 3 h2 ), ρ : A#H → (A#H) ⊗ H, ρ (a#h) = (x1 · a#x2 h1 ) ⊗ x3 h2 ,

H 9.5 The Structure of a Coalgebra in H H MH

363

for all a ∈ A and h ∈ H. Proof By Corollary 6.27 we know that (A, ρ , j) is a right H-comodule algebra. Also, since A is an algebra in H H YD and the functor F from the proof of Theorem H 9.11 is strong monoidal we deduce that F(A) = A ⊗ H is an algebra in H H MH . Firstly, H by (9.2.10), (9.2.11) and (9.2.12) we deduce that A ⊗ H is an object in H MHH with structure given by h · (a ⊗ h) · h = h1 · a ⊗ h2 hh ,

λ (a ⊗ h) = X 1 (x1 · a)[−1] x2 h1 ⊗ (X 2 · (x1 · a)[0] ⊗ X 3 x3 h2 ), ρ (a ⊗ h) = (x1 · a ⊗ x2 h1 ) ⊗ x3 h2 , for all a ∈ A and h, h , h ∈ H. Secondly, the multiplication on F(A) is given by (6.4.15)

(a ⊗ h)(b ⊗ h ) = ϕ2,A,A ((a ⊗ h) ⊗H (a ⊗ h )) = (x1 · a)(x2 h1 · a ) ⊗ x3 h2 h , for all a, a ∈ A and h, h ∈ H, and the unit is (IdH ⊗ 1A )ϕ0 (1H ) = 1H ⊗ 1A . In other H words, F(A) = A#H with the algebra structure in H H MH provided by j : H h → 1A #h ∈ A#H and λ , ρ as in the statement. The result below is a two-sided two-cosided version of Theorem 6.28. Theorem 9.14 For H a quasi-Hopf algebra with bijective antipode and (A, ρ , λ , i) H H an algebra in H H MH , there exists an algebra A in H YD such that A is isomorphic to H H A#H as algebras in H MH . Proof (A, ρ , i) is an algebra in H MHH . If A := Aco(H) then by Theorem 6.28 we know that A is a left H-module algebra and A ∼ = A#H, as algebras in H MHH . FurtherH H more, since A is actually an algebra in H MH and from the proof of Theorem 9.11 G is a strong monoidal functor, we get that A is an algebra in H H YD. Consequently, H via the structure described in Proposition 9.13. M A#H is an algebra in H H H Thus, the only thing that is left to check is the fact that the isomorphism χ in the proof of Theorem 6.28 intertwines the left H-coactions of A and A#H. But this H follows from Proposition 9.9, specialized for A = C = H, since any M ∈ H H MH deH composes as M coH ⊗ H in H H MM via an isomorphism, say χM , and χ = χA .

H 9.5 The Structure of a Coalgebra in H H MH

Theorem 6.31 does not say that, up to an isomorphism, a coalgebra in H MHH is some sort of smash product coalgebra. In other words, in the coalgebra case we cannot H produce a sort of dual result for Theorem 6.28. Nevertheless, when we pass to H H MH , because of the extra corner that we have in this case, this time we can characterize H coalgebras in H H MH as some sort of smash product coalgebras. To achieve this, we start by proving the following key result. In order to avoid any confusion, we denote by H M H the category of bimodules

364

Two-sided Two-cosided Hopf Modules

over a quasi-bialgebra H, endowed with the monoidal structure defined by the structure of H as in Lemma 4.13. Lemma 9.15

Let H be a quasi-bialgebra. Then the forgetful functor

H H H U :H H MH = (H MH , ⊗H , H) → H M H = (H MH , ⊗, k, a, l, r) H is opmonoidal under the structure given, for all M, N ∈ H H MH , by

ψ2,M,N : U (M ⊗H N) m ⊗H n → m(0) · n{−1} ⊗ m(1) · n(0) ∈ U (M) ⊗ U (N) and ψ0 = ε : U (H) = H → k. Proof

ψ2,M,N is well defined since, for all m ∈ M, h ∈ H and n ∈ N we have that ψ2,M,N (m · h ⊗H n) = (m · h)(0) · n{−1} ⊗ (m · h)(1) · n{0} = m(0) · h1 n{−1} ⊗ m(1) h2 · n{0} = m(0) · (h · n){−1} ⊗ m(1) · (h · n){0} = ψ2,M,N (m ⊗H h · n).

Also, it can be easily checked that ψ2,M,N is an H-bilinear map. We next show that ψ2 fulfills the corresponding relations in (1.3.2). Indeed, for H any M, N, P ∈ H H MH we have aU (M),U (N),U (P) (ψ2,M,N ⊗ IdU (P) )ψ2,M⊗H N,P (m ⊗H n ⊗h p) = aU (M),U (N),U (P) (ψ2,M,N ((m ⊗H n)(0) · p{−1} ) ⊗ (m ⊗H n)(1) · p{0} ) = aU (M),U (N),U (P) (ψ2,M,N (m(0) ⊗H n(0) · p{−1} ) ⊗ m(1) n(1) · p{0} ) = X 1 · m(0,0) · n(0){−1} p{−1}1 x1 ⊗ (X 2 m(0,1) · n(0){0} p{−1}2 x2 ⊗ X 3 m(1) n(1) · p{0} · x3 ) (6.1.2)

= m(0) · n{−1} X 1 p{−1}1 x1 ⊗ (m(1)1 · n{0}(0) · X 2 p{−1}2 x2

(9.1.2)

⊗ m(1)2 n{0}(1) X 3 · p{0} · x3 ) (9.1.1)

= m(0) · n{−1} p{−1} ⊗ (m(1)1 · n{0}(0) · p{0,−1} ⊗ m(1)2 n{0}(1) · p{0,0} )

= m(0) · n{−1} p{−1} ⊗ ((m(1) · n{0} )(0) · p{0,−1} ⊗ (m(1) · n{0} )(1) · p{0,0} ) = m(0) · (n ⊗H p){−1} ⊗ ψ2,N,P (m(1) · (n ⊗H p){0} ) = (IdU (M) ⊗ ψ2,N,P )ψ2,M,N⊗H P (m ⊗H n ⊗H p), for all m ∈ M, n ∈ N and p ∈ P, as required. We leave it to the reader to check that ψ makes the two corresponding square diagrams in (1.3.2) commutative. At this point we can prove one of the main results of this section. H Theorem 9.16 Let H be a quasi-bialgebra. Then giving a coalgebra in H H MH is equivalent to giving a pair (C, π ) consisting of an H-bimodule coalgebra C and an H-bimodule coalgebra morphism π : C → H.

H 9.5 The Structure of a Coalgebra in H H MH

365

H Proof Let (C, Δ, ε ) be a coalgebra in H H MH . The forgetful functor U in Lemma 9.15 is opmonoidal, so it follows that C is an H-bimodule coalgebra via the original H-bimodule structure, but with comultiplication ΔC and counit εC defined by

ψ2,C,C U (Δ) ΔC : C = U (C) −→ U (C ⊗H C) −→ U (C) ⊗ U (C) = C ⊗C U (ε ) ψ0 and εC : C = U (C) −→ U (H) −→ k. Explicitly, for all c ∈ C we have

ΔC (c) = c1 (0) · c2 {−1} ⊗ c1 (1) · c2 {0} and εC = εε : C → k,

(9.5.1)

where we denote Δ(c) := c1 ⊗ c2 . If we take π = ε : C → H, then π is a morphism in and so in particular H-bilinear. The left and right H-colinearity of π read

HM H, H H

Δ(π (c)) = c{−1} ⊗ π (c{0} ) = π (c(0) ) ⊗ c(1) , for all c ∈ C. These equalities allow us to compute that (π ⊗ π )ΔC (c) = π (c1 (0) · c2 {−1} ) ⊗ π (c1 (1) · c2 {0} ) = π (c1 (0) )c2 {−1} ⊗ c1 (1) π (c2 {0} ) = π (c1 )1 π (c2 )1 ⊗ π (c1 )2 π (c2 )2 = Δ(π (c1 )π (c2 )) = Δ(π (π (c1 )c2 )) = Δ(π (c)), for all c ∈ C, where we freely used that π is an H-bimodule morphism and the counit of Δ. Hence we have shown that C is a coalgebra in H M H , and that π : C → H is a morphism of coalgebras within H M H . Conversely, let (C, π ) be a pair consisting of an H-bimodule coalgebra C and an H-bimodule coalgebra morphism π : C → H. As above, denote by (ΔC , εC ) the coalH gebra structure of C in H M H . We claim that C becomes a coalgebra in H H MH via the original H-bimodule structure of C, H-coactions given by

λC : C c → c{−1} ⊗ c{0} := π (c1 ) ⊗ c2 ∈ H ⊗C ,

(9.5.2)

ρC : C c → c(0) ⊗ c(1) := c1 ⊗ π (c2 ) ∈ C ⊗ H,

(9.5.3)

for all c ∈ C, and coalgebra structure determined by Δ(c) = E(c1 ) ⊗H c2

and ε = π ,

(9.5.4)

for all c ∈ C, where E is the projection in (6.4.1) specialized for the object C, considered in H MHH with the above structure. H Indeed, the fact that C is an object in H H MH via its regular H-actions and Hcoactions (9.5.2), (9.5.3) follows easily from the defining properties of the pair (C, π ), H as well as the fact that π : C → H becomes a morphism in H H MH . The comultiplication Δ in (9.5.4) is an H-bimodule morphisms since (6.4.3)

Δ(h · c) = E(h1 · c1 ) ⊗H h2 · c2 = E(h1 · c1 ) · h2 ⊗H c2 = h · E(c1 ) ⊗H c2 = h · Δ(c)

366

Two-sided Two-cosided Hopf Modules (6.4.2)

and Δ(c ⊗ h) = E(c1 · h1 ) ⊗H c2 · h2 = E(c1 ) ⊗H c2 · h = Δ(c) · h, for all c ∈ C and h ∈ H. The computation

ρC⊗H C Δ(c) = E(c1 )(0) ⊗H c(2,1) ⊗ E(c1 )(1) π (c(2,2) ) (6.4.11)

E(x1 · E(c1 )) ⊗H c(2,1) ⊗ x3 π (c(2,2) )

(6.4.2)

=

E(x1 · c1 ) ⊗H x2 · c(2,1) ⊗ π (x3 · c(2,2) )

=

E(c(1,1) · x1 ) ⊗H c(1,2) · x2 ⊗ π (c2 · x3 )

=

(6.4.2)

=

E(c(1,1) ) ⊗H c(1,2) ⊗ π (c2 )

=

(Δ ⊗ IdH )ρC (c),

valid for all c ∈ C, shows that Δ in (9.5.4) is right H-colinear. It is also left H-colinear. To see this, observe that, for all c ∈ C, we have E(X 1 · c1 )1 · X 2 π (c2 ) ⊗ E(X 1 · c1 )2 · X 3 = q11 · c(1,1) · p1 ⊗ q12 · c(1,2) · p2 S(q2 π (c2 )).

(9.5.5)

Indeed, since q1 · c(1,1) ⊗ S(q2 π (c(1,2) ))π (c2 ) = c1 · X 1 ⊗ S(π (c(2,1) )X 2 )απ (c(2,2) )X 3 = c · q1 ⊗ S(q2 ), for all c ∈ C we compute that E(X 1 · c1 )1 · X 2 π (c2 ) ⊗ E(X 1 · c1 )2 · X 3 =

(q1 X11 · c(1,1) · β S(q2 X21 π (c(1,2) )))1 · X 2 π (c2 ) ⊗ (q1 X11 · c(1,1) · β S(q2 X21 π (c(1,2) )))2 · X 3

(3.2.13)

=

(3.2.14) (3.2.26)

=

1 (q1 X11 · c(1,1) )1 · δ 1 S(q22 X(2,2) π (c(1,2)2 )) f 1 X 2 π (c2 ) 1 ⊗ (q1 X11 · c(1,1) )2 · δ 2 S(q21 X(2,1) π (c(1,2)1 )) f 2 X 3

(q1 · (Q1 · c(1,1) )1 )1 · x11 δ 1 S(Q2 π (c(1,2) )x3 )π (c2 ) ⊗ (q1 · (Q1 · c(1,1) )1 )2 · x21 δ 2 S(q2 π ((Q1 · c(1,1) )2 )x2 )

(3.2.6)

=

(q1 · c1 )1 · Q1(1,1) p1 β S(Q2 ) ⊗ (q1 · c1 )2 · Q1(1,2) p2 S(q2 π (c2 )Q12 )

(3.2.21),(3.2.2) 1 = q1 · c(1,1) · p1 ⊗ q12 · c(1,2) · p2 S(q2 π (c2 )),

as desired. With the help of this relation we have that

λC⊗H C Δ(c) = λC⊗H C (E(c1 ) ⊗H c2 ) =

π (E(c1 )1 )π (c(2,1) ) ⊗ E(c1 )2 ⊗H c(2,2)

=

π (E(X 1 · c(1,1) · x1 )1 · π (X 2 · c(1,2) · x2 )) ⊗ E(X 1 · c(1,1) · x1 )2 ⊗H X 3 · c2 · x3

(6.4.2)

=

π (E(X 1 · c(1,1) )1 · X 2 π (c(1,2) )) ⊗ E(X 1 · c(1,1) )2 · X 3 ⊗H c2

=

π (q11 · (c1 )(1,1) · p1 ) ⊗ q12 · (c1 )(1,2) · p2 S(q2 π ((c1 )2 )) ⊗H c2

H 9.5 The Structure of a Coalgebra in H H MH

367

(3.2.23)

π (q11 x1 · (c1 )1 ) ⊗ q12 x2 · (c1 )(2,1) · β S(q2 x3 π ((c1 )(2,2) )) ⊗H c2

(5.5.17)

π (X 1 · c(1,1) ) ⊗ E(X 2 · c(1,2) ) ⊗H X 3 · c2

(6.4.2)

π (c1 ) ⊗ E(c(2,1) ) ⊗H c(2,2) = (IdH ⊗ Δ)λC (c),

= = =

for all c ∈ C, and therefore Δ in (9.5.4) is left H-colinear, as stated. So it remains to H show that Δ is coassociative in H H MH , and that ε is a counit for it. To this end, note that, for all c ∈ C, E(E(c)1 ) ⊗ E(c)2 = E(q11 · c(1,1) · (β S(q2 π (c2 )))1 ) ⊗ q12 · c(1,2) · (β S(q2 π (c2 )))2 (6.4.2)

= E(q11 · c(1,1) ) ⊗ q12 · c(1,2) · β S(q2 π (c2 )).

Therefore, we get that, for all c ∈ C, (Δ ⊗ IdC )Δ(c) =

E(E(c1 )1 ) ⊗H E(c1 )2 ⊗H c2

=

E(q11 · (c1 )(1,1) ) ⊗H q12 · (c1 )(1,2) · β S(q2 π ((c1 )2 )) ⊗H c2

(6.4.2)

E(q11 x1 · (c1 )1 ) ⊗H q12 x2 · (c1 )(2,1) · β S(q2 x3 · π ((c1 )(2,2) )) ⊗H c2

(5.5.17)

E(X 1 · c(1,1) ) ⊗H E(X 2 · c(1,2) ) ⊗H X 3 · c2

(6.4.2)

E(c1 ) ⊗H E(c(2,1) ) ⊗H c(2,2) = (IdC ⊗ Δ)Δ(c),

= = =

H that is, Δ is coassociative in H H MH , as required. Finally, π is a counit for Δ since (6.4.4)

E(c1 ) · π (c2 ) = E(c(0) ) · c(1) = c,

π (E(c1 )) · c2 = q1 π (c(1,1) )β S(q2 π (c(1,2) )) · c2 (3.2.2)

= X 1 β S(X 2 )α X 3 · εC (c1 )c2 = c, for all c ∈ C. One can check that the two correspondences defined above are inverses of each other, so we are done. Denote by H–BimCoalg(π ) the category whose objects are pairs (C, π ) consisting of an H-bimodule coalgebra C and an H-bimodule coalgebra morphism π : C → H. A morphism τ : (C, π ) → (C , π  ) in H–BimCoalg(π ) is a morphism of coalgebras H τ : C → C within H M H such that π  τ = π . Also, by Coalg(H H MH ) we denote the H H category of coalgebras and coalgebra morphisms within H MH . Corollary 9.17

H H–BimCoalg(π ) and Coalg(H H MH ) are isomorphic categories.

Proof By Theorem 9.16, the desired isomorphism is given by the functors T : H H H H–BimCoalg(π ) → Coalg(H H MH ) and V : Coalg(H MH ) → H–BimCoalg(π ) deH fined as follows. T sends (C, π ) to C, viewed as coalgebra in H H MH under the structure given by (9.5.2) and (9.5.4). T sends a morphism to itself. If (C, Δ, ε ) is a H coalgebra in H H MH then V (C) = C, considered as a coalgebra in H M H with the structure in (9.5.1). V acts as identity on morphisms. We leave the verification of all these details to the reader.

368

Two-sided Two-cosided Hopf Modules

Definition 9.18 For a coalgebra B in H H YD denote by B < H the k-vector space B ⊗ H endowed with the comultiplication Δ(b < h) = y1 X 1 · b1 < y2Y 1 (x1 X 2 · b2 )[−1] x2 X13 h1 ⊗ y31Y 2 · (x1 X 2 · b2 )[0] < y32Y 3 x3 X23 h2 ,

(9.5.6)

and counit ε (b < h) = εB (b)ε (h), for all b ∈ B and h ∈ H. As before, b → b[−1] ⊗ b[0] is the left coaction of H on B, ΔB (b) = b1 ⊗ b2 is the comultiplication of B in H H YD and εB is its counit. We call B < H the smash product coalgebra of B and H. We now have all the necessary ingredients for the proof of the main result of this section. In particular, the result says that a smash product coalgebra is indeed a coalgebra, but within H M H . Note that in the Hopf case we do not need the H-module structure on B, and that B < H is an ordinary k-coalgebra, too. Theorem 9.19 Let H be a quasi-Hopf algebra with bijective antipode, C an Hbimodule coalgebra and π : C → H an H-bimodule coalgebra morphism. Then there exists a coalgebra B in H H YD such that C " B < H as H-bimodule coalgebras. H Proof Consider C = T ((C, π )) as a coalgebra in H H MH with the structure given by (9.5.2) and (9.5.4). Then B = Cco(H) is a coalgebra in H H YD and C is isomorphic to H . The fact that C and B ⊗ H are isomorphic objects in M B ⊗ H as coalgebras in H H H H M H follows from the structure theorem for two-sided two-cosided Hopf modules H H H over H. That they are, moreover, isomorphic as coalgebras in H H MH is a consequence of a more general result. Namely, if the functors S : C → D and R : D → C define a monoidal category equivalence then RS (C) ∼ = C is a coalgebra isomorphism in C , for any coalgebra C within C , where RS (C) has the coalgebra structure provided by the monoidal structure of RS and the coalgebra structure of C. H The structure that makes B ⊗ H an object in H H MH is the one in (9.2.10)–(9.2.12), H H while the coalgebra structure of B ⊗ H in H MH is obtained from (6.6.2). With these H structures, T ((C, π )) and B ⊗ H are isomorphic as coalgebras in H H MH . By Corollary 9.17 we deduce that (C, π ) = V T (C) is isomorphic to V (B ⊗ H) as objects in H–BimCoalg(π ), and consequently as H-bimodule coalgebras. To end the proof it suffices to show that V (B ⊗ H) = (B < H, εB ⊗ IdH ). As a byproduct, we get that B < H is indeed a coalgebra in H M H , as claimed. The latest assertion follows from the following computation:

Δ(b ⊗ h)

(9.5.1)

(b ⊗ h)1(0) · (b ⊗ h)2{−1} ⊗ (b ⊗ h)1(1) · (b ⊗ h)2{−1}

(6.6.2)

(X 1 · b1 ⊗ 1H )(0) · (X 2 · b2 ⊗ X 3 h){−1}

= =

⊗ (X 1 b1 ⊗ 1H )(1) · (X 3 · b2 ⊗ X 3 h){0} (9.2.11)

=

(9.2.12) (9.2.10)

=

(y1 X 1 · b1 ⊗ y2 ) ·Y 1 (x1 X 2 · b2 )[−1] x2 X13 h1 ⊗ y3 · (Y 2 · (x1 X 2 · b2 )[0] ⊗Y 3 x3 X23 h2 ) (y1 X 1 · b1 ⊗ y2Y 1 (x1 X 2 · b2 )[−1] x2 X13 h1 )

H 9.6 A Braided Monoidal Structure on H H MH

369

⊗ (y31Y 2 · (x1 X 2 · b2 )[0] ⊗ y32Y 3 x3 X23 h2 ), valid for any b ∈ B and h ∈ H, and the fact that ε (b ⊗ h) = εε (b ⊗ h) = εB (b)ε (h). This finishes the proof of the theorem.

H 9.6 A Braided Monoidal Structure on H H MH H H We show that H H MH is braided monoidally equivalent to H YD. To this end, we apply Proposition 1.56 to the strong monoidal equivalence in Proposition 9.9. Recall that the category H H YD is braided via the braiding given by (8.2.15). H If H is a quasi-Hopf algebra with bijective antipode and M ∈ H H MH , then by EM : M → M we denote the projection to the space of coinvariants of M of the second type, defined by EM (m) = X 1 · m(0) · β S(X 2 m(1) )α X 3 = q1 · E M (m) · S(q2 ), for all m ∈ M. Here M m → ρM (m) := m(0) ⊗ m(1) ∈ M ⊗ H denotes the right coaction of H on M and qR = q1 ⊗ q2 . In what follows, we need the following property of E M .

Lemma 9.20 Let H be a quasi-Hopf algebra with bijective antipode and M a twosided two-cosided Hopf module over H. Then, for all m ∈ M, we have that m{−1} ⊗ E M (m{0} ) = X 1Y11 E M (m(0) ){−1} g1 S(q2Y22 )Y 3 m(1) ⊗ X 2Y21 · E M (m(0) ){0} · g2 S(X 3 q1Y12 ). Proof

(9.6.1)

H For all m ∈ M ∈ H H MH we have

X 1Y11 E M (m(0) ){−1} g1 S(q2Y22 )Y 3 m(1) ⊗ X 2Y21 · E M (m(0) ){0} · g2 S(X 3 q1Y12 ) (3.2.13)

=

X 1Y11 m(0,0){−1} β1 g1 S(q2Y22 m(0,1)2 )Y 3 m(1) ⊗ X 2Y21 · m(0,0){0} · β2 g2 S(X 3 q1Y12 m(0,1)1 )

(3.2.14)

=

X 1 m(0){−1} Y11 δ 1 S(q2 m(1)(1,2) Y22 )m(1)2 Y 3 ⊗ X 2 · m(0){0} ·Y21 δ 2 S(X 3 q1 m(1)(1,1) Y12 )

(3.2.21)

m{−1} X 1Y11 δ 1 S(q2Y22 )Y 3 ⊗ m{0}(0) · X 2Y21 δ 2 S(m{0}(1) X 3 q1Y12 )

(5.5.17)

m{−1} X 1 q1(1,1) x11 δ 1 S(q2 x3 ) ⊗ m{0}(0) · X 2 q1(1,2) x21 δ 2 S(m{0}(1) X 3 q12 x2 )

= =

(3.2.6),(3.1.7)

m{−1} q11 β S(q2 ) ⊗ m{0}(0) · q1(2,1) β S(m{0}(1) q1(2,2) )

(3.2.1),(3.2.2)

m{−1} ⊗ m{0}(0) · β S(m{0}(1) ) = m{−1} ⊗ E M (m{0} ),

= =

as needed. H We are now able to define a braiding for H H MH . H Theorem 9.21 If H is a quasi-Hopf algebra with bijective antipode then H H MH is a braided category with the braiding defined by

dM,N : M ⊗H N m ⊗H n → EN (m{−1} · n(0) ) ⊗H m{0} · n(1) ∈ N ⊗H M,

(9.6.2)

370

Two-sided Two-cosided Hopf Modules

H for all M, N ∈ H H MH . H Furthermore, if we consider H H MH as a braided category with the braiding d, then H M H is braided monoidally equivalent to H YD, where the braiding on H YD is c H H H H as in (8.2.15).

Proof

Let

H H YD

F G

H H H MH

be the strong monoidal equivalence functors de-

H fined in Proposition 9.9. We have that ν defined, for all M ∈ H H MH , by

ν M : M co(H) ⊗ H m ⊗ h → X 1 · m · S(X 2 )α X 3 h ∈ M,

(9.6.3)

is a natural monoidal isomorphism between F G and IdH M H , while ζ given, for all H H M∈H H YD, by

ζM : (M ⊗ H)co(H) m ⊗ h → ε (h)m ∈ M

(9.6.4)

is a natural monoidal isomorphism between G F and IdH YD . H

We show that (F , ν , ζ ) obeys the condition in (1.5.12), that is, for all M ∈ H H YD,

ν F (M) = F (ζM ) : (M ⊗ H)co(H) ⊗ H → M ⊗ H. To this end, by Remark 6.17 we have (M ⊗ H)co(H) = {p1 · m ⊗ p2 | m ∈ M}, where pR = p1 ⊗ p2 is the element defined in (3.2.19). Thus ζM (p1 · m ⊗ p2 ) = m, for all m ∈ M, and therefore F (ζM )((p1 · m ⊗ p2 ) ⊗ h) = m ⊗ h, for all m ∈ M and h ∈ H. If qR is the element in (3.2.19) we compute that

ν F (M) ((p1 · m ⊗ p2 ) ⊗ h)

(9.6.3),(3.2.19) 1

=

q · (p1 · m ⊗ p2 ) · S(q2 )h

(9.2.10)

q11 p1 · m ⊗ q12 p2 S(q2 )h

(3.2.23)

m ⊗ h,

= =

for all m ∈ M and h ∈ H. We conclude that ν F (M) = F (ζM ), as stated. It follows by Proposition 1.56 and Corollary 1.57 that the braiding c for H H YD can H such that F becomes a braided M be transferred along F to a braiding d on H H H monoidal equivalence. It only remains to show that d is as in (9.6.2). Using (1.5.13), we see that  ν −1 ⊗H ν −1 m ⊗H n M −→ N (E M (m(0) ) ⊗ m(1) ) ⊗H (E N (n(0) ) ⊗ n(1) ) dM,N = ϕ2,G (M),G (N)

−→

= (6.4.2)

=

(x1  E M (m(0) ) ⊗ x2 m(1)1  E N (n(0) )) ⊗ x3 m(1)2 n(1) (E M (x1 · m(0) ) ⊗ x2 m(1)1  E N (n(0) )) ⊗ x3 m(1)2 n(1) (E M (m(0,0) ) ⊗ m(0,1)  E N (n(0) )) ⊗ m(1) n(1)

cG (M),G (N) ⊗IdH

−→

(9.2.14)

=

(E M (m(0,0) )[−1] m(0,1)  E N (n(0) ) ⊗ E M (m(0,0) )[0] ) ⊗ m(1) n(1)

(X 1Y11 E M (m(0,0) ){−1} g1 S(q2Y22 )Y 3 m(0,1)  E N (n(0) ) ⊗ X 2Y21 · E M (m(0,0) ){0} · g2 S(X 3 q1Y12 )) ⊗ m(1) n(1)

H 9.7 Hopf Algebras within H H MH (9.6.1)

=

−1 ϕ2,G (N),G (M)

−→

(6.4.2)

=

371

(m(0){−1}  E N (n(0) ) ⊗ E N (m(0){0} )) ⊗ m(1) n(1) (X 1 m(0){−1}  E N (n(0) ) ⊗ 1H ) ⊗H (X 2  E N (m(0){0} ) ⊗ X 3 m(1) n(1) ) (m{−1}  E N (n(0) ) ⊗ 1H ) ⊗H (E M (m{0}(0) ) ⊗ m{0}(1) n(1) )

ν N ⊗ν M

−→ q1 · E N (m{−1} · n(0) ) · S(q2 ) ⊗H Q1 · E M (m{0}(0) ) · S(Q2 )m{0}(1) n(1) (3.2.19)

=

X 1 m{−1}1 · E N (n(0) ) · S(X 2 m{−1}2 )α ⊗ H X 3 Q1 · E M (m{0}(0) ) · S(Q2 )m{0}(1) n(1)

(3.2.21)

=

X 1 m{−1}1 · E N (n(0) ) · S(X 2 m{−1}2 )α ⊗ H Q1 · E M (X13 · m{0}(0) ) · S(Q2 )X23 m{0}(1) n(1)

(6.4.2)

m{−1} X 1 · E N (n(0) ) · S(m{0,−1} X 2 )α ⊗H EM (m{0,0}(0) ) · m{0,0}(1) X 3 n(1)

(6.4.4)

=

m{−1} X 1 · E N (n(0) ) · S(m{0,−1} X 2 )α ⊗H m{0,0} · X 3 n(1)

=

q1 · E N (m{−1} · n(0) ) · S(q2 ) ⊗H m{0} · n(1)  EN (m{−1} · n(0) ) ⊗H m{0} · n(1) ,

=

=

for all m ∈ M and n ∈ N, as desired.

H 9.7 Hopf Algebras within H H MH

Let H be a quasi-Hopf algebra with bijective antipode. The aim of this section is to H characterize the bialgebras and the Hopf algebras in H H MH . We show that giving a H H Hopf algebra in H MH is equivalent to giving a quasi-Hopf algebra projection for H. Later, we will obtain that quasi-Hopf algebra projections are characterized by the biproduct quasi-Hopf algebras, and therefore by Hopf algebras in H H YD, too. We denote by H–qBialgProj (resp. H–qHopfProj) the category whose objects are triples (A, i, π ) consisting of a quasi-bialgebra (resp. quasi-Hopf algebra) A and two quasi-bialgebra (resp. quasi-Hopf algebra) morphisms H

i

π

A such that

π i = IdH . A morphism in H–qBialgProj (resp. H–qHopfProj) between (A, i, π ) and (A , i , π  ) is a quasi-bialgebra (resp. quasi-Hopf algebra) morphism τ : A → A such that τ i = i and πτ = π  . The objects of H–qBialgProj (resp. H–qHopfProj) will be called quasi-bialgebra (resp. quasi-Hopf algebra) projections for H. H H H We also denote by Bialg(H H MH ) (resp. Hopf(H MH )) the category of bialgebras H (resp. Hopf algebras) and bialgebra morphisms within H H MH . H As announced, we next prove that the categories Bialg(H H MH ) and H–qBialgProj H H (resp. Hopf(H MH ) and H–qHopfProj) are isomorphic. We first need some lemmas. Lemma 9.22

H   Take M, N ∈ H H MH , and the elements m, m ∈ M and n, n ∈ N. Then

m ⊗H n = m ⊗H n ⇔ E(m(0) ) ⊗ m(1) · n = E(m(0) ) ⊗ m(1) · n .

(9.7.1)

372

Two-sided Two-cosided Hopf Modules

−1 Proof From Section 6.4 we know that νM : M m → EM (m(0) ) ⊗ m(1) ∈ M co(H) ⊗ H H H co(H) is the image of E , H is an isomorphism in H M H MH , for all M ∈ H MH . Here M a left H-module via the structure given by h¬EM (m) = EM (h · m), for all h ∈ H and m ∈ M. For a k-vector space U and V ∈ H M denote by ϒU,V : (U ⊗H)⊗H V → U ⊗V the canonical isomorphism. We then have that m ⊗H n = m ⊗H n if and only if −1 −1 ϒMco(H) ,N co(H) ⊗H (νM ⊗H νN−1 )(m ⊗H n) = ϒMco(H) ,N co(H) ⊗H (νM ⊗H νN−1 )(m ⊗H n ),

if and only if EM (m(0) ) ⊗ EN (m(1)1 · n(0) ) ⊗ m(1)2 n(1) = EM (m(0) ) ⊗ EN (m(1)1 · n(0) ) ⊗ m(1)2 n(1) . Thus, if m ⊗H n = m ⊗H n then EM (m(0) ) ⊗ EN (m(1)1 · n(0) ) · m(1)2 n(1) = EM (m(0) ) ⊗ EN (m(1)1 · n(0) ) · m(1)2 n(1) (6.4.3)

⇔ EM (m(0) ) ⊗ m(1) · EN (n(0) ) · n(1) = EM (m(0) ) ⊗ m(1) · EN (n(0) ) · n(1)

(6.4.4)

⇔ EM (m(0) ) ⊗ m(1) · n = EM (m(0) ) ⊗ m(1) · n .

The converse follows easily from (6.4.4), and we are done. Now we construct the functor that gives the desired isomorphism of categories. Proposition 9.23 there is a functor

Let H be a quasi-Hopf algebra with bijective antipode. Then H V : Bialg(H H MH ) → H–qBialgProj.

H On objects, V sends a bialgebra (B, mB , i : H → B, ΔB , π : B → H) in H H MH to the triple (B, i, π ), where B is considered as a quasi-bialgebra via mB := mB qB,B (where qB,B : B ⊗ B → B ⊗H B is the canonical surjection), 1B = i(1H ),

ΔB (b) = b1(0) · b2{−1} ⊗ b1(1) · b2{0}

and εB = επ : B → k

as in (9.5.1), and ΦB = (i ⊗ i ⊗ i)(Φ). V acts as identity on morphisms. Proof We must check that (B, mB , 1B , ΔB , εB , ΦB ) is indeed a quasi-bialgebra and, moreover, that i, π become quasi-bialgebra morphisms. H We know that (B, mB , i) is an algebra in H H MH if and only if (B, mB , 1B ) is a kalgebra and at the same time an H-bicomodule algebra via the original left and right H-coactions and reassociators Φλ = X 1 ⊗ X 2 ⊗ i(X 3 ), Φρ = i(X 1 ) ⊗ X 2 ⊗ X 3 and Φλ ,ρ = X 1 ⊗ i(X 2 ) ⊗ X 3 , such that, for all h ∈ H,

λ (i(h)) = h1 ⊗ i(h2 ) and ρ (i(h)) = i(h1 ) ⊗ h2 .

(9.7.2)

In other words, i is an H-bicomodule algebra morphism. Furthermore, the H-bimodule structure on B is nothing but the one induced by the restriction of scalars functor defined by i.

H 9.7 Hopf Algebras within H H MH

373

Similarly, we have that (B, ΔB , εB = επ ) is a coalgebra within the monoidal category H M H := (H MH , ⊗, k, a , l  , r ), that is, an H-bimodule coalgebra, and π : B → H is a coalgebra morphism in H M H . If we denote ΔB (b) = b1 ⊗ b2 we have i(X 1 )b(1,1) i(x1 ) ⊗ i(X 2 )b(1,2) i(x2 ) ⊗ i(X 3 )b2 i(x3 ) = b1 ⊗ b(2,1) ⊗ b(2,2) ,

(9.7.3)

for all b ∈ B, επ = εB and Δ(π (b)) = π (b1 ) ⊗ π (b2 ), for all b ∈ B. The left and right H-coactions on B can be recovered from ΔB and π as

λ (b) = π (b1 ) ⊗ b2

and ρ (b) = b1 ⊗ π (b2 ), ∀ b ∈ B.

(9.7.4)

H Since i is the unit and π is the counit of the bialgebra B within H H MH it follows that π i = IdH , and therefore π is surjective. Furthermore, π : B → H is an algebra H morphism in H H MH , so π is a k-algebra morphism as well. As we have seen, π intertwines the comultiplications ΔB and Δ of B and H, too. If we define ΦB := (i ⊗ i ⊗ i)(Φ), it is clear that (π ⊗ π ⊗ π )(ΦB ) = Φ. Combining (9.7.2) and (9.7.4) we get

λ (i(h)) = π (i(h)1 ) ⊗ i(h)2 = h1 ⊗ i(h2 ), ∀ h ∈ H, and therefore π (i(h)1 )⊗i(h)2 = π (i(h1 ))⊗i(h2 ), for all h ∈ H. As π is surjective, we obtain that i intertwines the comultiplications Δ and ΔB of H and B, and so ΔB (1B ) = ΔB (i(1H )) = i(1H ) ⊗ i(1H ) = 1B ⊗ 1B . It is also an algebra morphism such that (i ⊗ i ⊗ i)(Φ) = ΦB and εB i = ε . The most difficult part is to show that ΔB is multiplicative, that is, ΔB (bb ) = (b1(0) · b2{−1} )(b1(0) · b2{−1} ) ⊗ (b1(1) · b2{0} )(b1(1) · b2{0} ),

(9.7.5)

for all b, b ∈ B. Towards this end, observe first that by (9.6.2) and (9.7.1) we have H that ΔB is multiplicative in H H MH if and only if E((bb )1(0) ) ⊗ (bb )1(1) · (bb )2 = E(b1(0) E(b2{−1} · b1(0) )(0) ) ⊗ b1(1) E(b2{−1} · b1(0) )(1) · (b2{0} · b1(1) )b2 , (9.7.6) for all b, b ∈ B, where, for simplicity, from now on we denote EB by E. This allows us to compute that ΔB (bb ) =

(bb )1(0) · (bb )2{−1} ⊗ (bb )1(1) · (bb )2{0}

(6.4.4)

E((bb )1(0,0) ) · (bb )1(0,1) (bb )2{−1} ⊗ (bb )1(1) · (bb )2{0}

(6.4.2)

x1 ¬E((bb )1(0) ) · x2 ((bb )1(1) · (bb )2 ){−1} ⊗ x3 · ((bb )1(1) · (bb )2 ){0}

(9.7.6)

E(x1 · b1(0) E(b2{−1} · b1(0) )(0) ) · x2 b1(1) E(b2{−1} · b1(0) )(1)1

= = =

1

((b2{0} · b1(1) )b2 ){−1} ⊗ x3 b1(1) E(b2{−1} · b1(0) )(1)2 · ((b2{0} 2 (6.4.2) = E(b1(0,0) E(b2{−1} · b1(0) )(0,0) ) · b1(0,1) E(b2{−1} · b1(0) )(0,1)

· b1(1) )b2 ){0}

(b2{0} · b1(1) ){−1} b2{−1} ⊗ b1(1) E(b2{−1} · b1(0) )(1) · (b2{0} · b1(1) ){0} b2{0}

374

Two-sided Two-cosided Hopf Modules

(6.4.4)

=

b1(0) E(b2{−1} · b1(0) )(0) · b2{0,−1} b1(1) b2{−1} 1

⊗b1(1) E(b2{−1} · b1(0) )(1) · (b2{0,0} · b1(1) )b2{0} 2

(6.4.11)

b1(0) E(x1 b2{−1} · b1(0) ) · x2 b2{0,−1} b1(1) b2{−1} ⊗ (b1(1) x3 · b2{0,0} · b1(1) )b2{0}

(6.4.2)

b1(0) E(b2{−1} · b1(0,0) ) · b2{−1} b1(0,1) b2{−1} ⊗ b1(1) · (b2{0} · b1(1) )b2{0}

(6.4.3)

(b1(0) · b2{−1} )(E(b1(0,0) ) · b1(0,1) b2{−1} ) ⊗ (b1(1) · b2{0} )(b1(1) · b2{0} )

(6.4.4)

(b1(0) · b2{−1} )(b1(0) · b2{−1} ) ⊗ (b1(1) · b2{0} )(b1(1) · b2{0} ),

= = = =

1

1

2

2

for all b, b ∈ B, as needed. The remaining details are left to the reader. We can construct an inverse for V as follows. Proposition 9.24 Let H be a quasi-Hopf algebra and (B, i, π ) a quasi-bialgebra H projection for it. If the antipode of H is bijective then B is a bialgebra in H H MH with   the structure given, for all h, h ∈ H and b, b ∈ B, by h · b · h = i(h)bi(h );

λ : B b → π (b1 ) ⊗ b2 ∈ H ⊗ B, 

ρ : B b → b1 ⊗ π (b2 ) ∈ B ⊗ H; 

mB (b ⊗H b ) = bb , ΔB (b) = E(b1 ) ⊗H b2

i : H → B; and ε B = π .

(9.7.7) (9.7.8) (9.7.9) (9.7.10)

H In this way we have a well-defined functor T : H–qBialgProj → Bialg(H H MH ). T acts as identity on morphisms. H Proof It is easy to see that B is an object in H H MH with the structure as in (9.7.7) and (9.7.8). Since (b · h)b = b(h · b ), for all b, b ∈ B and h ∈ H, it follows that mB : B ⊗H B → B given by mB (b ⊗H b ) = bb , for all b, b ∈ B, is well defined. Thus H (B, mB , i) is an algebra in H H MH , since

λ (i(h)) = π (i(h)1 ) ⊗ i(h)2 = h1 ⊗ i(h2 ) and ρ (i(h)) = i(h)1 ⊗ π (i(h)2 ) = i(h1 ) ⊗ h2 , for all h ∈ H, that is, i is an H-bicomodule morphism, where the H-bicomodule structure of B is (B, λ , ρ , Φλ = X 1 ⊗ X 2 ⊗ i(X 3 ), Φρ = i(X 1 ) ⊗ X 2 ⊗ X 3 , Φλ ,ρ = X 1 ⊗ i(X 2 ) ⊗ X 3 ). We should point out that all these facts follow because i : H → B is a quasi-bialgebra morphism. H We proved that B is a coalgebra in H H MH with the structure in (9.7.10). It remains to show that ΔB is an algebra morphism, where the algebra structure on B ⊗H B is the d-tensor product algebra one, via the braiding d in (9.6.2). We compute ΔB i(h)

=

E(i(h)1 ) ⊗H i(h)2

=

q1 · i(h1 )(0) · β S(q2 i(h1 )(1) ) ⊗ i(h2 )

=

i(q1 h(1,1) β S(q2 h(1,2) )h2 ) ⊗H 1H

(3.2.21)

=

(3.2.2)

i(hq1 β S(q2 )) ⊗H 1H = i(h) ⊗H 1H ,

H 9.7 Hopf Algebras within H H MH

375

and this shows that, up to the identification given by the unit constraints of the monoidal category (H MH , ⊗H , H), ΔB i = i ⊗H i. Owing to (9.7.6) and (9.7.10), the fact that ΔB is multiplicative is equivalent to ΔB (bb )

E(b1 )E(b2{−1} · E(b1 )(0) ) ⊗H (b2{0} · E(b1 )(1) )b2

= (6.4.11)

E(b1 )E(π (b(2,1) ) · E(x1 · b1 ) · x2 ) ⊗H (b(2,2) · x3 )b2

(6.4.2)

E(X 1 · b(1,1) )E(X 2 π (b(1,2) ) · b1 ) · X 3 ⊗H b2 b2 ,

= =

for all b, b ∈ B. Since, for all b, b ∈ B, we have that E(X 1 · b1 )E(X 2 π (b2 ) · b ) · X 3 =

i(q1 X11 )b(1,1) i(β S(q2 X21 π (b(1,2) )Q1 X12 π (b(2,1) ))b1 i(β S(Q2 X22 π (b(2,2) )π (b2 ))X 3 )

(5.5.17)

=

i(q1 Q1(1,1) )(x1 · b1 )1 i(β S(q2 Q1(1,2) π ((x1 · b1 )2 ))Q12 π (x2 · b(2,1) )) b1 i(β S(Q2 π (x3 · b(2,2) )π (b2 ))

(3.2.21)

=

i(Q1 q1 )(b1 )(1,1) i(β S(q2 π ((b1 )(1,2) ))π ((b1 )2 ))b1 i(β S(Q2 π (b2 )π (b2 )))

=

i(Q1 )b1 i(q1 β S(q2 ))b1 i(β S(Q2 π (b2 b2 ))) = E(bb ),

(3.2.2)

it follows that ΔB is multiplicative if and only if E((bb )1 ) ⊗H (bb )2 = E(b1 b1 ) ⊗H b2 b2 , ∀ b, b ∈ B. The last equivalence is immediate since ΔB is multiplicative. At this point we can prove the main result of this section. Theorem 9.25 Let H be a quasi-Hopf algebra with bijective antipode. Then H Bialg(H H MH )

V T

H–qBialgProj

define an isomorphism of categories. They also produce an isomorphism of catH egories between Hopf(H H MH ) and H–qHopfProj. Proof One can check directly that V and T are inverse to each other; it is a straightforward computation left to the reader. Take (B, i, π ) ∈ H–qHopfProj, and denote by SB the antipode of B. We claim that H T ((B, i, π )) = B is a Hopf algebra in H H MH with antipode determined by S(b) = q1 π (b(1,1) )β · SB (q2 · b(1,2) ) · π (b2 ), ∀ b ∈ B. H A technical but straightforward computation ensures that S is a morphism in H H MH . 1 2 Then one can check that S(E(b)) = q π (b1 )β · SB (q · b2 ), for all b ∈ B, and this fact allows us to compute, for all b ∈ B, that

S(b1 )b2 = S(E(b1 ))b2 = i(π (X 1 · b(1,1) )β )SB (X 2 · b(1,2) )i(α )(X 3 · b2 )

376

Two-sided Two-cosided Hopf Modules = iπ (b)i(X 1 β S(X 2 )α X 3 ) = iπ (b),

as required. Similarly, one can see that i(S(π (b1 ))α )S(b2 ) = εB (b)i(α ), for all b ∈ B, and from here we get that b1 S(b2 ) = E(b1 )SB (b2 ) = i(π (X 1 · b(1,1) )β S(π (X 2 · b(1,2) ))α )S(X 3 · b2 ) = i(π (b1 )X 1 β S(π (b(2,1) )X 2 )α )S(b(2,2) )i(X 3 ) = εB (b2 )i(π (b1 )i(X 1 β S(X 2 )α X 3 ) = iπ (b), for all b ∈ B. Hence our claim is proved. In a similar manner one can prove that if S is the antipode for the bialgebra B in H M H then the quasi-bialgebra V (B) is actually a quasi-Hopf algebra with antipode H H determined by SB (b) = S(b(0){−1} p1 )α · S(b(0){0} ) · p2 S(b(1) ), ∀ b ∈ B,

(9.7.11)

and distinguished elements αB = i(α ) and βB = i(β ).

9.8 Biproduct Quasi-Hopf Algebras Let H be a quasi-Hopf algebra with bijecive antipode. We keep the same notation for the various functors that appear in the previous sections of this chapter. H We present a second characterization for bialgebras and Hopf algebras in H H MH . H H H By Theorem 9.21, the categories H MH and H YD are braided monoidally equivH alent. Therefore bialgebras (resp. Hopf algebras) in H H MH are in a one-to-one corH respondence to bialgebras (resp. Hopf algebras) in H YD. More precisely, if B is a H co(H) is a bialgebra (resp. Hopf albialgebra (resp. Hopf algebra) in H H MH then A := B H gebra) in H YD. The inverse of this correspondence associates to any bialgebra (resp. H H Hopf algebra) A in H H YD the bialgebra (resp. Hopf algebra) F (A) = A ⊗ H in H MH . H Thus, B and A ⊗ H are isomorphic as bialgebras (resp. Hopf algebras) in H H MH . Consequently, V (B) and V (A ⊗ H) are isomorphic as objects in H–qBialgProj (resp. H–qHopfProj). H Firstly, A ⊗ H is an object in H H MH with the structure as in (9.2.10)–(9.2.12). As an algebra V (A ⊗ H) = A#H, the smash product algebra of A and H. Recall that the multiplication of A#H is given by (a#h)(a #h ) = (x1 · a)(x2 h1 · a )#x3 h2 h ,

(9.8.1)

for all a, a ∈ A and h, h ∈ H, and its unit is 1A ⊗ 1H . This contributes to the structure of V (A ⊗ H) with j : H h → 1A ⊗ h ∈ A#H, so far an H-bicomodule algebra morphism, provided that A is an algebra in H H YD. Secondly, as a coalgebra V (A ⊗ H) = A < H, the smash product coalgebra of A

9.8 Biproduct Quasi-Hopf Algebras

377

and H. More precisely, the comultiplication is defined by Δ(a < h) = (y1 X 1 · a1 < y2Y 1 (x1 X 2 · a2 )[−1] x2 X13 h1 ) ⊗ (y31Y 2 · (x1 X 2 · a2 )[0] < y32Y 3 x3 X23 h2 ),

(9.8.2)

and the counit is ε (a ⊗ h) = εA (a)ε (h), for all a ∈ A and h ∈ H. This contributes to the structure of V (A ⊗ H) with p : A < H a < h → εA (a)h ∈ H, so far an Hbimodule coalgebra morphism, provided that A is a coalgebra in H H YD. As before, a → a[−1] ⊗ a[0] is the left coaction of H on A, ΔA (a) = a1 ⊗ a2 is the comultiplication of A in H H YD and εA is its counit. Summing up, we denote V (F (A)) = (A × H, j, p), where A × H is the k-vector space A ⊗ H endowed with multiplication and comultiplication defined by (9.8.1) and (9.8.2). Proposition 9.26 Let H be a quasi-Hopf algebra with bijective antipode and B an H object of H H YD which is at the same time an algebra and a coalgebra in H YD. Then the smash product algebra and the smash product coalgebra afford a quasi-bialgebra (resp. quasi-Hopf algebra) structure on A ⊗ H if and only if A is a bialgebra (resp. Hopf algebra) in H H YD. Proof Everything follows from the above comments and the fact that F : H H YD → H M H is a braided monoidal equivalence, and that T , V are inverse isomorphism H H functors. Note that the antipode s of the quasi-Hopf algebra A × H can be obtained from the antipode SA of A in H H YD and the antipode S of H as follows. The antipode S of H is F (S ) = S ⊗ Id , and so we have that M F (A) in H H A A H H s(a × h)

(9.7.11)

S((a × h)(0){−1} p1 )α · S((a × h)(0){0} ) · p2 S((a × h)(1) )

(9.2.12)

S((x1 · a × x2 h1 ){−1} p1 )α · S((x1 · a × x2 h1 ){0} ) · p2 S(x3 h2 )

= =

(9.2.10),(9.2.11)

=

S(X 1 (y1 x1 · a)[−1] y2 x12 h(1,1) p1 )α · S(X 2 · (y1 x1 · a)[0] × X 3 y3 x22 h(1,2) p2 S(x3 h2 ))

(3.2.21),(5.5.16)

=

S(X 1 (p11 · a)[−1] p12 h)α · S(X 2 · (p11 · a)[0] × X 3 p2 )

(8.2.3)

S(X 1 p11 a[−1] h)α · (SA (X 2 p12 · a[0] ) × X 3 p2 )

(9.2.10)

(1A × S(X 1 p11 a[−1] h)α )(X 2 p12 · SA (a[0] ) × X 3 p2 ),

= =

for all a ∈ A and h ∈ H. Clearly, the distinguished elements that together with s define the antipode for A × H are j(α ) = 1A × α and j(β ) = 1A × β . In the above computation we wrote a × h in place of a ⊗ h in order to distinguish the quasi-bialgebra structure on A ⊗ H defined above. Definition 9.27 V (F (A)) = (A×H, j, p) will be called in what follows the biproduct quasi-bialgebra (resp. quasi-Hopf algebra) between a bialgebra (resp. Hopf algebra) A in H H YD and H.

378

Two-sided Two-cosided Hopf Modules

Remark 9.28 The formulas (9.8.1) and (9.8.2) define a quasi-bialgebra structure on A ⊗ H even if H is only a quasi-bialgebra, not necessarily a quasi-Hopf algebra. Collecting the results proved so far we get the following. Theorem 9.29 Let H be a quasi-Hopf algebra with bijective antipode. Then there is a one-to-one correspondence between: • • • •

H bialgebras (resp. Hopf algebras) in H H MH ; quasi-bialgebra (resp. quasi-Hopf algebra) projections for H; bialgebras (resp. Hopf algebras) in H H YD; biproduct quasi-bialgebra (resp. quasi-Hopf algebra) structures for H.

We end this section by studying the invariance under twisting of the biproduct. If F = F 1 ⊗ F 2 is a gauge transformation for a quasi-bialgebra or quasi-Hopf algebra H then there is a monoidal isomorphism between the tensor categories H M and HF M ; see Proposition 3.5. This functor is the identity on objects and morphisms with the monoidal structure given by multiplication by F −1 (the inverse of F), that is, for any two left H-modules V,W , ϕ2,V,W : V ⊗W → V ⊗W , ϕ2,V,W (v ⊗ w) = G1 · v ⊗ G2 · w, where v ∈ V , w ∈ W and F −1 = G1 ⊗ G2 . Moreover, this functor induces HF a monoidal isomorphism between the pre-braided categories H H YD and HF YD as follows: it is the identity on objects and morphisms and if M ∈ H H YD with h ⊗ m → F h · m and λM (m) = m(−1) ⊗ m(0) , h ∈ H, m ∈ M, then M becomes an object in H HF YD with the same H-action and coaction given by:

λMF (m) = F 1 (G1 · m)(−1) G2 ⊗ F 2 · (G1 · m)(0) , ∀ m ∈ M.

(9.8.3)

The above assertion follows easily from the quasi-bialgebra (quasi-Hopf algebra) structure of HF and Definition 8.7; the details are left to the reader. Under this isomorphism a (co)algebra, bialgebra, Hopf algebra object B corresponds to a (co)algebra etc. object BF . Note that if B is a bialgebra (or Hopf algebra) in the first category then BF is a bialgebra (Hopf algebra) in the second category with the HF -coaction (9.8.3), multiplication and comultiplication as follows: b  b = (G1 · b)(G2 · b ) ,

˜ Δ(b) = F 1 · b1 ⊗ F 2 · b2 ,

(9.8.4)

and with the same unit and counit as B, where Δ(b) = b1 ⊗ b2 is the comultiplication of B in the first category (with the same antipode as B in case that B is a Hopf algebra). Therefore, we have two biproducts B × H and BF × HF which are quasibialgebras (quasi-Hopf algebras). We will prove that these biproducts are isomorphic in the following sense: Theorem 9.30 Let H be a quasi-bialgebra, F a gauge transformation for H and B a 1 2 bialgebra in H H YD. If we denote by F = 1B × F ⊗ 1B × F the gauge transformation for B × H induced by F, then the map ν : (B × H)F → BF × HF given by

ν (b × h) = F 1 · b × F 2 h , ∀ b ∈ B, h ∈ H

(9.8.5)

9.9 Notes

379

is a quasi-bialgebra isomorphism. Moreover, if H is a quasi-Hopf algebra and B is a Hopf algebra in H H YD then ν is a quasi-Hopf algebra isomorphism. Proof We only describe the structures involved and leave the verification of some details to the reader. Because the biproduct considered as an algebra is a smash product and the new quasi-bialgebra HF has the same algebra structure as H, we know by Proposition 5.10 that ν defined above is an algebra isomorphism with inverse given by:

ν −1 (b × h) = G1 · b × G2 h , ∀ b ∈ B, h ∈ H. Because ν (1B × h) = 1B × h, for all h ∈ H, it follows that (ν ⊗ ν ⊗ ν )(Φ(B×H)F ) = ΦBF ×HF . Thus, for the first statement, we only have to show that ν respects the comultiplications. This follows from the quasi-bialgebra structures of a biproduct and of a twisted quasi-bialgebra. Now, for the second assertion observe that the elements α , β for (B × H)F , denoted by αF and βF , respectively, are in fact αF = 1B × αF , βF = 1B × βF , so ν (αF ) = αBF ×HF and ν (βF ) = βBH ×HF . Thus the proof will be complete once we show that sBF ×HF ◦ ν = ν ◦ s, because the antipode for (B × H)F is the same as the antipode for B × H, say s. This equation follows mostly from (1B × h)(b × h ) = F 1 h1 G1 · b × F 2 h2 G2 h , ∀ b ∈ B and h, h ∈ H, which holds in the smash product BF × HF .

9.9 Notes Two-sided two-cosided Hopf modules over a Hopf algebra H were introduced by Woronowicz [220] under the name of bicovariant bimodules, as a tool in the study of non-commutative differential calculus on quantum groups. He also extended the structure theorem for Hopf modules to the category of Hopf bimodules H MHH and H two-sided two-cosided Hopf modules H H MH over H. Later on, Schauenburg proved in [194] that the structure theorems provide the classification of Hopf bimodules and two-sided two-cosided Hopf modules in the form of category equivalences H MHH ∼ = H M H ∼ H YD. These equivalences are even monoidal and they can be M and = H H H H regarded as a coordinate-free version of the classifications in [220]. Using categorical techniques, Schauenburg [195] also proved that all the results mentioned above remain valid in the setting provided by quasi-Hopf algebras. The content of this chapter is based on the following papers. The connection between the categories of Yetter–Drinfeld modules and two-sided two-cosided Hopf modules is from [63, 195], the monoidal equivalence between them is from [195, 44], while the fact that they are, moreover, equivalent as braided monoidal categories is taken from [45]. The content of Section 9.4 is from [44, 75], while that of Section 9.5 is from [44].

380

Two-sided Two-cosided Hopf Modules

The structure of a Hopf algebra with a projection is due to Radford [186]. A second characterization of Hopf algebras with a projection is due to Bespalov and Drabant [31], where Hopf algebras with a projection are identified with Hopf algebras within H M H . Their techniques were adapted to quasi-Hopf algebras in [45], which was our H H source of inspiration for Section 9.7. Note that the comultiplication on B ⊗ H defined in (9.5.6) and its counit appeared for the first time in [54] as the quasi-coalgebra part of the Radford biproduct construction for quasi-Hopf algebras. At that time there was no clue how to introduce a smash product coalgebra, and by hard computations it was proved in [54] that the comultiplication of the biproduct is coassociative up to conjugation by an invertible element. At this point it is clear that this quasi-coassociativity of the biproduct is nothing but a reformulation of the fact that B < H is a coalgebra in H M H , assuming that B is a coalgebra in H H YD. A proof for the assertion in Remark 9.28 can be found in [54].

10 Quasitriangular Quasi-Hopf Algebras

By using categorical tools, we introduce the concept of a quasitriangular (QT for short) quasibialgebra. For QT quasi-Hopf algebras we show that the square of the antipode is an inner automorphism, and therefore bijective. We uncover the QT structure of the quantum double D(H) of a finite-dimensional quasi-Hopf algebra H, and characterize D(H) as a biproduct quasi-Hopf algebra in the case when H itself is QT.

10.1 Quasitriangular Quasi-bialgebras and Quasi-Hopf Algebras We introduced the notion of quasi-bialgebra by investigating when the forgetful functor F to the category of vector spaces is quasi-monoidal. This led to the monoidal structure on the category of representations over a quasi-bialgebra, say H. We now go further: we will investigate when H M is a braided category. Proposition 10.1 Let H be a quasi-bialgebra, so H M is a monoidal category. Then H M is braided if and only if there exists an invertible element R = R1 ⊗ R2 = r1 ⊗ r2 ∈ H ⊗ H (formal notation, summation implicitly understood) such that the following relations hold: (Δ ⊗ IdH )(R) = X 2 R1 x1Y 1 ⊗ X 3 x3 r1Y 2 ⊗ X 1 R2 x2 r2Y 3 ,

(10.1.1)

(IdH ⊗ Δ)(R) = x R X r y ⊗ x X r y ⊗ x R X y ,

(10.1.2)

3 1

Δ

cop

2 1 1

1

1 2 2

2 2

3 3

(h)R = RΔ(h), ∀ h ∈ H.

(10.1.3)

Proof Suppose that H M is a braided category. Let c be a braiding for H M and regard H ∈ H M via its multiplication. If x ∈ X ∈ H M we define ϕx : H → X by ϕx (h) = h · x, for all h ∈ H, a left H-linear morphism. Since c is a natural transformation we have cX,Y (ϕx ⊗ ϕy ) = (ϕy ⊗ ϕx )cH,H , for all x ∈ X ∈ H M and y ∈ Y ∈ H M . By evaluating both sides of the above equality on 1H ⊗ 1H we obtain that cX,Y (x ⊗ y) = (ϕy ⊗ ϕx )cH,H (1H ⊗ 1H ). Thus cH,H (1H ⊗ 1H ) := R2 ⊗ R1 ∈ H ⊗ H determines completely the braiding c since cX,Y (x ⊗ y) = R2 · y ⊗ R1 · x, ∀ x ∈ X ∈ H M , y ∈ Y ∈ H M .

(10.1.4)

382

Quasitriangular Quasi-Hopf Algebras

Now, it can be easily checked that cX,Y is a morphism in H M if and only if (10.1.3) holds, (1.5.2) is equivalent to (10.1.1) and (1.5.1) is equivalent to (10.1.2). For instance, (10.1.3) is the consequence of the following facts: for all x ∈ X ∈ H M and y ∈ Y ∈ H M we have cX,Y (h · (x ⊗ y)) = cX,Y (h1 · x ⊗ h2 · y) = R2 h2 · y ⊗ R1 h1 · x, h · cX,Y (x ⊗ y) = h · (R2 · y ⊗ R1 · x) = h1 R2 · y ⊗ h2 R1 · x, so cX,Y is a morphism in H M if and only (10.1.3) is fulfilled (for the direct implication take x = y = 1H ∈ X = Y = H ∈ H M ). Since c is a natural isomorphism it follows that R := R1 ⊗ R2 is an invertible element in H ⊗ H with inverse R−1 = c−1 H,H (1H ⊗ 1H ). Conversely, if there is an R as in the statement then c defined by (10.1.4) is a braiding for H M , we leave the verification of this fact to the reader. Definition 10.2 A quasi-bialgebra or a quasi-Hopf algebra H is called quasitriangular (QT for short) if there exists an invertible element R ∈ H ⊗ H obeying (10.1.1), (10.1.2) and (10.1.3) (such an element is called an R-matrix). In other words, H is quasitriangular if and only if H M has a braided structure such that the forgetful functor F : H M → k M is quasi-monoidal. When we refer to a QT quasi-bialgebra or quasi-Hopf algebra we always indicate the R-matrix R that produces the QT structure, by pointing out the pair (H, R). By applying ε ⊗ ε ⊗ IdH to both sides of (10.1.1) we get that ε (R1 )R2 is an invertible idempotent of H, so it must be equal to 1H . Similarly, by using (10.1.2) we deduce that ε (R2 )R1 = 1H , and so we have (ε ⊗ IdH )(R) = (IdH ⊗ ε )(R) = 1H ,

(10.1.5)

in any QT quasi-bialgebra or quasi-Hopf algebra (H, R). Definition 10.3 A morphism of QT quasi-bialgebras ϕ : (H, R) → (H  , R ) is a morphism of the underlying quasi-bialgebras such that (ϕ ⊗ ϕ )(R) = R . Remarks 10.4 (1) Let H be a quasi-bialgebra. Then the pre-braided structures c on H M are given by (10.1.4), where R ∈ H ⊗ H is such that (10.1.1), (10.1.2) and (10.1.3) are satisfied. The invertibility of R is necessary only to turn c into a braiding on H M . −1 (2) If C is braided then C in is braided as well via cin X,Y = cY,X . This says that if (H, R) is QT then it admits a second QT structure, namely (H, R−1 21 ). (3) Let (H, R) be a QT quasi-bialgebra. If t denotes a permutation of {1, 2, 3}, then −1 −1 −1 we set Φt(1)t(2)t(3) = X t (1) ⊗ X t (2) ⊗ X t (3) , and by Ri j we denote the element obtained by acting with R non-trivially in the ith and jth positions of H ⊗ H ⊗ H. Then an immediate consequence of Proposition 1.51 and Proposition 10.1 is the fact that R satisfies the so-called quasi-Yang–Baxter equation: −1 R12 Φ312 R13 Φ−1 132 R23 Φ = Φ321 R23 Φ231 R13 Φ213 R12 .

(10.1.6)

10.1 Quasitriangular Quasi-bialgebras and Quasi-Hopf Algebras

383

Definition 10.5 A QT quasi-bialgebra or quasi-Hopf algebra (H, R) is called triangular if R−1 = R21 . In other words, (H, R) is triangular if and only if H M admits a symmetric structure such that the forgetful functor F : H M → k M is quasimonoidal. Example 10.6 We have shown in Proposition 1.44 that for an abelian group G the braided structures on VectG are given by the abelian 3-cocycles (φ , R) on G. Furthermore, if G is finite then the category VectG φ identifies as a monoidal category with the category of left representations over the quasi-Hopf algebra kφ [G]∗ ; see Proposition 3.51 and Example 3.50. Since VectG (φ ,R) is a braided category it follows that so is kφ [G]∗ M . Combined with Proposition 10.1 this tells us that kφ [G]∗ is a QT quasi-Hopf algebra, provided that (φ , R) is an abelian 3-cocycle of a finite abelian group G. Concrete examples of this type can be obtained by using Example 1.48. In the next result we describe all the QT structures of H(2). Example 10.7 Suppose that k is a field of characteristic different from 2 containing a primitive fourth root of unity i and let H(2) be the quasi-Hopf algebra constructed in Example 3.26. Then there are exactly two different R-matrices for H(2), namely R± = 1 − (1 ± i)p− ⊗ p− . Proof As we have seen before, H(2) = kφ [C2 ]∗ , where C2 is the cyclic group of order 2 and φ is the unique non-trivial normalized 3-cocycle on C2 as defined in Example 1.12. So by Example 10.6 and Example 1.48 we know that H(2) has at least two QT structures. We will prove that it has precisely two, namely the ones stated above. Suppose that R = a1 ⊗ 1 + b1 ⊗ g + cg ⊗ 1 + dg ⊗ g is an R-matrix for H(2), where a, b, c, d ∈ k. By (10.1.5) we have that a + c = a + b = 1 and b + d = c + d = 0, and therefore b = c = −d and a = 1 − b. Hence, R must be of the form R = (1 − b)1 ⊗ 1 + b1 ⊗ g + bg ⊗ 1 − bg ⊗ g = 1 − b(1 − g) ⊗ (1 − g) = 1 − ω p− ⊗ p− , where we denote 4b = ω . Now, one can easily see that Φ−1 = Φ = 1 − 2p− ⊗ p− ⊗ p− ,

(10.1.7)

and since X 2 ⊗ X 3 ⊗ X 1 = Φ, the above relation implies X 2 R1 x1 ⊗ X 3 x3 ⊗ X 1 R2 x2 = 1 − ω p− ⊗ 1 ⊗ p− , and therefore, after some computations, we get X 2 R1 x1Y 1 ⊗ X 3 x3 r1Y 2 ⊗ X 1 R2 x2 r2Y 3 = 1 − ω p− ⊗ p+ ⊗ p− − ω p+ ⊗ p− ⊗ p− − (2 − 2ω + ω 2 )p− ⊗ p− ⊗ p− . On the other hand, we have Δ(p− ) = p− ⊗ p+ + p+ ⊗ p− , so (Δ ⊗ Id)(R) = 1 − ω p− ⊗ p+ ⊗ p− − ω p+ ⊗ p− ⊗ p− .

384

Quasitriangular Quasi-Hopf Algebras

We conclude that (10.1.1) holds if and only if 2 − 2ω + ω 2 = 0, and this is equivalent to ω = 1 ± i. By using (10.1.7) for Φ−1 , we obtain in a similar way that x3 R1 X 2 ⊗ x1 X 1 ⊗ x2 R2 X 3 = 1 − ω p− ⊗ 1 ⊗ p− . By using this formula, it can be proved that x3 R1 X 2 r1 y1 ⊗ x1 X 1 r2 y2 ⊗ x2 R2 X 3 y3 = 1 − ω p− ⊗ p− ⊗ p+ − ω p− ⊗ p+ ⊗ p− − (2 − 2ω + ω 2 )p− ⊗ p− ⊗ p− . It is easy to see that (Id ⊗ Δ)(R) = 1 − ω p− ⊗ p− ⊗ p+ − ω p− ⊗ p+ ⊗ p− , so the relation in (10.1.2) holds if and only if 2 − 2ω + ω 2 = 0. The relation in (10.1.3) is automatically satisfied because of the commutativity and cocommutativity of H(2). Thus the R-matrices for H(2) are in bijective correspondence with the solutions of the equation 2 − 2ω + ω 2 = 0, from where we deduce that R± = 1 − (1 ± i)p− ⊗ p− are the only quasi-triangular structures on H(2). Remark 10.8 It is not difficult to show that H(2)+ = (H(2), R+ ) and H(2)− = (H(2), R− ) are non-isomorphic QT quasi-Hopf algebras, that is, there is no quasiHopf algebra isomorphism ν : H(2) → H(2) satisfying (ν ⊗ ν )(R+ ) = R− . Indeed, if such a ν exists then (1 + i)ν (p− ) ⊗ ν (p− ) = (1 − i)p− ⊗ p− . If we write ν (p− ) = ap− + bp+ , for some scalars a, b ∈ k, then from the above relation we obtain that a2 = −i and b = 0. Since p2± = p± and ν is an algebra map we get that ap− = ν (p− ) = ν (p2− ) = (ap− )2 = −ip− , and we conclude that a = −i. But a2 = −i, so i ∈ {−1, 0}, a contradiction. A remarkable fact is that, in the quasi-Hopf case, the conditions in (10.1.5) can replace the invertibility of R in the definition of a QT quasi-Hopf algebra H. As we shall see, this follows basically from the properties of a left rigid (pre-)braided category. Lemma 10.9 Let H be a quasi-Hopf algebra with antipode S and R ∈ H ⊗ H such that (10.1.1), (10.1.2), (10.1.3) and (10.1.5) are satisfied. Then R is invertible with inverse given by R−1 = X 1 β S(Y 2 R1 x1 X 2 )α Y 3 x3 X23 ⊗Y 1 R2 x2 X13 .

(10.1.8)

Proof The category H M fd is pre-braided and left rigid; see Remarks 10.4 and Proposition 3.33. By Theorem 1.77 we get that H M fd is braided, and so c defined by (10.1.4) is a natural isomorphism. Furthermore, by (1.8.1) and the left rigid monoidal structure on H M fd we have that the inverse of cX,Y on objects of H M fd is given by  coevX ⊗Id i c−1 X,Y = y ⊗ x −→ (β · xi ⊗ x ) ⊗ (y ⊗ x) a−1 X,X ∗ ,Y ⊗X

−→

X 1 β · xi ⊗ (X 2 · xi ⊗ (X13 · y ⊗ X23 · x))

10.1 Quasitriangular Quasi-bialgebras and Quasi-Hopf Algebras IdX ⊗a−1 X ∗ ,Y,X

−→

385

X 1 β · xi ⊗ ((x1 X 2 · xi ⊗ x2 X13 · y) ⊗ x3 X23 · x)

IdX ⊗(cX ∗ ,Y ⊗IdX )

−→

X 1 β · xi ⊗ ((R2 x2 X13 · y ⊗ R1 x1 X 2 · xi ) ⊗ x3 X23 · x)

IdX ⊗aY,X ∗ ,X

X 1 β · xi ⊗ (Y 1 R2 x2 X13 · y ⊗ (Y 2 R1 x1 X 2 · xi ⊗Y 3 x3 X23 · x))  IdX ⊗(IdY ⊗evX ) 1 −→ X β S(Y 2 R1 x1 X 2 )α Y 3 x3 X23 · x ⊗Y 1 R2 x2 X13 · y . −→

When H is finite dimensional this gives the definition of R−1 in (10.1.8) since, as we have seen in the proof of Proposition 10.1, R−1 = c−1 H,H (1H ⊗ 1H ). In the general case (i.e. H not necessarily finite dimensional), the fact that R−1 and R are inverse to each other can be proved by direct computations. On the one hand, if we denote by r = r1 ⊗ r2 another copy of R then we have: R−1 R

=

X 1 β S(Y 2 R1 y1 X 2 )α Y 3 y3 X23 r1 ⊗Y 1 R2 y2 X13 r2

(10.1.3)

X 1 β S(Y 2 R1 y1 X 2 )α Y 3 y3 r1 X13 ⊗Y 1 R2 y2 r2 X23

(10.1.1)

X 1 β S(R11 y1 X 2 )α R12 y2 X13 ⊗ R2 y3 X23

= =

(3.2.1),(10.1.2),(10.1.5)

=

(3.1.9)

=

(3.2.1),(3.1.10),(3.1.11)

=

X 1 β S(y1 X 2 )α y2 X13 ⊗ y3 X23 Y 1 Z 1 y11 β S(Y12 Z 2 y12 )α Y22 Z 3 y2 ⊗Y 3 y3 (3.2.2)

Z 1 β S(Z 2 )α Z 3 ⊗ 1H = 1H ⊗ 1H .

On the other hand, to show that RR−1 = 1H ⊗ 1H , we first observe that (10.1.8), (3.1.9), (3.2.1) and (3.1.11) imply R−1 = X 1 Z 1 β S(Y 2 X22 R1 Z 2 )α Y 3 X 3 ⊗Y 1 X12 R2 Z 3 ,

(10.1.9)

and by again using (3.1.9), (3.2.1) and (3.1.11) we obtain that R−1 = X11 x1Y 1 β S(X 2 x3 R1Y 2 )α X 3 ⊗ X21 x2 R2Y 3 .

(10.1.10)

Now, we calculate: RR−1

=

r1 X11 x1Y 1 β S(X 2 x3 R1Y 2 )α X 3 ⊗ r2 X21 x2 R2Y 3

(10.1.3)

X21 r1 x1Y 1 β S(X 2 x3 R1Y 2 )α X 3 ⊗ X11 r2 x2 R2Y 3

(10.1.1)

X21 x2 R11 β S(X 2 x3 R12 )α X 3 ⊗ X11 x1 R2

= =

(3.2.1),(10.1.2),(10.1.5)

=

(3.1.9)

=

(3.2.1),(3.1.10),(3.1.11)

=

X21 x2 β S(X 2 x3 )α X 3 ⊗ X11 x1 x2 X 1Y12 β S(x13 X 2Y22 )α x23 X 3Y 3 ⊗ x1Y 1 (3.2.2)

X 1 β S(X 2 )α X 3 ⊗ 1H = 1H ⊗ 1H ,

and this finishes the proof of the lemma. We show now that a twisting of a QT quasi-bialgebra by a gauge transformation produces another QT quasi-bialgebra. Proposition 10.10 Let (H, R) be a QT quasi-bialgebra and F ∈ H ⊗ H a gauge transformation. Define RF := F21 RF −1 ∈ H ⊗ H, where, if F = F 1 ⊗ F 2 , then F21 :=

386

Quasitriangular Quasi-Hopf Algebras

F 2 ⊗ F 1 . Then (HF , RF ) is a QT quasi-bialgebra as well, and the categories H M and HF M are braided monoidally isomorphic. Proof We know from Proposition 3.5 that we have a monoidal isomorphism between H M and HF M , given by the identity functor H M →HF M with (strong) monoidal structure defined by ϕ0 = Idk and ϕ2,X,Y : X ⊗Y → X ⊗Y , ϕ2,X,Y (x ⊗ y) = G1 · x ⊗ G2 · y, where F −1 = G1 ⊗ G2 . Since H is QT, we know that H M is braided with braiding cX,Y : X ⊗Y → Y ⊗ X, cX,Y (x ⊗ y) = R2 · y ⊗ R1 · x. Thus, there exists a unique braiding on HF M such that the above functor becomes braided monoidal and, in view of Definition 1.52, this braiding, denoted by cFX,Y , is defined by cFX,Y : X ⊗Y → −1 Y ⊗ X, cFX,Y = ϕ2,X,Y ◦ cX,Y ◦ ϕ2,X,Y , that is cFX,Y (x ⊗ y) = F 1 R2 G2 · y ⊗ F 2 R1 G1 · x, for all x ∈ X ∈HF M and y ∈ Y ∈HF M . But then, in view of Proposition 10.1 and its proof, HF has to be quasitriangular and its R-matrix has to be defined by RF = F 2 R1 G1 ⊗ F 1 R2 G2 , that is, RF = F21 RF −1 .

10.2 Further Examples of Monoidal Algebras Let H be a quasi-bialgebra and denote by H MH cop the category of H-bimodules. In this category we introduce a tensor product, as follows. If V,W ∈ H MH cop then V ⊗W ∈ H MH cop with h · (v ⊗ w) · h = Δ(h) · (v ⊗ w) · Δcop (h ) = h1 · v · h2 ⊗ h2 · w · h1 . H MH cop becomes a monoidal category, with associativity constraint aU,V,W ((u ⊗ v) ⊗ w) = Φ · (u ⊗ (v ⊗ w)) · Φ321 = X 1 · u ·Y 3 ⊗ (X 2 · v ·Y 2 ⊗ X 3 · w ·Y 1 ), for U,V,W ∈ H MH cop (the unit constraints are the usual ones). Suppose now that (H, R) is a QT quasi-bialgebra. One can easily see that H MH cop becomes a braided category, the braiding being given by cV,W (v ⊗ w) = R2 · w ·U 1 ⊗ R1 · v ·U 2 , for V,W ∈ H MH cop , where U = U 1 ⊗U 2 is the inverse of R. Suppose again that (H, R) is a QT quasi-bialgebra and consider the left and right regular actions of H on H ∗ , that is, (h  p)(h ) = p(h h), (p  h)(h ) = p(hh ), for p ∈ H ∗ and h, h ∈ H, turning H ∗ into an H-bimodule. On H ∗ we can consider the convolution product, given by ( f g)(h) = f (h1 )g(h2 ), for all f , g ∈ H ∗ and h ∈ H. We introduce another product on H ∗ , by f · g = (R2  g)(R1  f ), ∀ f , g ∈ H ∗ .

(10.2.1)

Denote by HR∗ the pair (H ∗ , ·). Then we have the following result: Theorem 10.11 Let (H, R) be a QT quasi-bialgebra. Then: (i) HR∗ is an algebra in the monoidal category H MH cop (we say that it is an H–H cop bimodule algebra), that is, for all f , g, l ∈ H ∗ and h, h ∈ H we have h  ( f · g)  h = (h1  f  h2 ) · (h2  g  h1 ),

10.2 Further Examples of Monoidal Algebras

387

( f · g) · l = (X 1  f  Y 3 ) · ((X 2  g  Y 2 ) · (X 3  l  Y 1 )),

ε · f = f · ε = f , h  ε  h = ε (h)ε (h )ε . (ii) HR∗ is commutative as an algebra in the braided monoidal category H MH cop , that is, for all f , g ∈ H ∗ , we have f · g = (R2  g  U 1 ) · (R1  f  U 2 ). Proof We denote by R = R1 ⊗ R2 = r1 ⊗ r2 = ρ 1 ⊗ ρ 2 several copies of R. For f , g ∈ H ∗ , the product f · g is given by ( f · g)(h) = g(h1 R2 ) f (h2 R1 ), for all h ∈ H. First we prove (ii). We compute: ((R2  g  U 1 ) · (R1  f  U 2 ))(h) =

(R1  f  U 2 )(h1 r2 )(R2  g  U 1 )(h2 r1 )

=

f (U 2 h1 r2 R1 )g(U 1 h2 r1 R2 )

(10.1.3)

=

f (U 2 r2 h2 R1 )g(U 1 r1 h1 R2 )

=

f (h2 R1 )g(h1 R2 ) = ( f · g)(h),

as desired. Now we prove (i); we have: ((h1  f  h2 ) · (h2  g  h1 ))(h ) =

(h2  g  h1 )(h1 R2 )(h1  f  h2 )(h2 R1 )

=

g(h1 h1 R2 h2 ) f (h2 h2 R1 h1 )

(10.1.3)

=

g(h1 h1 h1 R2 ) f (h2 h2 h2 R1 )

=

( f · g)(h h h) = (h  f · g  h )(h ).

For the second relation, we compute: (( f · g) · l)(h) =

l(h1 R2 )( f · g)(h2 R1 )

=

l(h1 R2 )g(h(2,1) R11 r2 ) f (h(2,2) R12 r1 )

(10.1.1)

=

l(h1 X 1 R2 x2 ρ 2Y 3 )g(h(2,1) X 2 R1 x1Y 1 r2 ) f (h(2,2) X 3 x3 ρ 1Y 2 r1 )

(10.1.6)

l(h1 X 1 r2 x2 ρ 2Y 3 )g(h(2,1) R2 X 3 x3 ρ 1Y 2 ) f (h(2,2) R1 X 2 r1 x1Y 1 ).

=

We now compute the right-hand side evaluated in h: ((X 1  f  Y 3 ) · ((X 2  g  Y 2 ) · (X 3  l  Y 1 )))(h) =

((X 2  g  Y 2 ) · (X 3  l  Y 1 ))(h1 R2 )(X 1  f  Y 3 )(h2 R1 )

=

l(Y 1 h(1,1) R21 r2 X 3 )g(Y 2 h(1,2) R22 r1 X 2 ) f (Y 3 h2 R1 X 1 )

(3.1.7)

l(h1Y 1 R21 r2 X 3 )g(h(2,1)Y 2 R22 r1 X 2 ) f (h(2,2)Y 3 R1 X 1 )

(10.1.2)

l(h1Y 1 y1 T 1 R2 x2 r2 X 3 )g(h(2,1)Y 2 y2 ρ 2 T 3 x3 r1 X 2 )

= =

f (h(2,2)Y 3 y3 ρ 1 T 2 R1 x1 X 1 ) =

l(h1 T 1 R2 x2 r2 X 3 )g(h(2,1) ρ 2 T 3 x3 r1 X 2 ) f (h(2,2) ρ 1 T 2 R1 x1 X 1 ),

388

Quasitriangular Quasi-Hopf Algebras

and this is obviously equal to the expression obtained for (( f · g) · l)(h). The relations ε · f = f · ε = f and h  ε  h = ε (h)ε (h )ε are obvious, using the fact that ε (h1 )h2 = h = h1 ε (h2 ) and ε (R1 )R2 = R1 ε (R2 ) = 1H .

10.3 The Square of the Antipode of a QT Quasi-Hopf Algebra We show that the square of the antipode S of a QT quasi-Hopf algebra (H, R) is an inner automorphism of H. At first sight the formula of the invertible element u that defines S2 as an inner automorphism of H looks “unnatural”. But as in the case of the inverse of R, we shall see that it comes naturally from some canonical isomorphisms in a rigid braided category, as well as the formula for its inverse. Proposition 10.12 Let (H, R) be a finite-dimensional QT quasi-Hopf algebra. If we define the elements 2

1

u := S2 (q˜2 R p˜1 )q˜1 R p˜2

and u−1 := S2 (q˜2 R1 p˜1 )q˜1 R2 p˜2 ,

where qL = q˜1 ⊗ q˜2 and pL = p˜1 ⊗ p˜2 are as in (3.2.20), and R = R1 ⊗ R2 and 1 2 R−1 = R ⊗ R , then for any finite-dimensional left H-module V the map V ∗ v∗ → u−1  v∗ ∈ ∗V is a left H-linear isomorphism with inverse given by ∗V ∗ v → u  ∗ v ∈ V ∗ . Here, in order to avoid ambiguities, we have denoted by  the left Hmodule structures of both left and right duals V ∗ and ∗V of V in H M fd . Consequently, S2 (h) = uhu−1 , for all h ∈ H. Proof Everything follows from the fact that in a braided rigid category the left and right dual objects are isomorphic; see Corollary 1.76. More precisely, for C = H M fd , a rigid braided category, we have an isomorphism  ΘV : V ∗ → ∗V in H M , for any finite-dimensional left H-module V . As was explained in the proof of Corollary 1.76, ΘV is defined by the composition: aV−1∗ ,∗V,V

IdV ∗ ⊗coev ΘV = v∗ −→ V v∗ ⊗ (i v ⊗ S−1 (β ) · i v) −→ (x1 · v∗ ⊗ x2  i v) ⊗ x3 S−1 (β ) · i v cV ∗ ,∗V ⊗IdV

−→

(R2 x2  i v ⊗ R1 x1 · v∗ ) ⊗ x3 S−1 (β ) · i v

a∗V,V ∗ ,V

−→ X 1 R2 x2  i v ⊗ (X 2 R1 x1 · v∗ ⊗ X 3 x3 S−1 (β ) · i v)   Id∗V ⊗evV ∗

−→ v S(X 2 R1 x1 )α X 3 x3 S−1 (β ) · i v X 1 R2 x2  i v ,

where {i v, i v}i are dual bases in V and V ∗ . In other words, we have ΘV (v∗ ) = v∗ (S(X 2 R1 x1 )α X 3 x3 S−1 (X 1 R2 x2 β ) · vi )i v = v∗ (S−1 (u−1 ) · i v)i v = u−1  v∗ , for all v∗ ∈ V ∗ . In a similar manner, by the proof of Corollary 1.76 we have that its

10.3 The Square of the Antipode of a QT Quasi-Hopf Algebra

389

inverse ΘV−1 is given by  aV,V ∗ ,∗V coevV ⊗Id∗  −1 Θ V = ∗ v −→ V (β · i v ⊗ i v) ⊗ ∗ v −→ X 1 β · i v ⊗ (X 2 · i v ⊗ X 3  ∗ v) IdV ⊗cV−1∗ ,∗V

−→

X 1 β · i v ⊗ (R X 3  ∗ v ⊗ R X 2 · i v) 1

−1 aV, ∗ V,V ∗

2

−→ (x1 X 1 β · i v ⊗ x2 R X 3  ∗ v) ⊗ x3 R X 2 · i v

evV ⊗IdV ∗ ∗

−→

1

2

 1 2 v(S−1 (α x2 R X 3 )x1 X 1 β · i v)x3 R X 2 · i v ,

for all ∗ v ∈ ∗V . Thus we have shown that −1

ΘV (∗ v) = ∗ v(S−1 (q˜1 R p˜2 )S(q˜2 R p˜1 ) · i v)i v = ∗ v(S−1 (u) · i v)i v = u  ∗ v, 1

2

for all ∗ v ∈ ∗V . Hence, by using the fact that ΘV and ΘV−1 are inverses of each other we get that uu−1  v∗ = u−1 u  v∗ = v∗ , for all v∗ ∈ V ∗ . Since H is finite dimensional this applies to V = H, and therefore S−1 (uu−1 ) = ∑ hi (S−1 (uu−1 ))hi = ∑(uu−1  hi )(1H )hi = ∑ hi (1H )hi = 1H , i

i

i

where {hi , hi }i are dual bases in H and H ∗ . The antipode S is bijective, so we conclude that uu−1 = 1H . In a similar manner one can show that u−1 u = 1H , and this finishes the first part of the proof. Now, to prove that u defines S2 as an inner automorphism of H, observe that, for all h ∈ H and v∗ ∈ V ∗ , ΘV (h  v∗ ) = u−1  (h  v∗ ) = ∑ u−1  (h  v∗ )(i v)i v i

= ∑ v∗ (S(h) · i v)u−1  i v i −1 2

= u S (h)  v∗ , and similarly h  ΘV (v∗ ) = hu−1  v∗ . Taking V = H, by arguments similar to the ones above we get that u−1 S2 (h) = hu−1 , for all h ∈ H, or equivalently S2 (h) = uhu−1 , for all h ∈ H. We next see that the results in Proposition 10.12 remain valid for any QT quasiHopf algebra, not necessarily finite dimensional. As we have explained, we gave a detailed proof for Proposition 10.12 in order to obtain, in a canonical way, the form of the element u that defines S2 as an inner automorphism of H, as well as the form of its inverse. Furthermore, in the forthcoming proofs we will not make use of the bijectivity of the antipode. Consequently, we will get that the antipode of a QT quasi-Hopf algebra is always bijective. We start by proving a second formula for u. Note that other equivalent definitions for u and u−1 can be obtained by interchanging the definitions of the maps ΘV , ΘV−1

390

Quasitriangular Quasi-Hopf Algebras

and ΘV , ΘV−1 respectively, given in the proof of Corollary 1.76, specialized of course for C = H M fd . Lemma 10.13 Let (H, R) be a finite-dimensional QT quasi-Hopf algebra. If we 1 2 denote R−1 = R ⊗ R then the following relations hold: 1

2

R = q1 R x2 p˜21 ⊗ S(q2 R x1 p˜1 )x3 p˜22 ,

(10.3.1)

u = S(R p )α R p .

(10.3.2)

2 2

1 1

Proof By Remarks 10.4 we have that (H, R−1 21 ) is a QT quasi-Hopf algebra. By Lemma 10.9 it then follows that 2

1

R21 = S(q2 R x1 p˜1 )x3 p˜22 ⊗ q1 R x2 p˜21 . Switching the order of the factors in the tensor product we get the first formula stated above. We can use it together with x1 p˜1 ⊗ x2 p˜21 ⊗ x3 p˜22 = X12 p˜1 S−1 (X 1 ) ⊗ X22 p˜2 ⊗ X 3 ,

(10.3.3)

which can be viewed as the “op”-version of (5.2.7), to prove the second one as follows: S(R2 p2 )α R1 p1

2

1

S(x3 p˜22 p2 )S2 (q2 R x1 p˜1 )α q1 R x2 p˜21 p1

= (10.3.3)

S(X 3 p2 )S2 (q2 R X12 p˜1 S−1 (X 1 ))α q1 R X22 p˜2 p1

(10.1.3)

S(p2 )S2 (S−1 (X 3 )q2 X22 R p˜1 )S(X 1 )α q1 X12 R p˜2 p1

(5.5.17)

S(p2 )S2 (q2 x3 R p˜1 )S(q11 x1 )α q12 x2 R p˜2 p1

2

=

1

2

=

1

2

=

(3.2.1),(3.2.20)

2

1

1

=

S(α p2 )S2 (q˜2 R p˜1 )q˜1 R p˜2 p1

=

S(α p2 )up1 = S(S(p1 )α p2 )u = u.

(3.2.2)

Note that in the penultimate equality we used the fact that S2 is inner via u. So our proof is complete. From now on (H, R) is an arbitrary QT quasi-Hopf algebra, so neither finite dimensional nor with bijective antipode. Also, u is the element of H defined by u = S(R2 x2 β S(x3 ))α R1 x1 .

(10.3.4)

We record the obvious fact that ε (u) = 1. In what follows we prove formulas that connect the R-matrix R with the quasiHopf algebra structure of H. Lemma 10.14 Let (H, R) be a QT quasi-Hopf algebra with antipode S. Then the following relations hold: (S ⊗ S)(R)γ = γ21 R,

(10.3.5)

where γ = γ 1 ⊗ γ 2 is the element defined by (3.2.5) and γ21 = γ 2 ⊗ γ 1 , and f21 R f −1 = (S ⊗ S)(R),

(10.3.6)

10.3 The Square of the Antipode of a QT Quasi-Hopf Algebra

391

where f = f 1 ⊗ f 2 is the element defined in (3.2.15) with its inverse f −1 as in (3.2.16) and f21 = f 2 ⊗ f 1 . Proof

If R = r1 ⊗ r2 is another copy of R, then by (10.1.9) and (3.2.5) we have:

(S ⊗ S)(R)γ R−1 =

S(T 2 y12 R1 )α T 3 y2 X 1 Z 1 β S(Y 2 X22 r1 Z 2 )α Y 3 X 3 ⊗ S(T 1 y11 R2 )α y3Y 1 X12 r2 Z 3

(3.2.1)

=

S(T 2 y12 R1 )α T 3 y2 X 1 Z 1 β S(y3(2,1)Y 2 X22 r1 Z 2 )α y3(2,2)Y 3 X 3 ⊗ S(T 1 y11 R2 )α y31Y 1 X12 r2 Z 3

(3.1.7)

=

S(T 2 y12 R1 )α T 3 y2 X 1 Z 1 β S(Y 2 (y31 X 2 )2 r1 Z 2 )α Y 3 y32 X 3 ⊗ S(T 1 y11 R2 )α Y 1 (y31 X 2 )1 r2 Z 3

(3.1.9)

=

1 S(T 2 X(1,2) z12 y12 R1 )α T 3 X21 z2 y21 Z 1 β S(Y 2 X22 z32 y2(2,2) r1 Z 2 )α Y 3 X 3 y3 1 ⊗ S(T 1 X(1,1) z11 y11 R2 )α Y 1 X12 z31 y2(2,1) r2 Z 3

(3.1.7)

=

(3.2.1)

S(T 2 z12 y12 R1 )α T 3 z2 y21 Z 1 β S(Y 2 X22 z32 y2(2,2) r1 Z 2 )α Y 3 X 3 y3 ⊗ S(X 1 T 1 z11 y11 R2 )α Y 1 X12 z31 y2(2,1) r2 Z 3

(10.1.3)

=

S(T 2 z12 y12 R1 )α T 3 z2 y21 Z 1 β S(Y 2 X22 z32 r1 y2(2,1) Z 2 )α Y 3 X 3 y3 ⊗ S(X 1 T 1 z11 y11 R2 )α Y 1 X12 z31 r2 y2(2,2) Z 3

(3.1.7)

=

(3.2.1) (3.1.9)

=

S(T 2 z12 y12 R1 )α T 3 z2 Z 1 β S(Y 2 X22 z32 r1 Z 2 )α Y 3 X 3 y3 ⊗ S(X 1 T 1 z11 y11 R2 )α Y 1 X12 z31 r2 Z 3 y2 S(T 2 z12 y12 R1 )α T 3 z2 Z 1 β S(X13Y 2 x3 z32 r1 Z 2 )α X23Y 3 y3 ⊗ S(X 1Y11 x1 T 1 z11 y11 R2 )α X 2Y21 x2 z31 r2 Z 3 y2

(3.2.1),(3.1.11)

=

(10.1.3) (3.1.9)

=

S(T 2 z12 y12 R1 )α T 3 z2 Z 1 β S(x3 r1 z31 Z 2 )α y3 ⊗ S(x1 T 1 z11 y11 R2 )α x2 r2 z32 Z 3 y2 1 S(T 2 X(1,2) z12t21 y12 R1 )α T 3 X21 z2t12 β S(x3 r1 X 2 z3t22 )α y3 1 ⊗ S(x1 T 1 X(1,1) z11t11 y11 R2 )α x2 r2 X 3t 3 y2

(3.1.7),(3.2.1)

=

(3.1.10) (3.1.9)

=

(10.1.3) (3.2.1),(3.1.10)

=

(3.1.11)

S(T 2 z12 y12 R1 )α T 3 z2 β S(x3 r1 X 2 z3 )α y3 ⊗ S(x1 X 1 T 1 z11 y11 R2 )α x2 r2 X 3 y2 S(z21t 1 T 2 R1 y11 )α z22t 2 T13 β S(x3 r1 X 2 z3t 3 T23 )α y3 ⊗ S(x1 X 1 z1 T 1 R2 y12 )α x2 r2 X 3 y2 S(R1 y11 )S(t 1 )α t 2 β S(t 3 )S(x3 r1 X 2 )α y3 ⊗ S(x1 X 1 R2 y12 )α x2 r2 X 3 y2

(3.2.2)

S(x3 r1 X 2 R1 y11 )α y3 ⊗ S(x1 X 1 R2 y12 )α x2 r2 X 3 y2

(10.1.2)

=

S(R1 X 1 y11 )α y3 ⊗ S(R21 X 2 y12 )α R22 X 3 y2

(3.2.1),(10.1.5)

S(X 1 y11 )α y3 ⊗ S(X 2 y12 )α X 3 y2 = γ21 .

= =

(3.2.5)

392

Quasitriangular Quasi-Hopf Algebras

The proof of the relation (10.3.6) is now immediate since by (10.3.5), (3.2.15) and (10.1.3) we have (S ⊗ S)(R) f = (S ⊗ S)(Δcop (x1 )R)γ Δ(x2 β S(x3 )) = (S ⊗ S)(Δ(x1 ))(S ⊗ S)(R)γ Δ(x2 β S(x3 )) = (S ⊗ S)(Δ(x1 ))γ21 RΔ(x2 β S(x3 )) = (S ⊗ S)(Δ(x1 ))γ21 Δcop (x2 β S(x3 ))R = f21 R, as stated. This completes the proof of the lemma. Lemma 10.15 Let (H, R) be a QT quasi-Hopf algebra with antipode S, and u the element defined by (10.3.4). Then the following relations hold: S2 (h)u = uh, ∀ h ∈ H,

(10.3.7)

S(α )u = S(R )α R .

(10.3.8)

2

Proof

1

By (10.1.3) and (3.2.1) it is not hard to see that for all h ∈ H we have S2 (h)u = S(R2 h(1,2) x2 β S(h2 x3 ))α R1 h(1,1) x1 ,

and then, by (3.1.7) and (3.2.1), it follows that S2 (h)u = uh. To prove (10.3.8), one performs the following substitution in u x1 ⊗ x2 ⊗ x3 ⊗ 1H = (Δ ⊗ IdH ⊗ IdH )(Φ−1 )(IdH ⊗ IdH ⊗ Δ)(Φ−1 ) (1H ⊗ Φ)(IdH ⊗ Δ ⊗ IdH )(Φ) and simplifies in several steps the resulting expression for S(α )u by using (3.2.1), (3.1.10), (3.1.11) and (3.2.2). A last preliminary result that we need is the following: Lemma 10.16 Let (H, R) be a QT quasi-Hopf algebra with antipode S. If u is the element defined by (10.3.4), then S2 (u) = u. Proof We set pR = p1 ⊗ p2 = x1 ⊗ x2 β S(x3 ) and denote by F 1 ⊗ F 2 another copy of f . Then we compute: S2 (u)

=

S2 (S(R2 p2 )α R1 p1 )

=

S(S(p1 )S(R1 )S(α )S(S(p2 )S(R2 )))

(10.3.6)

S(S(p1 ) f 2 R1 g1 S(α )S(S(p2 ) f 1 R2 g2 ))

(8.7.7)

=

S(S(p1 ) f 2 R1 β S(S(p2 ) f 1 R2 ))

=

S(S(R2 )S( f 1 )S2 (p2 ))S(β )S(R1 )S( f 2 )S2 (p1 )

=

(10.3.6)

S(F 1 R2 g2 S( f 1 )S2 (p2 ))S(β )F 2 R1 g1 S( f 2 )S2 (p1 )

(8.7.7)

=

S(R2 g2 S( f 1 )S2 (p2 ))α R1 g1 S( f 2 )S2 (p1 )

(10.3.8)

S(g2 S( f 1 )S2 (p2 ))S(α )ug1 S( f 2 )S2 (p1 )

(10.3.7)

S(g2 S( f 1 )S2 (p2 ))S(α )S2 (g1 )uS( f 2 )S2 (p1 )

= = =

10.3 The Square of the Antipode of a QT Quasi-Hopf Algebra (10.3.7)

S(S(g1 )α g2 S( f 1 )S2 (p2 ))S3 ( f 2 )uS2 (p1 )

(8.7.7)

S(S( f 1 β S( f 2 ))S2 (p2 ))uS2 (p1 )

(8.7.7)

=

S2 (S(α p2 ))uS2 (p1 ) = uS(α p2 )S2 (p1 )

=

uS(S(x1 )α x2 β S(x3 )) = u,

= =

393

(10.3.7)

(3.2.2)

as needed. Now, we can prove the main result of this section. Theorem 10.17 Let H be a QT quasi-Hopf algebra with antipode S. If u is the element defined by (10.3.4) then u is invertible with inverse given by u−1 = X 1 R2 p2 S(S(X 2 R1 p1 )α X 3 ),

(10.3.9)

where pR = p1 ⊗ p2 = x1 ⊗ x2 β S(x3 ), and S2 (h) = uhu−1 , ∀ h ∈ H.

(10.3.10)

In particular, the antipode S is bijective. Proof By the previous results we only have to check that u and u−1 are inverses. If r1 ⊗ r2 is another copy of R, we calculate: u−1 u

(10.3.9)

X 1 R2 p2 S(α X 3 )S2 (X 2 R1 p1 )u

(10.3.7)

X 1 R2 p2 S(α X 3 )uX 2 R1 p1

(10.3.8)

=

X 1 R2 p2 S(r2 X 3 )α r1 X 2 R1 p1

=

X 1 R2 x2 β S(r2 X 3 x3 )α r1 X 2 R1 x1

= =

(10.1.2)

=

X 1 R21 β S(X 2 R22 )α X 3 R1

(3.2.1),(10.1.5)

X 1 β S(X 2 )α X 3 = 1H ,

=

(3.2.2)

and thus u−1 is a left inverse for u. It is also a right inverse. Indeed, Lemma 10.16, (10.3.7) and the fact that S2 is an algebra morphism imply uu−1 = S2 (u−1 )u = S2 (u−1 )S2 (u) = S2 (u−1 u) = S2 (1H ) = 1H , and the proof is complete. Since S is always bijective we can give a second formula for the inverse of R. Namely, as in the proof of Lemma 10.9 (see also (10.3.6)) we have that R−1 = x13 X 2 R1 p1 ⊗ x23 X 3 S−1 (S(x1 )α x2 X 1 R2 p2 ).

(10.3.11)

Furthermore, by (3.1.9), (3.2.1) and (3.1.11) it follows that R−1 = x3 y22 R1 p1 ⊗ y3 S−1 (S(x1 y1 )α x2 y21 R2 p2 ).

(10.3.12)

We leave the verification of all these details to the reader. Corollary 10.18 Let (H, R) be a QT quasi-Hopf algebra with antipode S and u ∈ H defined in (10.3.4). Then uS(u) = S(u)u and this element is central in H.

394

Quasitriangular Quasi-Hopf Algebras

Proof Let h ∈ H. We apply S to the equality uh = S2 (h)u and we obtain S(h)S(u) = S(u)S3 (h). By replacing h with S−1 (h) we obtain hS(u) = S(u)S2 (h) = S(u)uhu−1 , so hS(u)u = S(u)uh, which means that S(u)u is central in H. By taking h = u, we obtain uS(u) = S(u)u. We now prove that the canonical element u of a QT quasi-Hopf algebra is invariant under twisting. Proposition 10.19 Let (H, R) be a QT quasi-Hopf algebra and F ∈ H ⊗ H a gauge transformation. If we denote by u and uF the canonical elements of the QT quasiHopf algebras (H, R) and (HF , RF ), respectively, then u = uF . 1

2

Proof We denote F = F 1 ⊗ F 2 = F 1 ⊗ F 2 = F ⊗ F and F −1 = G1 ⊗ G2 = G 1 ⊗ G 2 . We have −1 ΦF = F23 (IdH ⊗ Δ)(F)Φ(Δ ⊗ IdH )(F −1 )F12 ,

so with the above notation we obtain 1 1 1 1 2 1 2 2 1 2 3 2 2 Φ−1 F = F F1 x G ⊗ F F2 x G1 G ⊗ F x G2 G .

Similarly, αF = S(G1 )α G2 , βF = F 1 β S(F 2 ), RF = F 2 R1 G1 ⊗ F 1 R2 G2 . Thus, we can compute the element uF given by (10.3.4) for HF : uF

1

2

=

S(R2F F 2 F21 x2 G21 G 1 F β S(F )S(F 2 x3 G22 G 2 ))αF R1F F 1 F11 x1 G1

=

S(R2F F 2 F21 x2 G21 β S(F 2 x3 G22 ))αF R1F F 1 F11 x1 G1

=

S(R2F F 2 F21 x2 G21 β S(G22 )S(F 2 x3 ))αF R1F F 1 F11 x1 G1

(3.2.1)

=

S(R2F F 2 F21 x2 β S(F 2 x3 ))αF R1F F 1 F11 x1

=

S(F R2 G2 F 2 F21 x2 β S(F 2 x3 ))S(G 1 )α G 2 F R1 G1 F 1 F11 x1

=

S(R2 F21 x2 β S(F 2 x3 ))α R1 F11 x1

(10.1.3)

S(F11 R2 x2 β S(F 2 x3 ))α F21 R1 x1

(3.2.1)

S(R2 x2 β S(x3 ))α R1 x1 = u,

= =

1

2

finishing the proof.

10.4 The QT Structure of the Quantum Double Throughout this section H is a finite-dimensional quasi-Hopf algebra and {ei , ei }i are dual bases in H and H ∗ . Also, D(H) is the quantum double of H, the quasi-Hopf algebra considered in Theorem 8.33. We recall that the quasi-Hopf algebra structure of D(H) was obtained by using the reconstruction theorem for quasi-Hopf algebras and the category isomorphism H YD H ∼ = D(H) M proved in Corollary 8.31. Since H YD H ∼ = Zr (H M ) is a braided category, by Proposition 10.1 it follows that D(H) has a unique QT structure such that the monoidal category isomorphism

10.4 The QT Structure of the Quantum Double

395

∼ = D(H) M becomes braided monoidal. We compute this QT structure of D(H) as follows.

H YD

H

Theorem 10.20 Let H be a finite-dimensional quasi-Hopf algebra. Then D(H), the quantum double of H, is a QT quasi-Hopf algebra with the R-matrix given by n

RD = ∑ (ε  S−1 (p2 )ei p11 ) ⊗ (ei  p12 ),

(10.4.1)

i=1

where pR = p1 ⊗ p2 is as in (3.2.19). This QT structure of D(H) turns the isomorphism H YD H ∼ = D(H) M from Corollary 8.31 into a braided monoidal one. Proof From the above comments we only have to transfer the braided structure of H YD H to the category D(H) M through the categorical isomorphism in Corollary 8.31, and then to apply Proposition 10.1 in order to get the explicit form of the Rmatrix of D(H). We have a strict monoidal functor F : H YD H → D(H) M that acts as identity on objects and morphisms; see (8.5.6) for the explicit definition of F. Note also that any left D(H)-module M is a left-right Yetter–Drinfeld module via the structure given by h · m = (ε  h)m and M m → ∑(ei  p12 )m ⊗ S−1 (p2 )ei p11 ∈ M ⊗ H. i

For the general case see Lemmas 8.27 and 8.28. By using these correspondences we get that the braided structure of given by the braiding c defined, for all M, N ∈ D(H) M , by cM,N (m ⊗ n)

=

cM,N (m ⊗ n)

(8.2.18)

=

n(0) ⊗ n(1) · m

=

∑(ei  p12 )n ⊗ S−1 (p2 )ei p11 · m

D(H) M

is

i

=

∑(ei  p12 )n ⊗ (ε  S−1 (p2 )ei p11 )m, i

for all m ∈ M and n ∈ N. Now, by the proof of Proposition 10.1 we have that (RD )21 = cD(H),D(H) (1D ⊗ 1D ) and this leads to the formula for RD in (10.4.1). As an application of the quantum double construction we next describe the QT quasi-Hopf algebra structure of D(H(2)). Proposition 10.21 The quantum double of H(2) is the associative unital algebra generated by X and Y with relations X 2 = 1, Y 2 = X, XY = Y X. The quasi-coalgebra structure on D(H(2)) is given by the formulas: Δ(X) = X ⊗ X, ε (X) = 1, 1 Δ(Y ) = − (Y ⊗Y + XY ⊗Y +Y ⊗ XY − XY ⊗ XY ), 2

ε (Y ) = −1.

396

Quasitriangular Quasi-Hopf Algebras

If we denote pX± := 12 (1 ± X) then the reassociator, the distinguished elements α and β and the antipode are given by ΦX = 1 − 2pX− ⊗ pX− ⊗ pX− , α = X, β = 1, S(X) = X, S(Y ) = Y, respectively. Moreover, D(H(2)) is a QT quasi-Hopf algebra with R-matrix given by R = pX+ ⊗ 1 − pX− ⊗ XY. Proof By using the commutativity and cocommutativity of H(2), (3.2.1) and the fact that β = 1, we find that the multiplication rule (8.5.2) takes the following form on D(H(2)): (ϕ  h)(ϕ   h ) = (Ω1 Ω5  ϕ )(Ω2 Ω4  ϕ  )  Ω3 hh , for all ϕ , ϕ  ∈ H(2)∗ and h, h ∈ H(2). From the definition (8.5.1) of Ω we find 1 1 Ω1 Ω5 ⊗ Ω2 Ω4 ⊗ Ω3 = X(1,1) X 3 y1 x1 f 2 ⊗ X(1,2) X 2 y2 x12 x3 f 1 ⊗ X21 y3 x22 .

By using the expressions of Φ and Φ−1 in (10.1.7) we easily compute that 1 1 X(1,1) X 3 ⊗ X(1,2) X 2 ⊗ X21 = 1 − 2p− ⊗ p− ⊗ p− ,

x1 ⊗ x12 x3 ⊗ x22 = 1 − 2p− ⊗ p− ⊗ p+ , Φ−1 ( f 2 ⊗ f 1 ⊗ 1) = p− ⊗ 1 ⊗ p− + p− ⊗ g ⊗ p+ + p+ ⊗ 1 ⊗ 1, where f = g ⊗ p− + 1 ⊗ p+ is the Drinfeld twist of H(2). By the above relations the multiplication of D(H(2)) comes out explicitly as (ϕ  h)(ϕ   h ) = ϕϕ   hh − 2(p−  ϕ )(p−  ϕ  )  p− hh . Let {P1 , Pg } be the dual basis of H(2)∗ corresponding to the basis {1, g} of H(2). Then ε = P1 +Pg and {ε , μ = P1 −Pg } is clearly a basis for H(2)∗ . Now let X = ε  g and Y = μ  1. Since p−  P1 = 12 μ and p−  Pg = − 12 μ , we obtain X 2 = (ε  g)(ε  g) = ε  1 − 2(p−  ε )(p−  ε ) = 1, XY = Y X = μ  g, Y 2 = μ 2  1 − 2(p−  μ )2  p− = ε  1 − ε  2p− = ε  g = X, which are the multiplication rules that we stated. A direct computation ensures that the reassociator of D(H(2)) has the desired form. Since H(2) is commutative and cocommutative, β = 1 and Φ−1 = Φ = Y 2 ⊗Y 1 ⊗ 3 Y , by (8.5.9) we have Δ(ϕ  h) = p11 p2  ϕ2  p12 X 1 h1 ⊗ X12 X 3  ϕ1  X22 h2 , for all ϕ ∈ H(2)∗ and h ∈ H(2). On the other hand, p11 p2 ⊗ p12 X 1 ⊗ X12 X 3 ⊗ X22 = (p+ ⊗ 1 ⊗ 1 ⊗ 1 − p− ⊗ g ⊗ 1 ⊗ 1)(1 − 2 ⊗ p− ⊗ p− ⊗ p+ ) = p+ ⊗ 1 ⊗ 1 ⊗ 1 − p− ⊗ g ⊗ 1 ⊗ 1 − 2 ⊗ p− ⊗ p− ⊗ p+ .

10.4 The QT Structure of the Quantum Double

397

Then we have Δ(X) = Δ(ε  g) = ε  g ⊗ ε  g = X ⊗ X, and since Δ(P1 ) = P1 ⊗ P1 + Pg ⊗ Pg and Δ(Pg ) = P1 ⊗ Pg + Pg ⊗ P1 we get that Δ(μ ) = Δ(P1 − Pg ) = (P1 − Pg ) ⊗ (P1 − Pg ) = μ ⊗ μ , and therefore Δ(Y ) = p+  μ  1 ⊗ μ  1 − p−  μ  g ⊗ μ  1 − 2μ  p− ⊗ p−  μ  p+ 1 = −XY ⊗Y − (Y − XY ) ⊗ (Y + XY ) 2 1 = − (Y ⊗Y + XY ⊗Y +Y ⊗ XY − XY ⊗ XY ), 2 as needed. It follows from (8.5.10) that ε (X) = 1 and ε (Y ) = −1. In our particular situation, (8.5.14) takes the form S(ϕ  h) = p11 p2U 1 f 2  ϕ  p12U 2 f 1 h for all ϕ ∈ H(2)∗ and h ∈ H. But p11 p2 f 2 ⊗ p12 f 1 = U 1 ⊗U 2 = g ⊗ 1, so the antipode for D(H(2)) is the identity map. Obviously, α = ε  g = X, β = ε  1 = 1. Finally, since p11 p2 ⊗ p12 = p+ ⊗ 1 − p− ⊗ g we have from (10.4.1) that the canonical R-matrix for D(H(2)) is R = pX+ ⊗ 1 − pX− ⊗ XY , so our proof is complete. The quantum double construction yields also a QT structure on the quasi-Hopf algebras Dω (H) and Dω (G) constructed in Section 8.6. Proposition 10.22

The quasi-Hopf algebra Dω (H) is QT with R-matrix n

R = ∑ (ei #1) ⊗ (ε #ei ) ∈ Dω (H) ⊗ Dω (H), i=1

where

{ei , ei }i

are dual bases in H and H ∗ . The inverse of R is n

R−1 = ∑ (ei #1H ) ⊗ (σ −1 (S(ei ), ei )#S(ei )), i=1

where Proof

σ −1

is the convolution inverse of σ .

This is a direct consequence of Theorem 8.34.

Remarks 10.23 matrix

(1) It follows that Dω (G) is a QT quasi-Hopf algebra with R R=

∑ (pg ⊗ e) ⊗ ∑ ph ⊗ g



g∈G

.

h∈G

(2) The map T described in Proposition 8.36 is an isomorphism of QT quasi bialgebras between Dω (H) and Dω (H)F −1 . In the rest of this section we specialize Theorem 10.17 for the QT quasi-Hopf algebra D(H). We prove first an explicit formula for the element uD of D(H). In what follows we denote by S the antipode of the dual quasi-Hopf algebra H ∗ , that −1 is, the endomorphism of H ∗ defined by S(ϕ ) = ϕ ◦ S, for all ϕ ∈ H ∗ , and by S its composition inverse.

398

Quasitriangular Quasi-Hopf Algebras

Proposition 10.24 Let H be a finite-dimensional quasi-Hopf algebra and uD the corresponding element u for D(H), the quantum double of H, as in (10.3.4). Then n

uD = ∑ β  S

−1

(ei )  ei .

(10.4.2)

i=1

Proof

Let us start by noting that (3.2.17), (8.7.7) and (3.2.1) imply f11 p1 ⊗ f21 p2 S( f 2 ) = g1 S(q˜2 ) ⊗ g2 S(q˜1 ).

(10.4.3)

Secondly, observe that the definition (8.5.14) of the antipode SD of D(H) can be reformulated as follows: SD (ϕ  h)

  −1 (ε  S(h)) ( f11 p1 )1U 1  S (ϕ )  f 2 S−1 ( f21 p2 )  ( f11 p1 )2U 2   (10.4.3) −1 = (ε  S(h)) g11 S(q˜2 )1U 1  S (ϕ )  q˜1 S−1 (g2 )  g12 S(q˜2 )2U 2   (7.3.1),(3.2.13) −1 = (ε  S(h)) g11 G1 S(q2 q˜22 )  S (ϕ )  q˜1 S−1 (g2 )  g12 G2 S(q1 q˜21 ) , (8.5.2)

=

where we denote by G1 ⊗ G2 another copy of f −1 . Now, we claim that n

SD (R 2 )αD R 1 = ∑ β  S

−1

(ei )  α  ei ,

(10.4.4)

i=1

where RD = R 1 ⊗ R 2 is the R-matrix of D(H) defined in (10.4.1). Indeed, one can easily check that S

−1

(h  ϕ ) = S

−1

(ϕ )  S(h) and S

−1

(ϕ  h) = S(h)  S

−1

(ϕ ),

(10.4.5)

for all ϕ ∈ H ∗ and h ∈ H. Now, we calculate: SD (R 2 )αD R 1 (10.4.1),(8.5.13)

=

(8.5.2)

=

n

∑ SD (ei  p12 )(ε  α )(ε  S−1 (p2 )ei p11 )

i=1 n

∑

i=1

(3.2.13)

=

=

 1

G1 S(q2 q˜22 )  S

−1

(ei )  q˜1 S−1 (S(p12 )2 g2 )

  S(p12 )1 g1 2 G2 S(q1 q˜21 )α S−1 (p2 )ei p11   n −1 1 1 2 2 1  S (p11  ei )  q˜1 p1(2,1) S−1 (g2 ) g G S q ( q ˜ p ) 2 ∑ 1 (2,2)

i=1

(10.4.5),(3.2.22)

S(p12 )1 g1

   g12 G2 S q1 (q˜2 p1(2,2) )1 α S−1 (p2 )ei

n

∑ g11 G1 S(q2 p12 q˜22 )  S

−1

(ei  S−1 (p2 ))  q˜1 S−1 (g2 )

i=1

 g12 G2 S(q1 p11 q˜21 )α ei (10.4.5),(3.2.23)

=

n

∑ g11 G1 S(q˜22 )  S

i=1

−1

(ei )  q˜1 S−1 (g2 )  g12 G2 S(q˜21 )α ei

10.4 The QT Structure of the Quantum Double n

∑ g11 G1  S

(10.4.5)

=

(3.2.1),(3.2.20)

=

(8.7.7),(3.2.1)

=

−1

399

(ei )  q˜1 S−1 (g2 )  g12 G2 S(q˜21 )α q˜22 ei

i=1 n

∑ ei  g12 G2 α S−1 (α S−1 (g2 )ei g11 G1 )

i=1 n

n

i=1

i=1

∑ ei  S−1 (α ei β ) = ∑ β  S

−1

(ei )  α  ei .

We are now able to calculate the element uD . Since H can be viewed as a quasi-Hopf subalgebra of D(H) via the morphism iD it follows that the corresponding element pR for D(H) is (pR )D = p1D ⊗ p2D = ε  p1 ⊗ ε  p2 . Therefore: uD

(10.3.4)

=

(10.4.4),(8.5.2)

=

SD (R 2 p2D )αD R 1 p1D = (ε  S(p2 ))SD (R 2 )αD R 1 (ε  p1 ) n

∑ S(p2 )(1,1) β  S

−1

i=1 n

∑ ei  S(p2 )(1,2) S−1

= (3.2.1)

=

(3.2.19),(3.2.2)

=

(ei )  α S−1 (S(p2 )2 )  S(p2 )(1,2) ei p1

 α S−1 (S(p2 )2 )ei S(p2 )(1,1) β p1

i=1 n

∑ ei  S−1 (S(p1 )α p2 ei β )

i=1 n

n

i=1

i=1

∑ ei  S−1 (ei β ) = ∑ β  S

−1

(ei )  ei ,

as claimed. For the proof of the next proposition we need the formula S(U 1 )q˜1U12 ⊗ q˜2U22 = f ,

(10.4.6)

where qL = q˜1 ⊗ q˜2 and U = U 1 ⊗ U 2 are the elements defined in (3.2.20) and (7.3.1), and f = f 1 ⊗ f 2 is the Drinfeld’s twist defined in (3.2.15). The formula (10.4.6) follows easily from the axioms and the basic properties of a quasi-Hopf algebra. Proposition 10.25 Let H be a finite-dimensional quasi-Hopf algebra and D(H) its quantum double. Then −2

(ϕ )  F 1 S−1 (g2 )  g12 G2 S( f 1 F12 )S2 (h), (10.4.7) for all ϕ ∈ H ∗ and h ∈ H, where f = f 1 ⊗ f 2 = F 1 ⊗ F 2 is the Drinfeld element defined in (3.2.15) and f −1 = g1 ⊗ g2 = G1 ⊗ G2 is its inverse as in (3.2.16). Fur2 as an inner automorphism of D(H). thermore, the element uD in (10.4.2) defines SD 2 (ϕ  h) = g11 G1 S( f 2 F22 )  S SD

Proof

For all ϕ ∈ H ∗ and h ∈ H we compute:

2 (ϕ  h) SD

= (8.5.14)

=

SD (p11U 1  S

−1

(ϕ )  f 2 S−1 (p2 )  p12U 2 )(ε  S(S(h) f 1 ))

(ε  S(p12U 2 )F 1 )(P11 U 1  S

−1

(p11U 1  S

−1

(ϕ )  f 2 S−1 (p2 ))

400

Quasitriangular Quasi-Hopf Algebras  F 2 S−1 (P2 )  P21 U 2 )(ε  S( f 1 ))(ε  S2 (h)) (8.5.2)

(S(p12U 2 )1 g1 S(q˜2 ))1 U 1 p2 S( f 2 )  S

=

(10.4.3)

(g11 S(p1 q˜2U22 )1 U 1 p2 S( f 2 )  S

=

(3.2.22)

−2

(ϕ )  S(U 1 )q˜1U12 S−1 (g2 )

 g12 S(p1 q˜2U22 )2 U 2 S( f 1 ))(ε  S2 (h))

(7.3.1),(3.2.13)

=

(ϕ )  S(p11U 1 )q˜1

S−1 (S(p12U 2 )2 g2 )  (S(p12U 2 )1 g1 S(q˜2 ))2 U 2 S( f 1 ))(ε  S2 (h))

(3.2.13)

(3.2.21)

−2

2 (g11 G1 S( f 2 q˜22U(2,2) )S

−2

(ϕ )  S(U 1 )q˜1U12 S−1 (g2 )

2  g12 G2 S( f 1 q˜21U(2,1) ))(ε  S2 (h)) (10.4.6)

=

g11 G1 S( f 2 F22 )  S

−2

(ϕ )  F 1 S−1 (g2 )  g12 G2 S( f 1 F12 )S2 (h),

where U 1 ⊗ U 2 is a second copy of the element U defined in (7.3.1). Everything follows now from Theorem 10.17 and Proposition 10.24.

10.5 The Quantum Double D(H) when H is Quasitriangular We show that a finite-dimensional quasi-Hopf algebra H is QT if and only if its quantum double D(H) is a quasi-Hopf algebra with a projection. In this case, D(H) is isomorphic to a biproduct quasi-Hopf algebra between a certain Hopf algebra Bi in i ∗ the braided category H H YD and H. Also, we will show that B equals H as a vector space, but with a different multiplication and comultiplication: the structures of H ∗ in H H YD are induced by the R-matrix and the quasi-Hopf algebra structure of H. Lemma 10.26 Let H be a finite-dimensional quasi-Hopf algebra. Then there exists a quasi-Hopf algebra projection π : D(H) → H covering the canonical inclusion iD : H → D(H) if and only if H is QT. Proof First assume that there is a quasi-Hopf algebra morphism π : D(H) → H such that π ◦ iD = IdH . Then it is not hard to see that R = π (RD1 ) ⊗ π (RD2 ) is an R-matrix for H, where RD = RD1 ⊗ RD2 is the canonical R-matrix of D(H) defined in (10.4.1). Thus H is QT. Conversely, let H be a QT quasi-Hopf algebra with R-matrix R = R1 ⊗ R2 , and define π : D(H) → H by

π (ϕ  h) = ϕ (q2 R1 )q1 R2 h,

(10.5.1)

where qR = q1 ⊗ q2 is as in (3.2.19). We have to show that π is a quasi-Hopf algebra morphism and π ◦ iD = IdH . As before, we write qR = Q1 ⊗ Q2 and R = r1 ⊗ r2 , and then compute for all ϕ , ψ ∈ H ∗ and h, h ∈ H that

π ((ϕ  h)(ψ  h )) (8.5.2)

=

π ((Ω1  ϕ  Ω5 )(Ω2 h(1,1)  ψ  S−1 (h2 )Ω4 )  Ω3 h(1,2) h )

=

ϕ (Ω5 q21 R11 Ω1 )ψ (S−1 (h2 )Ω4 q22 R12 Ω2 h(1,1) )q1 R2 Ω3 h(1,2) h

10.5 The Quantum Double D(H) when H is Quasitriangular (8.5.1)

401

1 1 ϕ (S−1 ( f 2 X 3 )q21 R11 X(1,1) y1 x1 )ψ (S−1 ( f 1 X 2 x3 h2 )q22 R12 X(1,2) y2 x12 h(1,1) )

=

q1 R2 X21 y3 x22 h(1,2) h (10.1.3),(3.2.26)

=

ϕ (q2 Q12 z2 R11 y1 x1 )ψ (S−1 (x3 h2 )Q2 z3 R12 y2 x12 h(1,1) ) q1 Q11 z1 R2 y3 x22 h(1,2) h

(10.1.1)

ϕ (q2 Q12 R1 y1 x1 )ψ (S−1 (x3 h2 )Q2 y3 r1 x12 h(1,1) )q1 Q11 R2 y2 r2 x22 h(1,2) h

=

(3.2.19),(10.1.3)

ϕ (q2 R1 X11 y1 x1 )ψ (S−1 (α X 3 x3 h2 )X 2 y3 x22 h(1,2) r1 )q1 R2 X21 y2 x12 h(1,1) r2 h

(3.1.9),(3.2.19)

ϕ (q2 R1 )ψ (S−1 (h2 )Q2 h(1,2) r1 )q1 R2 Q1 h(1,1) r2 h

= =

(3.2.21)

π (ϕ  h)π (ψ  h ).

=

From (10.1.5) and (3.2.19), it follows that π (ε  h) = h, for any h ∈ H. Thus we have shown that π is an algebra map, and that π ◦ iD = IdH . Since π ◦ iD = IdH , we have that (π ⊗ π ⊗ π )(ΦD ) = Φ. π preserves the comultiplication, since

π ((ϕ  h)1 ) ⊗ π ((ϕ  h)2 ) (8.5.9)

=

π ((ε  X 1Y 1 )(p11 x1  ϕ2  Y 2 S−1 (p2 )  p12 x2 h1 )) ⊗ π (X12  ϕ1  S−1 (X 3 )  X22Y 3 x3 h2 )

ϕ2 (Y 2 S−1 (p2 )q2 R1 p11 x1 )ϕ1 (S−1 (X 3 )Q2 r1 X12 )X 1Y 1 q1 R2 p12 x2 h1

=

⊗ Q1 r2 X22Y 3 x3 h2 (10.1.3)

=

ϕ (S−1 (X 3 )Q2 r1 X12Y 2 S−1 (p2 )q2 p12 R1 x1 ) X 1Y 1 q1 p11 R2 x2 h1 ⊗ Q1 r2 X22Y 3 x3 h2

(3.2.23),(10.1.3)

=

ϕ (S−1 (X 3 )Q2 X22 r1Y 2 R1 x1 )X 1Y 1 R2 x2 h1 ⊗ Q1 X12 r2Y 3 x3 h2

(5.5.17)

ϕ (q2 y3 r1Y 2 R1 x1 )q11 y1Y 1 R2 x2 h1 ⊗ q12 y2 r2Y 3 x3 h2

(10.1.2)

ϕ (q2 R1 )Δ(q1 R2 h) = Δ(π (ϕ  h)).

= =

π also preserves the counit, since (ε ◦ π )(ϕ  h) = ϕ (S−1 (α ))ε (h) = εD (ϕ  h). One can easily see that π (αD ) = α and π (βD ) = β , so we are done if we can show that S ◦ π = π ◦ SD . This follows from the next computation: (π ◦ SD )(ϕ  h) −1

(8.5.14)

=

π ((ε  S(h) f 1 )(p11U 1  S (ϕ )  f 2 S−1 (p2 )  p12U 2 ))

=

S

−1

(ϕ ), f 2 S−1 (p2 )q2 R1 p11U 1 S(h) f 1 q1 R2 p12U 2

(10.1.3),(3.2.23),(7.3.1)

=

S

−1

(ϕ ), f 2 R1 g1 S(q2 )S(h) f 1 R2 g2 S(q1 )

(10.3.6)

S

−1

(ϕ ), S(q2 R1 )S(q1 R2 h) = (S ◦ π )(ϕ  h),

=

and this completes our proof. Remark 10.27

Observe that, in the case where (H, R) is QT, the map π given by

402

Quasitriangular Quasi-Hopf Algebras

(10.5.1) is a quasitriangular morphism, that is, (π ⊗ π )(RD ) = R. Indeed, (π ⊗ π )(RD )

(10.4.1)

=

(10.5.1)

=

n

∑ π ⊗ π , ε  S−1 (p2 )ei p11 ⊗ ei  p12 

i=1 n

∑ ei (q2 R1 )S−1 (p2 )ei p11 ⊗ q1 R2 p12

i=1 (3.2.23) (10.1.3) −1 2 2 1 1 = S (p )q p2 R ⊗ q1 p11 R2 =

R.

Now we will apply the structure theorem of a quasi-Hopf algebra with a projection (Theorem 9.29) in our case, namely when (H, R) is a finite-dimensional QT quasiHopf algebra, B = D(H), i = iD is the canonical inclusion and π is the map defined by (10.5.1). For this, we show first that A := Bco(H) = Im(E D(H) ) is isomorphic to H ∗ as k-vector spaces, where E D(H) is the projection associated to D(H) as in (6.3.7). This follows from E D(H) (ϕ  h)

= (8.5.9)

=

(10.5.1) (8.5.2)

=

(3.2.1)

=

(ϕ  h)1 (ε  β S(π ((ϕ  h)2 )))

ϕ1 (S−1 (X 3 )q2 R1 X12 )(ε  X 1Y 1 )(p11 y1  ϕ2  Y 2 S−1 (p2 )  p12 y2 h1 )(ε  β S(q1 R2 X22Y 3 y3 h2 ))

ε (h)ϕ1 (S−1 (X 3 )q2 R1 X12 )(ε  X 1Y 1 )(p11 y1  ϕ2  Y 2 S−1 (p2 )  p12 y2 β S(q1 R2 X22Y 3 y3 ))

ε (h)E D(H) (ϕ  1H ),

for all ϕ ∈ H ∗ and h ∈ H, so A = E D(H) (D(H)) = E D(H) (H ∗  1H ); this means that we have a surjective k-linear map H ∗ → A, ϕ → E D(H) (ϕ  1H ). This map is also injective since Id ⊗ ε , E D(H) (ϕ  1H ) =

ϕ1 (S−1 (X 3 )q2 R1 X12 )Id ⊗ ε , (ε  X 1Y 1 ) (p11 y1  ϕ2  Y 2 S−1 (p2 )  p12 y2 β S(q1 R2 X22Y 3 y3 )

=

Id ⊗ ε , (ε  X 1Y 1 )(p11 y1  ϕ  S−1 (X 3 )q2 R1 X12Y 2 S−1 (p2 )  p12 y2 β S(q1 R2 X22Y 3 y3 ))

(8.5.2)

=

Id ⊗ ε , (X 1Y 1 )(1,1) p11 y1  ϕ  S−1 (X 3 )q2 R1 X12Y 2 S−1 ((X 1Y 1 )2 p2 )  (X 1Y 1 )(1,2) p12 y2 β S(q1 R2 X22Y 3 y3 )

(10.1.5)

=

X11 p1  ϕ  S−1 (α X 3 )X 2 S−1 (X21 p2 )

(3.2.19),(3.2.23)

=

ϕ.

In fact we have shown that the map

μ : A = E D(H) (D(H)) → H ∗ , μ (E D(H) (ϕ  h)) = ε (h)ϕ is an isomorphism of k-vector spaces, with inverse μ −1 (ϕ ) = E D(H) (ϕ  1H ). From now on, H ∗ will be the k-vector space H ∗ , with the structure of Hopf algebra in the braided category H H YD induced from A via μ . Let us compute the structure maps of YD. H ∗ in H H

10.5 The Quantum Double D(H) when H is Quasitriangular Proposition 10.28 formulas

The structure of H ∗ as an object in

H YD H

is given by the

h · ϕ = h1  ϕ  S−1 (h2 ),

λH ∗ ( ϕ ) = R ⊗ R · ϕ . 2

Proof

403

1

(10.5.2) (10.5.3)

(10.5.2) is easy, and left to the reader. Observe that (5.5.16) yields

E D(H) (ϕ ⊗ 1H ) (ε  X 1Y 1 )(p11 y1  ϕ  S−1 (X 3 )q2 R1 X12Y 2 S−1 (p2 )

=

 p12 y2 β S(q1 R2 X22Y 3 y3 )) (10.1.3),(5.5.17)

=

(3.2.19)

(ε  q11 x1Y 1 )(p11 P1  ϕ  q2 x3 R1Y 2 S−1 (p2 )  p12 P2 S(q12 x2 R2Y 3 ))

(8.5.2),(3.2.25)

=

(ε  q11 x1 ) (y1 p1  ϕ  q2 x3 R1 S−1 (y3 p22 g2 )  y2 p21 g1 S(q12 x2 R2 ))

(10.3.6),(10.1.3)

=

(8.5.2)

 (q11 x1 )(1,2) y2 R2 p22 g2 S(q12 x2 )

(3.1.7),(10.1.3)

=

(3.2.25)

y1 (q11 )1 X 1 p11 P1  ϕ  q2 S−1 (y3 R1 (q11 )(2,1) X 2 p12 P2 )  y2 R2 (q11 )(2,2) X 3 p2 S(q12 )

(3.1.7),(3.2.21)

=

(q11 x1 )(1,1) y1 p1  ϕ  q2 x3 S−1 ((q11 x1 )2 y3 R1 p21 g1 )

y1 X 1 p11 q11 P1  ϕ  S−1 (y3 R1 X 2 p12 q12 P2 S(q2 ))  y2 R2 X 3 p2

(3.2.23),(5.5.16) 1 1

=

y x  ϕ  S−1 (y3 R1 x12 p1 )  y2 R2 x22 p2 S(x3 ).

(10.5.4)

A similar computation, using (10.5.4), leads to 2 π (E D(H) (ϕ  1H )1 ) ⊗ E D(H) (ϕ  1H )2 = X 1Y 1 r2 z2 y21 R21 x(2,1) p21 S(x3 )1 2 ⊗ X12Y 2 r1 z1 y1 x1  ϕ  S−1 (X 3 y3 R1 x12 p1 )  X22Y 3 z3 y22 R22 x(2,2) p22 S(x3 )2 . (10.5.5)

Here R = r1 ⊗ r2 is another copy of R. Since Id ⊗ ε , E D(H) ((ϕ  h)(Ψ  h )) = Id ⊗ ε , (ϕ  h)(Ψ  h )

ε (h )(X11 x1  ϕ  S−1 ( f 2 X 3 ))(X21 x2 h1  Ψ  S−1 ( f 1 X 2 x3 h2 )), (10.5.6) for all ϕ , Ψ ∈ H ∗ and h, h ∈ H, we obtain, using (9.2.13), that

λH ∗ ( ϕ )

=

q11 X 1 r2 y2 R2 x22 p2 S(q2 x3 ) q1(2,1) X 2 r1 y1 x1  ϕ  S−1 (q1(2,2) X 3 y3 R1 x12 p1 )

(3.1.7),(10.1.3)

=

X 1 r2 q1(1,2) y2 R2 x22 p2 S(q2 x3 ) ⊗X 2 r1 q1(1,1) y1 x1  ϕ  S−1 (X 3 q12 y3 R1 x12 p1 )

(3.1.7),(10.1.3)

=

(5.5.17)

2 X 1 r2 y2 R2 q12Y(1,2) p2 S(q2Y22 )Y 3 2 ⊗ X 2 r1 y1Y 1  ϕ  S−1 (X 3 y3 R1 q11Y(1,1) p1 )

(3.2.21),(3.2.23)

X 1 r2 y2 R2Y 3 ⊗ X 2 r1 y1Y 1  ϕ  S−1 (X 3 y3 R1Y 2 )

(10.1.1),(10.5.2)

R2 ⊗ R1 · ϕ ,

= =

404

Quasitriangular Quasi-Hopf Algebras

as stated. Our next goal is to compute the algebra structure of H ∗ . Lemma 10.29 Let H be a finite-dimensional quasi-Hopf algebra and A a quasibialgebra. Let π : A → H be a quasi-bialgebra map that is a left inverse of the quasi-bialgebra map i : H → A. Then E D(H) satisfies the equation E D(H) (E D(H) (a) ◦ a ) = E D(H) (a) ◦ E D(H) (a ),

(10.5.7)

for all a, a ∈ D(H), where i and ◦ are as in Proposition 4.3, specialized to iD . Proof From the definition of an alternative projection we have that E D(H) (ai(h)) = ε (h)E D(H) (a), E D(H) (i(h)a) = h i E D(H) (a) and E D(H) (aa ) = a1 E D(H) (a )i(S(π (a2 ))), for all a, a ∈ D(H), h ∈ H. Then we compute: E D(H) (E D(H) (a) ◦ a ) = (3.2.19)

E D(H) (i(X 1 )Π(a)i(S(x1 X 2 )α x2 X13 )a i(S(x3 X23 )))

=

q1 i E D(H) (E D(H) (a)i(S(q2 ))a )

=

q1 i [E D(H) (a)1 E D(H) (i(S(q2 ))a )i(S(π (E D(H) (a)2 )))]

(∗)

=

q1 i [i(x1 )E D(H) (a)i(S(x23 X 3 ) f 1 )E D(H) (i(S(q2 ))a ) i(S(x2 X 1 β S(x13 X 2 ) f 2 ))]

(3.2.13)

=

q1 i [i(x1 )E D(H) (a)i(S(q22 x23 X 3 ) f 1 )E D(H) (a ) i(S(x2 X 1 β S(q21 x13 X 2 ) f 2 ))]

(3.1.9),(3.2.1)

=

q1 X 1 i [i(x1 )E D(H) (a)i(S(q22 X 3 ) f 1 )E D(H) (a ) i(S(x2 β S(q21 X 2 x3 ) f 2 ))]

(3.2.26),(3.2.19) 1

=

q Q11 y1(1,1) i [i(p1 )E D(H) (a)i(S(Q2 y12 )y2 )E D(H) (a ) i(S(p2 S(q2 Q12 y1(1,2) )y3 ))]

(3.2.21)

i(q11 p1 Q1 y11 )E D(H) (a)i(S(Q2 y12 )y2 )E D(H) (a )i(S(q12 p2 S(q2 )y3 ))

(3.2.23)

i(Q1 y11 )E D(H) (a)i(S(Q2 y12 )y2 )Π(a )i(S(y3 ))

= =

(3.2.19),(3.1.9)

=

(3.2.1)

E D(H) (a) ◦ E D(H) (a ),

as desired, where (*) refers to the fact that E D(H) (a) ∈ D(H)co(H) . Proposition 10.30

The structure of H ∗ as a Hopf algebra in H H YD is defined by

ϕ ◦ Ψ = (x1 X 1  ϕ  S−1 ( f 2 x23Y 3 R1 X 2 )) (x2Y 1 R21 X13  Ψ  S−1 ( f 1 x13Y 2 R22 X23 )), Δ(ϕ ) = X11 p1  ϕ2  S−1 (X21 p2 ) ⊗ X 2 ε (ϕ ) = ϕ (S−1 (α )),

 ϕ1

 S−1 (X 3 ),

(10.5.8) (10.5.9) (10.5.10)

10.5 The Quantum Double D(H) when H is Quasitriangular S(ϕ ) = (R1  ϕ  u−1 S(R2 )) ◦ S.

405 (10.5.11)

The unit element is ε . It follows from (10.5.7) that the multiplication ◦ on H ∗ is

Proof

ϕ ◦ Ψ = Id ⊗ ε , E D(H) (ϕ  1H ) ◦ E D(H) (Ψ  1H ) = Id ⊗ ε , E D(H) (E D(H) (ϕ  1H ) ◦ (Ψ  1H )). Now we can see that E D(H) (ϕ  1H ) ◦ (Ψ  1H ) i(X 1 )E D(H) (ϕ  1H )i(S(z1 X 2 )α z2 X13 )(Ψ  1H )i(S(z3 X23 ))

= (3.1.9),(3.2.1)

=

(10.5.4)

i(q1 z11 )(y1 x1  ϕ  S−1 (y3 R1 x12 p1 )  y2 R2 x22 p2 S(x3 )) i(S(q2 z12 )z2 )(Ψ  S(z3 ))

(8.5.2),(3.1.7)

=

(10.1.3)

(y1 q11 z1(1,1) x1  ϕ  S−1 (y3 R1 q1(2,1) (z1(1,2) x2 )1 p1 )  y2 R2 q1(2,2) (z1(1,2) x2 )2 p2 S(q2 z12 x3 )z2 )(Ψ  S(z3 ))

(3.1.7),(3.2.21)

=

[y1 q11 x1 z11  ϕ  S−1 (y3 R1 q1(2,1) x12 p1 z12 )  y2 R2 q1(2,2) x22 p2 S(q2 x3 )z2 ](Ψ  S(z3 ))

(5.5.17)

2 2 [y1 X 1 z11  ϕ  S−1 (y3 R1 q11 X(1,1) p1 z12 )  y2 R2 q12 X(1,2) p2

=

S(q2 X22 )X 3 z2 ](Ψ  S(z3 )) (3.2.21),(3.2.23) 1

y X 1 z11  ϕ  S−1 (y3 R1 X 2 z12 )  y2 R2 X 3 z2 )(Ψ  S(z3 )).

=

By using (10.5.6) and (3.1.9) we obtain (10.5.8). It is easy to prove that ε is the unit element of H ∗ . (10.5.9–10.5.10) follow from the following formula for comultiplication in D(H) and the axioms of a quasi-bialgebra (we leave the details to the reader): Δ(E D(H) (ϕ  1H )) = E D(H) (i(X 1 )(ϕ  1H )1 ) ⊗ i(X 2 )(ϕ  1H )2 i(S(X 3 )). After a straightforward, but long and tedious, computation using (10.5.5), (8.5.14) and (10.5.6), we get that the antipode S of H ∗ is given by S(ϕ ) = Q1 q1 R2 x2 · [p1 P2 S(Q2 )  S

−1

(ϕ )  S(q2 R1 x1 P1 )x3 S−1 (p2 )],

for all ϕ ∈ H ∗ . This formula is equivalent to the one in (10.5.11) since it implies S(ϕ S)

(10.5.2)

=

Q11 q11 R21 x12 p1 P2 S(Q2 )  ϕ  S(q2 R1 x1 P1 ) x3 S−1 (Q12 q12 R22 x22 p2 )

(10.1.2)

=

Q11 q11 y1 X 1 r2 z2 x12 p1 P2 S(Q2 )  ϕ  S(q2 y3 R1 X 2 r1 z1 x1 P1 )x3 S−1 (Q12 q12 y2 R2 X 3 z3 x22 p2 )

(5.5.16),(5.5.17)

=

(10.1.3),(3.2.19) (3.1.9),(3.2.1)

=

Q11Y 1 X 1 y11 r2 P2 S(Q2 )  ϕ  S(q2 R1Y12 X 2 y12 r1 P1 )Y 3 y3 S−1 (Q12 q1 R2Y22 X 3 y2 β ) Q11 r2 P2 S(Q2 )  ϕ  S(q2 R1 x1 r1 P1 )S−1 (Q12 q1 R2 x2 β S(x3 ))

406

Quasitriangular Quasi-Hopf Algebras (3.2.19),(10.3.9)

=

Q11 r2 P2 S(Q2 )  ϕ  S−1 (Q12 u−1 S2 (r1 P1 ))

(10.3.10),(10.1.3) 2

=

r Q12 P2 S(Q2 )  ϕ  S−1 (r1 Q11 P1 u−1 )

(3.2.23),(10.3.10) 2

=

r  ϕ  S(r1 )S−1 (u−1 ) = (r1  ϕ S  u−1 S(r2 )) ◦ S,

for all ϕ ∈ H ∗ , where the last equality is based on the fact that S2 (u−1 ) = u−1 (see Lemma 10.16) and on the following computation, for h ∈ H: ϕ , S(r1 )S−1 (u−1 )hr2 

(10.3.10)

=

ϕ , S(r1 )S2 (hr2 )S−1 (u−1 )

=

ϕ S, S−2 (u−1 )S(hr2 )r1 .

Now (10.5.11) follows from the bijectivity of the antipode S of H. Remarks 10.31

(1) The quasi-Hopf algebra isomorphism χ : H ∗ × H → D(H) is

χ (ϕ × h) = x1 X 1  ϕ  S−1 (x3 R1 X 2 )  x2 R2 X 3 h.

(10.5.12)

(2) Let (H, R) be a QT quasi-Hopf algebra. By the left-handed version of Corollary 8.4 we have a braided monoidal functor Zl : H M → Zl (H M ) ≡ H H YD which sends algebras, coalgebras, bialgebras, etc. in H M to the corresponding objects in H H YD. If M ∈ H M then Zl (M) = M as left H-module, and together with the left H-coaction given by

λM : M → H ⊗ M,

λM (m) := R2 ⊗ R1 · m, ∀ m ∈ M,

(10.5.13)

it becomes a left Yetter–Drinfeld module over H. Zl maps a morphism to itself. We observe that H ∗ lies in the image of Zl , that is, the left H-coaction is given by (10.5.13), which is exactly what we proved in Proposition 10.28.

10.6 Notes QT quasi-Hopf algebras were introduced by Drinfeld in [82]. That the square of the antipode of a QT quasi-Hopf algebra (H, R) is inner was proved for the first time in [9], but under the condition S bijective and R invertible. An alternative definition for a QT quasi-Hopf algebra (H, R) was given in [55], and it was shown that with this definition R is invertible and the square of the antipode S of H is inner, so in particular bijective; therefore it is equivalent to the initial one given by Drinfeld. The techniques used in [55] are similar to those used in the Hopf case [81, 187]. The content of Section 10.2 is taken from [173]. The presentation in Section 10.4 is from [64], although the R-matrix of D(H) was found earlier by Hausser and Nill [108]. In Section 10.5 we presented the content of [48], were it was proved a quasiHopf algebra version of a result of Majid from [144].

11 Factorizable Quasi-Hopf Algebras

We introduce the notion of factorizable quasi-Hopf algebras by using a categorical point of view. We show that the quantum double D(H) of any finite-dimensional quasi-Hopf algebra H is factorizable, and we characterize D(H) when H itself is factorizable. Finally, we prove that any finite-dimensional factorizable quasi-Hopf algebra is unimodular. In particular, we obtain that the quantum double D(H) is a unimodular quasi-Hopf algebra.

11.1 Reconstruction in Rigid Monoidal Categories Throughout this section, except in Proposition 11.1, F is a strong monoidal functor between the monoidal categories C and D. We further assume that the functor Nat(− ⊗ F, F) : C opp → Set is representable, where Nat(− ⊗ F, F) stands for the set of natural transformations ξ : − ⊗ F → F, and where for any object M of D we denote by M ⊗ F : C → D the functor that sends N ∈ ObC to (M ⊗ F)(N) = M ⊗ F(N); if f : N → N  is a morphism in C then (M ⊗ F)( f ) = IdM ⊗ F( f ), a morphism in D. In other words, there exists an object B of D such that ∼ =

θM : HomD (M, B) → Nat(M ⊗ F, F), ∀ M ∈ Ob(D),

(11.1.1)

by functorial bijections. If this is the case then we say that F satisfies the representability assumption for modules. We show that under some suitable conditions B has a Hopf algebra structure in D, U and F factors as F : C → B D → D, where U is the forgetful functor. We start by showing that B has an algebra structure in D. Proposition 11.1 Let F : C → D be a functor between two monoidal categories, satisfying the representability assumption for modules (11.1.1). Then B becomes an U algebra in D, and F factors as F : C → B D → D, where U is the forgetful functor. Proof Take μ = θB (IdB ), a natural transformation between B ⊗ F and F. Then μ determines completely θM since, by the functoriality of θ− , for any morphism

408

Factorizable Quasi-Hopf Algebras

f : M → B in D the diagram θB

HomD (B, B)

Nat(B ⊗ F, F) Nat( f ⊗F,F)

HomD ( f ,B)

HomD (M, B)

θM

Nat(M ⊗ F, F)

is commutative, and so θM ( f ) = (μX ( f ⊗ IdF(X) ))X∈Ob(C ) . Thus, if we denote μX = B

F(X) μX

then, for any M ∈ Ob(D),

F(X)

⎛

M

F(X)

⎜ fh ⎜ μX θM ( f ) = ⎜ ⎜ ⎝

⎞ ⎟ ⎟ ⎟ ⎟ ⎠

F(X)

, ∀ f ∈ HomD (M, B).

(11.1.2)

X∈Ob(C )

Define ξ := (μX (IdB ⊗ μX ))X∈Ob(C ) , a natural transformation from B ⊗ B ⊗ F to F. Then there exists a unique morphism mB : B ⊗ B → B such that θB⊗B (mB ) = ξ . Hence mB is the unique morphism in D obeying B B F(X)

B B F(X)

μX

F(X)

It follows that

μX

, ∀ X ∈ Ob(C ).

=

(11.1.3)

μX

F(X)

⎛

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ θB⊗B⊗B (mB (mB ⊗ IdB )) = θB⊗B⊗B (mB (IdB ⊗ mB )) = ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

B B B F(X) μX

μX

μX

F(X)

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

.

X∈Ob(C )

By our assumption we get that mB is associative in D. A unit for it is the unique morphism η B : 1 → B such that θ1 (η B ) = 1F , the identity natural transformation of F. In other words, η B is the unique morphism from 1 to B in D such that

μX (η B ⊗ IdF(X) ) = IdF(X) , ∀ X ∈ Ob(C ).

(11.1.4)

11.1 Reconstruction in Rigid Monoidal Categories

409

By the above definitions one can see that θB (mB (η B ⊗ IdB )) = θB (mB (IdB ⊗ η B )) = μ = θB (IdB ), hence η B is indeed a unit for mB . So we have proved that (B, mB , η B ) is an algebra in D. Finally, observe that the definition of μ was designed in such a way that each μX : B ⊗ F(X) → F(X) provides a left B-module structure on F(X) in D. Therefore U F factors as F : C → B D → D, as stated. In order to build a coalgebra structure on B in D we need extra assumptions on s : F. Namely, we ask that D is braided and that for any M ∈ Ob(C ) the maps θM ⊗s ⊗s ⊗s HomD (M, B ) → Nat(M ⊗ F , F ), s ∈ {2, 3}, given by ⎞ ⎛ M

θM2

:

f

B B

μX

⎛

M

θM3 :

f

B B B

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜  → ⎜ − ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

f

μY

F(Y )

F(X)

and by

F(Y )

F(X)

M

⎜ ⎜ ⎜ ⎜ ⎜  → ⎜ − ⎜ ⎜ ⎜ ⎜ ⎝

M F(X)

X,Y ∈Ob(C )

F(Y )

F(Z)

f

μZ μY μX

F(X)

F(Y )

F(Z)

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ X,Y,Z∈Ob(C )

are bijections. This will be called the higher representability assumption for modules. Under these extra assumptions, the coalgebra structure on B can be defined as follows. Without loss of generality, we can assume (as before) that C , D are strict monoidal categories and that F is a strict monoidal functor, thus F(X ⊗ Y ) and F(X) ⊗ F(Y ) identify as objects of D, for all X,Y ∈ Ob(C ), and F(1) is just the unit object of D. Consequently, we can see μX⊗Y either as a morphism from B⊗F(X ⊗Y ) to F(X ⊗Y ) or as a morphism from B ⊗ F(X) ⊗ F(Y ) → F(Y ) ⊗ F(Y ) in D, for all X,Y ∈ Ob(C ), and μ1 as a morphism from B to 1 in D. Proposition 11.2 Let F : C → D be a strong monoidal functor satisfying the representability assumption for modules (11.1.1). If D is braided and F satisfies the higher representability assumption for modules then B has a bialgebra structure in U D, and F factors, as a monoidal functor, as F : C → B D → D.

410 Proof

Factorizable Quasi-Hopf Algebras By the above comments we have that

ξ = (μX⊗Y : B ⊗ F(X) ⊗ F(Y ) → F(X) ⊗ F(Y ))X,Y ∈Ob(C ) is a natural transformation between B ⊗ F ⊗2 and F ⊗2 . So by our higher assumption there exists a unique morphism ΔB : B → B ⊗ B in D such that θB2 (ΔB ) = ξ . B

Equivalently, ΔB =

 is the unique morphism in D from B to B ⊗ B such that B B B

F(Y )

F(X)



μX⊗Y =

, ∀ X, Y ∈ Ob(C ). μX

(11.1.5)

μY

F(Y )

F(X)

We have that, for all X,Y, Z ∈ Ob(C ), B F(X) F(Y ) F(Z)



B F(X) F(Y ) F(Z)



 = μY μX

= μ(X⊗Y )⊗Z

μX⊗Y μZ

μZ

F(Y )

F(X)

F(X) F(Y )

F(Z)

F(Z)

and similarly B

F(X) F(Y ) F(Z)

B F(X)





 μX

μY μX

F(X)

μY μZ

F(Y )

F(Z)



(1.5.6)

=

F(Y )

μZ

F(Z)

F(X)

F(Y )

F(Z)

B F(X) F(Y ) F(Z)



μY ⊗Z

= μX

F(X)

F(Y ) F(Z)

= μX⊗(Y ⊗Z) .

11.1 Reconstruction in Rigid Monoidal Categories

411

Since we assumed C strict monoidal we have θB3 ((ΔB ⊗IdB )ΔB ) = θB3 ((IdB ⊗ΔB )ΔB ), and so ΔB is coassociative in D. We claim that ε B := μ1 : 1 → B is a counit for ΔB . One can see that θB ((ε B ⊗ IdB )ΔB ) = (μ1⊗X )X∈Ob(C ) = μ = θB (IdB ) and θB ((IdB ⊗ ε B )ΔB ) = (μX⊗1 )X∈Ob(C ) = μ = θB (IdB ). By our assumptions we get (ε B ⊗ IdB )ΔB = IdB = (IdB ⊗ ε B )ΔB . For any two objects X,Y of D we compute that B B F(X) F(Y )



=

μX

μX⊗Y

=

μX⊗Y

μX⊗Y

μY

F(X) F(Y ) F(X)

F(X) F(Y )

B B

B B F(X) F(Y )

F(X) F(Y )

F(Y ) B

B F(X) F(Y )





B

B

F(Y )

F(X)



μX μY

(1.5.5)

=

μX

=

μX

μY

μX

μY

F(X)

F(Y )

μY

F(Y )

F(X) B

B

B

B

F(X) F(Y )



F(X) F(Y )



=



μX

μY

F(X)

F(Y )

(1.5.6)

=

. μX

μY

F(X)

F(Y )

2 Together with the fact that θB⊗B is a bijection this implies that B is a bialgebra in D. By (11.1.5) it follows that the bialgebra structure on B is given in such a way that F U factors, as a monoidal functor, as F : C → B D → D. So our proof is finished.

As in Section 3.5, to have an antipode for B we need further to assume that C is a left rigid monoidal category. Consequently, since F is strong monoidal, by Proposition 1.67 if follows that F(X ∗ ) is a left dual object for F(X) in D whenever X ∗ is a left dual object for X in C . Furthermore, if we assume F strict monoidal then F(X ⊗ Y ) and F(X) ⊗ F(Y ) can be identified, for all X,Y ∈ Ob(C ), hence in this

412

Factorizable Quasi-Hopf Algebras

case (F(coevX ), F(evX )) : F(X ∗ )  F(X) is an adjunction in D. In other words, the corestriction of F at its image is a left rigid monoidal functor, that is, we have a natural isomorphism F(X ∗ ) ∼ = F(X)∗ , indexed by X ∈ Ob(C ). For the adjunction (F(coevX ), F(evX )) : F(X ∗ )  F(X) we keep the same notation as in Section 1.6. Theorem 11.3 Let F : C → D be a strong monoidal functor, where C is a left rigid monoidal category and (D, d) is a braided category. If F satisfies the representability assumption (11.1.1) and the higher representability assumption for modules then B is a Hopf algebra in D and F factors, as a left rigid monoidal functor, as F : C → U B D → D. Proof From the above results and comments, we only have to show that B admits an antipode S. To this end, define

ζ = F(IdX ⊗ evX )(IdF(X) ⊗ μX ∗ ⊗ IdF(X) )  (dB,F(X) ⊗ IdF(X ∗ ⊗X) )(IdB ⊗ F(coevX ⊗ IdX )) X∈Ob(C ) . Clearly, ζ is a natural transformation between B⊗F and F, and since θB is a bijection there exists S : B → B such that θB (S) = ζ . Equivalently, S is uniquely determined by the equations B B F(X)

F(X)



h

S

μX

, ∀ X ∈ Ob(C ).

μX ∗

=

(11.1.6)



F(X) F(X)

We end by proving that S is an antipode for the bialgebra B in D. Observe that the naturality of μ and the fact that coevX : 1 → X ⊗ X ∗ is a morphism in C imply B



B

r  =

F(X)

μX⊗X ∗

F(X ∗ ) F(X) F(X ∗ )

This allows us to compute that B

F(X)

B

 Sh



F(X)

μX

=

F(X)

B

 Sh

 μX ∗

=

μX

μX

μX

F(X)

F(X)

F(X)



.

11.1 Reconstruction in Rigid Monoidal Categories B

F(X)

B F(X)



F(X)

B

μX⊗X ∗

=

413

r 

B F(X)

= r



=



F(X)

=

r r

,

μX

F(X)

F(X)

F(X)

for all X ∈ Ob(C ). In a similar manner, by using the equality ∗ B F(X )

F(X) ∗ B F(X ) F(X)

μX ∗ ⊗X

=

r

,





1

1

that is a consequence of the fact that μ is a natural transformation and evX : X ∗ ⊗X → 1 is a morphism in C , we deduce that B F(X)

B F(X)

 Sh



 Sh

μX



 =

F(X)



μX

(1.5.6)

=

(1.5.6)

=

μX ∗

μX ∗

μX ∗

μX

F(X)

B

μX





F(X)

F(X)



F(X) B

 =

F(X)

F(X)

μX ∗ ⊗X



B F(X)

r F(X)

B F(X)

B F(X)

 =

F(X)



μX

μX

=

B



F(X)

B

r =



B F(X)

= r

=

r r

μX

,

F(X)

F(X)

F(X)

F(X)

for all X ∈ Ob(C ). Since θB is bijective, S satisfies (2.7.1), as needed. As an application of Theorem 11.3, in the next section we will give the explicit structure of B as a braided Hopf algebra, if (H, R) is a QT quasi-Hopf algebra, C = H M fd and F = IdC . As we shall see next, in such a particular case the comultiplication of B has an extra property, which reduces to the braided cocommutativity relation in the case when C is, moreover, a symmetric monoidal category. Corollary 11.4 Let (C , c) be a braided category and assume that the identity functor of C satisfies the representability assumption (11.1.1) and the higher representability assumption for modules. Then B admits a bialgebra structure in C such

414

Factorizable Quasi-Hopf Algebras

that its comultiplication is weakly braided cocommutative, in the sense that B

X

B

= μX

μY

X

Y

X

Y



Y



, ∀ X, Y ∈ Ob(C ).

μX

(11.1.7)

μY

X

Y

Consequently, if C is a symmetric (resp. symmetric and rigid) category then B is a braided cocommutative bialgebra (resp. Hopf algebra) in C . Proof

The relations in (11.1.7) follow from the following computation: B

X

Y



(∗)

= μX⊗Y = μX

μY

X

Y

B

μY ⊗X

X Y



B X Y

μY

=

μX

X Y B

X

X Y



B

X Y

Y

(1.5.5)

=

=

μY

X Y

X



μX

(1.5.5)

=

B



μX

(1.5.5)

Y

μY

X

Y

,

μX μY

X

Y

where in (*) we used that μ is a natural transformation and cX,Y : X ⊗Y → Y ⊗ X is a morphism in C . Thus, the braided reconstruction theorem associates to any braided rigid category C a braided Hopf algebra B which acts on each object of C and is weakly braided cocommutative. We call B the automorphism braided group of C .

11.2 The Enveloping Braided Group of a QT Quasi-Hopf Algebra Throughout this section, (H, R) is a QT quasi-Hopf algebra with R-matrix R = R1 ⊗ R2 , C is the braided category H M , HL is the vector space H with the left regular action and B is the same vector space H but with the left adjoint action defined by (8.7.1). If M, N ∈ C , let HomH (M, N) be the set of H-linear maps between M and N.

11.2 The Enveloping Braided Group of a QT Quasi-Hopf Algebra

415

Our goal is to apply Corollary 11.4 to the above setting. The first step is to show that B as above is the object we need. Lemma 11.5 Let H be a quasi-Hopf algebra with bijective antipode, pR = p1 ⊗ p2 and qR = q1 ⊗ q2 the elements defined by (3.2.19) and M ∈ C . Then, if we define for all ξ ∈ Nat(M ⊗ Id, Id)

σM : Nat(M ⊗ Id, Id) → HomH (M, B), σM (ξ )(m) = ξHL (p1 · m ⊗ p2 ), ∀ m ∈ M, then σM is well defined and a functorial bijection with inverse θM given for all N ∈ C , λ ∈ HomH (M, B), m ∈ M and n ∈ N by:

θM (λ )N : M ⊗ N → N, Proof

θM (λ )N (m ⊗ n) = q1 λ (m)S(q2 ) · n.

We first have to check that σM (ξ ) is H-linear. For all h ∈ H we have: h  σM (ξ )(m)

=

h1 ξHL (p1 · m ⊗ p2 )S(h2 )

=

ξHL (h(1,1) p1 · m ⊗ h(1,2) p2 S(h2 ))

(3.2.21)

=

ξHL (p1 h · m ⊗ p2 ) = σM (ξ )(h · m),

where the second equality uses that ξHL is H-linear and ξ is functorial under the morphism HL → HL defined by right multiplication. It follows that σ is functorial and θM (λ ) is a natural transformation. To see that θM is well defined we have to check that each θM (λ )N is H-linear. That is true since for all h ∈ H

θM (λ )N (h · (m ⊗ n)) = θM (λ )N (h1 · m ⊗ h2 · n) =

q1 λ (h1 · m)S(q2 )h2 · n

(∗)

q1 h(1,1) λ (m)S(q2 h(1,2) )h2 · n

=

(3.2.21)

=

hq1 λ (m)S(q2 ) · n

=

h · θM (λ )N (m ⊗ n),

where in (*) we used the fact that λ is H-linear. Thus we only have to show that σM and θM are inverses. Now, (σM ◦ θM )(λ )(m) = θM (λ )HL (p1 · m ⊗ p2 ) = q1 λ (p1 · m)S(q2 )p2 (∗) 1

= q (p1  λ (m))S(q2 )p2 (3.2.23)

= q1 p11 λ (m)S(q2 p12 )p2 = λ (m), where in (*) we again used the fact that λ is H-linear. Similarly, (θM ◦ σM )(ξ )N (m ⊗ n)

=

q1 σM (ξ )(m)S(q2 ) · n

=

q1 ξHL (p1 · m ⊗ p2 )S(q2 ) · n

=

ξHL (q11 p1 · m ⊗ q12 p2 S(q2 )) · n

(3.2.23)

=

ξHL (m ⊗ 1) · n = ξN (m ⊗ n),

where the last equality uses that ξ is functorial under HL h → h · n ∈ N.

416

Factorizable Quasi-Hopf Algebras

By the above lemma, the natural transformation μ corresponding to the identity morphism IdB is μ = {μN | N ∈ C }, where each μN : B ⊗ N → N is given by

μN (h ⊗ n) = θB (IdB )N (h ⊗ n) = q1 hS(q2 ) · n, ∀ h ∈ H, n ∈ N.

(11.2.1)

Now, we are able to begin our reconstruction. The reconstruction of the algebra structure on B does not involve the braiding and is given as follows. Lemma 11.6 B = H0 as an algebra, where the algebra structure of H0 in C is given in Definition 4.4. Proof Following the proof of Proposition 11.1, the multiplication on B is obtained as mB = θB⊗B (ξ ), where for all N ∈ C

ξN : (B ⊗ B) ⊗ N → N,

ξN = μN ◦ (IdB ⊗ μN ) ◦ aB,B,N .





More precisely, ξN ((h ⊗ h ) ⊗ n) = Q1 (X 1  h)S(Q2 )q1 (X 2  h )S(q2 )X 3 · n, where we denote by Q1 ⊗ Q2 another copy of qR , and therefore: mB (h ⊗ h )



=

ξHL (p1  (h ⊗ h ) ⊗ p2 )

=

Q1 (X 1 p11  h)S(Q2 )q1 (X 2 p12  h )S(q2 )X 3 p2

(3.2.19)

=





Q1 (X 1 x11  h)S(Q2 )q1 (X 2 x21  h )S(q2 )X 3 x2 β S(x3 ) 

(3.2.19),(3.2.1)

Q1 (y1  h)S(Q2 )q1 y2(1,1) (x1  h )S(q2 y2(1,2) )y22 x2 β S(y3 x3 )

(3.2.21),(3.2.19)

X 1 y11 hS(X 2 y12 )α X 3 y2 q1 (x1  h )S(q2 )x2 β S(y3 x3 )

(3.1.9),(3.2.19)

X 1 hS(y1 X 2 )α y2 X13 q1 p11 h S(q2 p12 )p2 S(y3 X23 )

(3.2.23),(4.1.9)

h ◦ h .

= = = =





Finally, the unit for B is η (1) = θk (r)(1) = rHL (p1 · 1 ⊗ p2 ) = rHL (1 ⊗ β ) = β , where r denotes the right unit constraint, and this finishes the proof. We now assume that the higher representability assumption for modules holds. By the proof of Proposition 11.2, the coproduct Δ of B is characterized as being the unique morphism Δ : B → B ⊗ B in C such that:

μM⊗N = (μM ⊗ μN ) ◦ a−1 B,M,B⊗N ◦ (IdB ⊗ aM,B,N ) ◦ (IdB ⊗ (cB,M ⊗ IdN )) ◦ (IdB ⊗ a−1 B,M,N ) ◦ aB,B,M⊗N ◦ (Δ ⊗ IdM⊗N ), for all M, N ∈ C . Explicitly, the map Δ is characterized by (q1 hS(q2 ))1 · m ⊗ (q1 hS(q2 ))2 · n = q1 (y1 X 1  h1 )S(q2 )y2Y 1 R2 x2 X13 · m ⊗ Q1 (y31Y 2 R1 x1 X 2  h2 )S(Q2 )y32Y 3 x3 X23 · n, for all m ∈ M, n ∈ N, where qR = q1 ⊗ q2 = Q1 ⊗ Q2 . We used the braiding (10.1.4), computed h · (m ⊗ n) in the usual way and set Δ(h) = h1 ⊗ h2 . Since the above equality is true for all m ∈ M, n ∈ N, we conclude that Δ is the unique morphism in C satisfying the following equality, for all h ∈ H: Δ(q1 hS(q2 )) = q1 (y1 X 1  h1 )S(q2 )y2Y 1 R2 x2 X13

11.2 The Enveloping Braided Group of a QT Quasi-Hopf Algebra ⊗ Q1 (y31Y 2 R1 x1 X 2  h2 )S(Q2 )y32Y 3 x3 X23 .

417 (11.2.2)

Now, under our assumption, the explicit formula for Δ is given by the following: Lemma 11.7

B is a coalgebra in C with comultiplication

Δ(h) = x1 X 1 h1 g1 S(x2 R2 y3 X23 ) ⊗ x3 R1  y1 X 2 h2 g2 S(y2 X13 ),

(11.2.3)

for all h ∈ H, and counit ε = ε . Proof We prove that Δ defined by (11.2.3) is a morphism in C which satisfies (11.2.2). The fact that Δ is a H-linear map follows by applying (3.2.13), (3.1.7) and (10.1.3) several times. If we denote by r1 ⊗ r2 another copy of R then, for all h ∈ H, we have: q1 (y1 X 1  h1 )S(q2 )y2Y 1 R2 x2 X13 ⊗ Q1 (y31Y 2 R1 x1 X 2  h2 )S(Q2 )y32Y 3 x3 X23 =

q1 y11 X11 z1 Z 1 h1 g1 S(q2 y12 X21 z2 r2t 3 Z23 )y2Y 1 R2 x2 X13 ⊗ Q1 y3(1,1) (Y 2 R1 x1 X 2 z3 r1  t 1 Z 2 h2 g2 S(t 2 Z13 ))S(Q2 y3(1,2) )y32Y 3 x3 X23

(3.1.9),(3.2.21)

=

(3.1.7),(10.1.3)

q1 y11 z1 X 1 Z 1 h1 g1 S(q2 y12 z2 T 1 r2 X22t 3 Z23 )y2Y 1 R2 z3(1,2) x2 T13 X13 ⊗ y3 Q1 (Y 2 R1 z3(1,1) x1 T 2 r1 X12  t 1 Z 2 h2 g2 S(t 2 Z13 ))S(Q2 )Y 3 z32 x3 T23 X23

(3.1.9),(10.1.3)

=

(3.1.7),(10.1.3)

q1 y11 z1 X 1 Z 1 h1 g1 S(q2 y12 z2 T 1V 1 r2 v12 X22t 3 Z23 )y2 z31Y 1 T12 R2V 3 v2 X13 ⊗ y3 Q1 (z3(2,1)Y 2 T22 R1V 2 r1 v11 X12  t 1 Z 2 h2 g2 S(t 2 Z13 )) S(Q2 )z3(2,2)Y 3 T 3 v3 X23

(10.1.1),(3.2.21)

=

(3.2.19),(3.1.9) (3.2.21),(3.1.9)

=

(3.2.1),(10.1.5) (3.1.7),(3.1.9)

=

(3.2.1),(3.1.9)

Y 1 y11 z1 X 1 Z 1 h1 g1 S(Y 2 y12 z2V 1 T11 R21W 2 v12 X22t 3 Z23 )α Y 3 y2 z31V 2 T21 R22W 3 v2 X13 ⊗ y3 z32 Q1 (V13 T 2 R1W 1 v11 X12  t 1 Z 2 h2 g2 S(t 2 Z13 ))S(Q2 )V23 T 3 v3 X23 X 1 Z 1 h1 g1 S(W 2 v12 X22t 3 Z23 )α W 3 v2 X13 2 2 ⊗ Q1 (W 1 v11  X(1,1) t 1 Z 2 h2 g2 S(X(1,2) t 2 Z13 ))S(Q2 )v3 X23 3 X 1 Z11 h1 g1 S(y1Y 2t 3 x22 X(1,2) Z22 )α y2 (Y 3 x3 X23 Z 3 )1 3 ⊗ Q1Y11t 1 x1 X 2 Z21 h2 g2 S(Q2Y21t 2 x12 X(1,1) z21 )y3 (Y 3 x3 X23 Z 3 )2

(3.1.9),(3.1.7)

=

(3.2.1)

3 X 1 Z11 h1 g1 S(y1Y 2 X(1,2) Z22 )α y2Y13 (X23 Z 3 )1 3 ⊗ Q1 x1 X 2 Z21 h2 g2 S(Q2 x2Y 1 X(1,1) Z12 )x3 y3Y23 (X23 Z 3 )2

(3.2.19),(3.1.9)

=

(3.2.5),(3.2.13) (3.2.14)

=

(3.2.1),(3.2.19)

=

X 1 Z11 h1 S(X13 Z 2 )1 g1 γ 1 (X23 Z 3 )1 ⊗ X 2 Z21 h2 S(X13 Z 2 )2 g2 γ 2 (X23 Z 3 )2 X 1 (Z 1 hS(X13 Z 2 )α X23 Z 3 )1 ⊗ X 2 (Z 1 hS(X13 Z 2 )α X23 Z 3 )2 Δ(q1 hS(q2 )).

Finally, by the proof of Proposition 11.2, the counit for B is ε (h) = μk (h ⊗ 1) = q1 hS(q2 ) · 1 = ε (h), for all h ∈ H, where we used that ε (α ) = 1. Suppose now that M ∈ C is finite dimensional and let M ∗ be its left dual in C .

418

Factorizable Quasi-Hopf Algebras

Recall that M ∗ ∈ C via h · m∗ , m = m∗ (S(h) · m), for all h ∈ H, m∗ ∈ M ∗ , m ∈ M, and the morphisms evM and coevM are the ones defined in Proposition 3.33. According to the proof of Theorem 11.3, the reconstructed S is characterized by −1 μM ◦ (S ⊗ IdM ) = lM ◦ (IdM ⊗ evM ) ◦ aM,M∗ ,M ◦ ((IdM ⊗ μM∗ ) ⊗ IdM )

◦ (aM,B,M∗ ⊗ IdM ) ◦ ((cB,M ⊗ IdM∗ ) ⊗ IdM ) ◦ (a−1 B,M,M ∗ ⊗ IdM ) ◦ ((IdB ⊗ coevM ) ⊗ IdM ) ◦ (lB ⊗ IdM ), for all M ∈ C finite dimensional, where l, a and c are the left unit constraint, the associativity constraint and the braiding for C , respectively. Explicitly, S is characterized by the following equality that holds for all h ∈ H, m ∈ M: q1 S(h)S(q2 ) · m = Y 1 X 1 R2 x2 β S(Y 2 q1 (X 2 R1 x1  h)S(q2 )X 3 x3 )α Y 3 · m. So we conclude that S is the unique morphism in C which satisfies q1 S(h)S(q2 ) = Q1 X 1 R2 x2 β S(q1 (X 2 R1 x1  h)S(q2 )X 3 x3 )S(Q2 ), for all h ∈ H. Thus, by our assumption, the desired formula for S is S(h) = X 1 R2 x2 β S(q1 (X 2 R1 x1  h)S(q2 )X 3 x3 ),

(11.2.4)

for all h ∈ H (it is not hard to see that S as above is H-linear). Conversely, given these formulas for the braided group B (denoted in what follows by H), one can also check directly (by technical and tedious computations) that it satisfies the axioms for a Hopf algebra in a braided category. We call H the associated enveloping algebra braided group of H. We summarize all these facts in the following: Theorem 11.8 Let (H, R) be a QT quasi-Hopf algebra and denote by C the braided category of left H-modules, H M . Then H gives a braided Hopf algebra H in C , considering the left adjoint action (8.7.1), the same algebra structure as H0 defined in Definition 4.4, coproduct and counit as in Lemma 11.7 and antipode S as in (11.2.4). Moreover, H is weakly braided cocommutative, in the sense that q1 (y1 X 1  h1 )S(q2 )y2Y 1 R2 x2 X13 ⊗ y3 Q1 (Y 2 R1 x1 X 2  h2 )S(Q2 )Y 3 x3 X23 1

2

= Y 1 R y2 q1 (x1 X 2  h2 )S(q2 )x2 X13 ⊗ Q1 (Y 2 R y1 X 1  h1 )S(Q2 )Y 3 y3 x3 X23 , (11.2.5) 1

2

where qR = q1 ⊗ q2 = Q1 ⊗ Q2 , R = R1 ⊗ R2 and R−1 = R ⊗ R . Proof The relation (11.2.5) follows easily by using the equality between the first and the last but one terms of the computation performed in the proof of Corollary 11.4; we leave the details to the reader. An equivalent definition for S can be obtained with the help of the element u defined in (10.3.4).

11.3 Bosonisation for Quasi-Hopf Algebras

419

Proposition 11.9 Let (H, R) be a QT quasi-Hopf algebra with R-matrix R and let u be the element given by (10.3.4). Then the antipode S defined by (11.2.4) admits a second description: S(h) = u−1 S(R1 hS(R2 )), ∀ h ∈ H. Proof

It is based on the following computation: S(h)

=

X 1 R2 p2 S(q1 (X 2 R1 p1  h)S(q2 )X 3 )

=

X 1 R2 p2 S(q1 X12 R11 p11 hS(q2 X22 R12 p12 )X 3 )

(10.1.1)

X 1Y 1 R2 x2 r2 Z 3 p2 S(T 1 X12Y 2 R1 x1 Z 1

=

p11 hS(T 2 X22Y 3 x3 r1 Z 2 p12 )α T 3 X 3 ) (3.1.9)

Y11 R2 x2 r2 Z 3 p2 S(Y21 R1 x1 Z 1 p11 hS(Y 2 x3 r1 Z 2 p12 )α Y 3 )

=

(10.1.3),(3.1.9)

=

R2 q1 r2 X22 Z 3 p2 S(R1 X 1 Z 1 p11 hS(q2 r1 X12 Z 2 p12 )X 3 )

(3.1.9)

R2 q1 r2 p2 S2 (q2 r1 p1 )S(R1 h)

(10.3.9)

R2 u−1 S(R1 h)

(10.3.10)

u−1 S(R1 hS(R2 )),

= = =

valid for all h ∈ H. Here r1 ⊗ r2 is another copy of R. Remark 11.10

The braided antipode S is bijective: if we define S−1 : H → H by S−1 (h) = R S−1 (uh)S(R ), ∀ h ∈ H, 1

2

(11.2.6)

then by using (10.3.10) several times as well as the second definition of S proved above, it is not hard to see that S and S−1 are inverses.

11.3 Bosonisation for Quasi-Hopf Algebras In this section we introduce the bosonisation process. Let (H, R) be a QT quasi-Hopf algebra, so the category H M is braided. By Remark 10.31(2) we have a braided monoidal functor Zl : H M → Zl (H M ) ≡ H H YD. Because Zl is a braided monoidal functor, a braided bialgebra or Hopf algebra B ∈ H M can also be regarded in H H YD and then we can consider the biproduct quasi-Hopf algebra B × H (see Section 9.8); it will be denoted by bos(B) and called the bosonisation of B. Specializing the biproduct construction from Section 9.8 to the above setting we obtain the following result. Proposition 11.11 Under the above hypothesis, bos(B) is a quasi-bialgebra with algebra structure given by the smash product (b × h)(b × h ) = (x1 · b)(x2 h1 · b ) × x3 h2 h ,

(11.3.1)

for all b, b ∈ B, h, h ∈ H, with unit 1B × 1H , with comultiplication Δ(b × h) = y1 X 1 · b1 × y2Y 1 R2 x2 X13 h1 ⊗ y31Y 2 R1 x1 X 2 · b2 × y32Y 3 x3 X23 h2

(11.3.2)

420

Factorizable Quasi-Hopf Algebras

and counit ε (b × h) = ε B (b)ε (h), for all b ∈ B, h ∈ H, and with reassociator Φbos(B) = 1B × X 1 ⊗ 1B × X 2 ⊗ 1B × X 3 .

(11.3.3)

Moreover, if H is a quasi-Hopf algebra and B is a Hopf algebra in the braided category H H YD with antipode SB , then bos(B) is a quasi-Hopf algebra with the antipode s given by s(b × h) = (1B × S(X 1 x11 R2 h)α )(X 2 x21 R1 · SB (b) × X 3 x2 β S(x3 )),

(11.3.4)

for all b ∈ B, h ∈ H, and distinguished elements 1B × α and 1B × β . Because as an algebra bos(B) is a smash product, from Proposition 5.7 it follows that the modules over B in the braided category H M correspond to the ordinary modules over bos(B). Also, by Theorem 9.30 the bosonisation construction is invariant under a twist. Corollary 11.12 If H is a QT quasi-Hopf algebra then the smash product defined by the left adjoint action of H on H0 has a structure of a quasi-Hopf algebra. Proof Since H is a QT quasi-Hopf algebra, by Theorem 11.8 we can associate to H the braided Hopf algebra H in H M . Then bos(H) = H0 #H as an algebra, and is a quasi-Hopf algebra via the structure given by the bosonisation process. Our next goal is to compute the quasi-Hopf algebra structure on bos(H), in the case where H = H(2)+ or H = H(2)− , the QT quasi-Hopf algebras considered in Example 10.6. To this end, note that, in general, bos(H) is the k-vector space H ⊗ H with the following quasi-Hopf algebra structure: (b × h)(b × h ) = (x1  b) ◦ (x2 h1  b ) × x3 h2 h ,

(11.3.5)

Δ(b × h) = y1 X 1  b1 × y2Y 1 R2 x2 X13 h1 ⊗ y31Y 2 R1 x1 X 2  b2 × y32Y 3 x3 X23 h2 , Φbos(H0 ) = β × X ⊗ β × X ⊗ β × X , 1

s(b × h) = (β

2

3

(11.3.6) (11.3.7)

× S(X 1 x11 R2 h)α )(X 2 x21 R1  SH0 (b) × X 3 x2 β S(x3 )),

(11.3.8)

for all b, b , h, h ∈ H, where  stands for the left adjoint action of H on itself, and Δ(b) = b1 ⊗ b2 := x1 X 1 b1 g1 S(x2 R2 y3 X23 ) ⊗ x3 R1  y1 X 2 b2 g2 S(y2 X13 ), 1 2 2

1

2 1 1

2

3

SH0 (b) = X R p S(q (X R p  b)S(q )X ), f −1

(11.3.9) (11.3.10)

for all b ∈ H. Here while pR = and qR = q1 ⊗ q2 are the elements defined by (3.2.16) and (3.2.19), respectively. The unit for bos(H) is β × 1H , the counit is ε (b × h) = ε (b)ε (h), for all b, h ∈ H, and the distinguished elements α and β are given by β × α and β × β , respectively. R = R1 ⊗ R2 ,

= g1 ⊗ g2 ,

p1 ⊗ p2

Example 11.13 bos(H(2)+ ) = bos(H(2)− ) = k[C2 ×C2 ] as bialgebras, viewed as a quasi-Hopf algebra via the non-trivial reassociator Φx := 1 − 2px− ⊗ px− ⊗ px− , where x is one of the generators of C2 ×C2 , and where px− := 12 (1 − x). The antipode is the identity map and the distinguished elements α and β are given by α = x and β = 1.

11.4 The Function Algebra Braided Group

421

Proof Since H := H(2)± are commutative algebras and β = 1, it follows from (3.2.1) and (3.2.2) that the multiplication ◦ defined in (4.1.9) coincides with the original multiplication of H(2). Also, from the definition of H(2) it follows that the action  is trivial, that is, h  h = ε (h)h , for all h, h ∈ H(2). Using (11.3.5) we obtain that the multiplication on bos(H) is the componentwise multiplication, and by (11.3.9) we get that the comultiplication Δ on H reduces to Δ(b) = X 1 b1 g1 S(y3 X23 ) ⊗ y1 X 2 b2 g2 S(y2 X13 ) = Δ(b)(X 1 X23 y3 ⊗ X 2 X13 y1 y2 ) f −1 . We have that (Id ⊗ Id ⊗ Δ)(Φ) = 1 − 2p− ⊗ p− ⊗ p+ ⊗ p− − 2p− ⊗ p− ⊗ p− ⊗ p+ , and therefore X 1 X23 ⊗ X 2 X13 = 1. Also, by (10.1.7) we have y3 ⊗ y1 y2 = 1 − 2p− ⊗ p− = 1 ⊗ p+ + g ⊗ p− , and a straightforward computation ensures that the Drinfeld twist f and its inverse f −1 for H(2) are given by f = f −1 = g ⊗ p− + 1 ⊗ p+ . Combining all these facts we get Δ = Δ, and keeping in mind that the action  is trivial we conclude that the comultiplication in (11.3.6) is the componentwise comultiplication on H(2) ⊗ H(2). Thus bos(H(2)+ ) = bos(H(2)− ) = H(2) ⊗ H(2) as bialgebras. Hence bos(H(2)+ ) = bos(H(2)− ) is generated as an algebra by x = 1 × g and y = g × 1, with relations x2 = y2 = 1 and xy = yx. The elements x and y are grouplike elements, so bos(H(2)+ ) = bos(H(2)− ) = k[C2 × C2 ] as bialgebras. According to (11.3.7) the reassociator of bos(H(2)+ ) = bos(H(2)− ) is given by Φx = 1 × X 1 ⊗ 1 × X 2 ⊗ 1 × X 3 = 1 × 1 ⊗ 1 × 1 ⊗ 1 × 1 − 2 × p− ⊗ 1 × p− ⊗ 1 × p− = 1 − 2px− ⊗ px− ⊗ px− , since 1 × p− = 12 (1 × 1 − 1 × g) = 12 (1 − x) = px− . Finally, using that  is trivial, β = 1 and the axiom (3.2.2), we obtain SH = S, the antipode of H(2). From (11.3.8) and (3.2.2) we deduce that the antipode of bos(H(2)+ ) = bos(H(2)− ) is the identity map. Clearly, the elements α and β are 1 × g = x and 1 × 1 = 1, respectively.

11.4 The Function Algebra Braided Group We present a dual version of Theorem 11.3 and use it to associate to any co-quasitriangular (CQT for short) dual quasi-Hopf algebra A a braided weakly commutative Hopf algebra A in the category of right A-comodules. We will call A the function algebra braided group associated to A. This procedure is the formal dual of the one performed in Section 11.2 where to any QT quasi-Hopf algebra H is associated a weakly cocommutative braided group H in the braided category of left H-modules. We notice that, in the finite-dimensional case, A cannot be obtained

422

Factorizable Quasi-Hopf Algebras

from H by (usual) dualisation. In fact, we show that if H is finite dimensional then the function algebra braided group H ∗ associated to H ∗ is always isomorphic to the categorical left op-cop dual of H as braided Hopf algebra, and this will uncover the true meaning of the factorizable notion in the quasi-Hopf setting. Throughout this section, A will be a dual quasi-bialgebra or a dual quasi-Hopf algebra with structure as in Section 3.7. For a right A-comodule M with structure morphism ρM : M → M ⊗ A we denote ρM (m) = m(0) ⊗ m(1) . Recall from Section 3.7 that M A is a monoidal category. As we shall see, the braided structures on M A are in a one-to-one correspondence with the CQT structures on A. Definition 11.14 A dual quasi-bialgebra or dual quasi-Hopf algebra is called CQT if there exists a k-bilinear form σ : A × A → k such that the following relations hold:

σ (ab, c) = ϕ (c1 , a1 , b1 )σ (a2 , c2 )ϕ −1 (a3 , c3 , b2 )σ (b3 , c4 )ϕ (a4 , b4 , c5 ), σ (a, bc) = ϕ

−1

(b1 , c1 , a1 )σ (a2 , c2 )ϕ (b2 , a3 , c3 )σ (a4 , b3 )ϕ

−1

(11.4.1)

(a5 , b4 , c4 ), (11.4.2)

σ (a1 , b1 )a2 b2 = b1 a1 σ (a2 , b2 ),

(11.4.3)

σ (a, 1A ) = σ (1A , a) = ε (a),

(11.4.4)

for all a, b, c ∈ A. If (A, σ ) is a CQT dual quasi-bialgebra or dual quasi-Hopf algebra then M A is a (pre-)braided category. For any M, N ∈ M A and m ∈ M, n ∈ N, the (pre-)braiding is cM,N (m ⊗ n) = σ (m(1) , n(1) )n(0) ⊗ m(0) . Dual to the quasi-Hopf case, if A is a CQT dual quasi-Hopf algebra we can prove that the bilinear form σ is convolution invertible and the antipode S is bijective. Proposition 11.15 Let (A, σ ) be a CQT dual quasi-Hopf algebra. Then: (i) σ is convolution invertible. More precisely, its inverse σ −1 is given by

σ −1 (a, b) = ϕ (a1 , S(a3 ), b4 a10 )β (a2 )ϕ (b1 , S(a6 ), a8 ) σ (S(a5 ), b2 )ϕ −1 (S(a4 ), b3 , a9 )α (a7 ),

(11.4.5)

for all a, b ∈ A. (ii) The element u ∈ A∗ , given by u(a) = ϕ −1 (a7 , S(a3 ), S2 (a1 ))σ (a6 , S(a4 ))α (a5 )β (S(a2 ))

(11.4.6)

for all a ∈ A, is invertible. Its inverse is given, for all a ∈ A, by u−1 (a) = ϕ (a1 , S2 (a8 ), S(a6 ))β (a4 )σ (S2 (a9 ), a2 )α (S(a7 ))ϕ −1 (S2 (a10 ), a3 , S(a5 )). (11.4.7) (iii) For all a ∈ A, the following relation holds: S2 (a) = u(a1 )a2 u−1 (a3 ). In particular, the antipode S is bijective.

(11.4.8)

11.4 The Function Algebra Braided Group

423

Proof If A is finite dimensional then the proof follows mostly from the results in Sections 10.1 and 10.3, by duality. This is why we give only a sketch of the proof, leaving other details to the reader. (i) This follows by Lemma 10.9, by duality. (ii) First, one can prove that

σ (S(a1 ), S(b1 ))γ (a2 , b2 ) = γ (b1 , a1 )σ (a2 , b2 ),

(11.4.9)

for all a, b ∈ A, and then that f (b1 , a1 )σ (a2 , b2 ) f −1 (a3 , b3 ) = σ (S(a), S(b)).

(11.4.10)

Note that these formulas are the formal duals of Lemma 10.14. Secondly, by using (11.4.10) and the equalities u(a2 )S2 (a1 ) = u(a1 )a2

and α (S(a1 ))u(a2 ) = σ (a3 , S(a1 ))α (a2 )

(11.4.11)

one can show that u ◦ S2 = u (see Lemma 10.15 for the dual case). Now, by using (11.4.11), (11.4.2) and (11.4.4) it can be proved that u−1 defined by (11.4.7) is a left inverse of u. It is also a right inverse since u(a1 )u−1 (a2 ) = u−1 (S2 (a1 ))u(a2 ) = u−1 (S2 (a1 ))u(S2 (a2 )) = ε (S2 (a)) = ε (a), because of (11.4.11) and u ◦ S2 = u (for the dual case see Theorem 10.17). We will now use the dual braided reconstruction theorem in order to obtain the structure of A as a braided Hopf algebra in M A . Since it is the formal dual result of Theorem 11.3 we restrict ourselves to presenting the concepts and results, leaving the verification of the details to the reader. Let C and D be two monoidal categories with D braided. If F : C → D is a functor then for any M ∈ D we denote by F ⊗ M : C → D the functor given by (F ⊗ M)(N) = F(N) ⊗ M, for all N ∈ C . If N → N  is a morphism in C then (F ⊗ M)( f ) = F( f ) ⊗ IdM . Suppose that there is an object B ∈ D such that for all M ∈ D HomD (B, M) ∼ = Nat(F, F ⊗ M) by functorial bijections θM . This is the representability assumption for comodules. Let μ = {μN : F(N) → F(N) ⊗ B | N ∈ C } be the natural transformation corresponding to the identity morphism IdB . Then, by using μ and the braiding in D we have induced maps, for s ∈ {2, 3}: ΘsM : HomD (B⊗s , M) ∼ = Nat(Fs , Fs ⊗ M), and we assume that these are bijections. This is the higher representability assumption for comodules. −1 and θ1−1 we can define a multiplication, a unit, a By using (Θ2B )−1 , μ1 , θB⊗B comultiplication and a counit for B. The next result is the formal dual of the one in Section 11.1, and this is why we will omit its proof. Also, the weak commutativity of B will be uncovered at the end of this section.

424

Factorizable Quasi-Hopf Algebras

Theorem 11.16 Let C and D be monoidal categories with D braided and F : C → D a strong monoidal functor satisfying the representability and the higher representability assumptions for comodules. Then B as above is a bialgebra in D. If C is left rigid then B is a Hopf algebra in D. Let now (A, σ ) be a CQT dual quasi-Hopf algebra, AR the k-vector space A viewed as a right A-comodule via Δ and A the same k-vector space A but viewed now as an object of M A via the right coadjoint coaction:

ρA (a) = a2 ⊗ S(a1 )a3 ,

(11.4.12)

for all a ∈ A. We apply Theorem 11.16 for C = D = M A and F = IdC . The first step is to show that A is the representability object we need. Dual to the quasi-Hopf case, since the antipode S is bijective, we define the elements pL , qL ∈ (A ⊗ A)∗ by pL (a, b) = ϕ (S−1 (a3 ), a1 , b)β (S−1 (a2 )), qL (a, b) = ϕ −1 (S(a1 ), a3 , b)α (a2 ), (11.4.13) for all a, b ∈ A. Then, for all a, b ∈ A, the following relations hold: pL (a2 , b2 )S−1 (a3 )(a1 b1 ) = pL (a, b1 )b2 , qL (a2 , b1 )S(a1 )(a3 b2 ) = qL (a, b2 )b1 , (11.4.14) pL (S(a1 ), a3 b2 )qL (a2 , b1 ) = ε (a)ε (b), qL (S−1 (a3 ), a1 b1 )pL (a2 , b2 ) = ε (a)ε (b). (11.4.15) In what follows by HomA (N, M) we denote the space of right A-comodule morphisms between N, M ∈ M A . Lemma 11.17 Let A be a dual quasi-Hopf algebra and M ∈ M A . If we define

θM : HomA (A, M) → Nat(Id, Id ⊗ M), θM (χ )N (n) = pL (S(n(1) ), n(3) )n(0) ⊗ χ (n(2) ),

(11.4.16)

for all χ ∈ Hom(A, M), N ∈ M A and n ∈ N, then θM is well defined and a bijection. −1 : Nat(Id, Id ⊗ M) → HomA (A, M), is given by Its inverse, θM

θM−1 (ξ )(a) = qL (a1 , (a2 )1(1) )ε ((a2 )0 )(a2 )1(0) ,

(11.4.17)

for all ξ ∈ Nat(Id, Id ⊗ M) and a ∈ A, where we denote ξAR (a) = a0 ⊗ a1 . Proof We have to prove first that θM is well defined, meaning that θM (χ )N is a right A-colinear map and θM (χ ) is a natural transformation. Since χ : A → M is a morphism in M A we have

χ (a)(0) ⊗ χ (a)(1) = χ (a2 ) ⊗ S(a1 )a3 ,

(11.4.18)

for all a ∈ A. Now, if n ∈ N then:

ρN⊗M (θM (χ )N (n))

= (3.7.6)

=

pL (S(n(1) ), n(3) )ρN⊗M (n(0) ⊗ χ (n(2) )) pL (S(n(2) ), n(4) )n(0) ⊗ χ (n(3) )(0) ⊗ n(1) χ (n(3) )(1)

11.4 The Function Algebra Braided Group

425

(11.4.18)

pL (S(n(2) ), n(6) )n(0) ⊗ χ (n(4) ) ⊗ n(1) (S(n(3) )n(5) )

(11.4.14)

pL (S(n(1) ), n(3) )n(0) ⊗ χ (n(2) ) ⊗ n(4)

(11.4.16)

θM (χ )N (n(0) ) ⊗ n(1) = (θM (χ )N ⊗ idA )(ρN (n)),

= = =

as needed. It is not hard to see that θM (χ ) is a natural transformation, so we are left −1 −1 is also well defined, and that θM and θM are inverses. The first to show that θM assertion follows from the following. Since ξAR is a right A-comodule map we have (a1 )0 ⊗ (a1 )1 ⊗ a2 = a01 ⊗ a1(0) ⊗ a02 a1(1) ,

(11.4.19)

for all a ∈ A. On the other hand, for all a∗ ∈ A∗ the map λa∗ : AR → AR , λa∗ (a) := a∗ (a1 )a2 , is right A-colinear. Since ξ is functorial under the morphism λa∗ we get a∗ (a01 )a02 ⊗ a1 = a∗ (a1 )(a2 )0 ⊗ (a2 )1 , for all a∗ ∈ A∗ and a ∈ A, and this is equivalent to a01 ⊗ a02 ⊗ a1 = a1 ⊗ (a2 )0 ⊗ (a2 )1 ,

(11.4.20)

for all a ∈ A. Then for a ∈ A we have −1 (θM (ξ ) ⊗ IdA )(ρA (a))

=

θM−1 (ξ )(a2 ) ⊗ S(a1 )a3

=

qL (a2 , (a3 )1(1) )ε ((a3 )0 )(a3 )1(0) ⊗ S(a1 )a4

(11.4.19)

=

qL (a2 , (a3 )1(1) )ε ((a3 )01 )(a3 )1(0) ⊗ S(a1 )((a3 )02 (a3 )1(2) )

=

qL (a2 , (a3 )1(1) )ε ((a3 )02 )(a3 )1(0) ⊗ S(a1 )((a3 )01 (a3 )1(2) )

(11.4.20)

qL (a2 , (a4 )1(1) )ε ((a4 )0 )(a4 )1(0) ⊗ S(a1 )(a3 (a4 )1(2) )

(11.4.14)

=

qL (a1 , (a2 )1(2) )ε ((a2 )0 )(a2 )1(0) ⊗ (a2 )1(1)

=

−1 ( ρM ◦ θM (ξ ))(a),

=

−1 −1 so θM (ξ ) is a right A-comodule map. We show now that θM is a left inverse for θM . Indeed, from the definitions we have

θM (ξ )AR (a) = pL (S(a2 ), a4 )a1 ⊗ ξ (a3 ) := a0 ⊗ a1 , for all a ∈ A, and therefore −1 ( θM ◦ θM )(χ )(a)

=

qL (a1 , χ (a4 )(1) )ε (a2 )pL (S(a3 ), a5 )χ (a4 )(0)

=

qL (a1 , S(a3 )a4 )pL (S(a2 ), a5 )χ (a2 )

(11.4.15)

=

ε (a1 )ε (a3 )χ (a2 ) = χ (a),

−1 for all χ ∈ Hom(A, M) and a ∈ A. In order to prove that θM is a right inverse for A ∗ ∗ ∗ θM observe first that for any N ∈ M and n ∈ N , the map λn : N → AR , λn∗ (n) = n∗ (n(0) )n(1) , is right A-colinear. That ξ is functorial under the morphism λn∗ means

n∗ (n[0](0) )n[0](1) ⊗ n[1] = n∗ (n(0) )n(1)0 ⊗ n(1)1 ,

426

Factorizable Quasi-Hopf Algebras

where we denote ξN (n) := n[0] ⊗ n[1] . Since this is true for any n∗ ∈ N ∗ we obtain n[0](0) ⊗ n[0](1) ⊗ n[1] = n(0) ⊗ n(1)0 ⊗ n(1)1

(11.4.21)

for all n ∈ N. Now, again for all n ∈ N we compute: −1 (θM ◦ θM )(ξ )N (n)

=

θM (θM−1 (ξ ))N (n) = pL (S(n(1) ), n(3) )n(0) ⊗ θM−1 (ξ )(n(2) )

=

pL (S(n(1) ), n(4) )qL (n(2) , (n(3) )1(1) )ε ((n(3) )0 )n(0) ⊗ (n(3) )1(0)

(11.4.19)

=

pL (S(n(1) ), (n(3) )02 (n(3) )1(2) )qL (n(2) , (n(3) )1(1) )

ε ((n(3) )01 )n(0) ⊗ (n(3) )1(0) (11.4.21)

pL (S(n[0](1) ), n[0](3) n[1](2) )qL (n[0](2) , n[1](1) )n[0](0) ⊗ n[1](0)

(11.4.15)

ε (n[0](1) )ε (n[1](1) )n[0](0) ⊗ n[1](0) = n[0] ⊗ n[1] = ξN (n),

= =

as needed, and this finishes the proof. We are now able to begin our reconstruction. The natural transformation μ ∈ Nat(Id, Id ⊗ A) corresponding to the identity morphism IdA is given by

μN (n) = θA (IdA )N (n) = pL (S(n(1) ), n(3) )n(0) ⊗ n(2) for all N ∈ M A and n ∈ N. By the dual version of Proposition 11.1 the multiplication of A is characterized as being the unique morphism m : A ⊗ A → A in M A such that

μM⊗N = (IdM⊗N ⊗ m) ◦ a−1 M,N,A⊗A ◦ (IdM ⊗ aN,A,A ) ◦ (IdM ⊗ (cA,N ⊗ IdA )) ◦ (IdM ⊗ a−1 A,N,A ) ◦ aM,A,N⊗A ◦ ( μM ⊗ μN ), for any M, N ∈ M A . By using the braided categorical structure of M A and the definition of μ it is not hard to see that m is the unique morphism in M A which satisfies pL (S(m(1) n(1) ), m(3) n(3) )(m(0) ⊗ n(0) ) ⊗ m(2) n(2) = pL (S(m(3) ), m(15) )pL (S(n(5) ), n(13) )ϕ (m(2) , S(m(4) )m(14) , n(14) )

ϕ −1 (S(m(5) )m(13) , n(4) , S(n(6) )n(12) )σ (S(m(6) )m(12) , n(3) ) ϕ (n(2) , S(m(7) )m(11) , S(n(7) )n(11) )ϕ −1 (m(1) , n(1) , m(9) n(9) ) (m(0) ⊗ n(0) ) ⊗ (S(m(8) )m(10) )·(S(n(8) n(10) ) for all M, N ∈ M A and m ∈ M, n ∈ N, where we denote by a·b := m(a ⊗ b). One can easily check that the above equality is equivalent to pL (S(a1 b1 ), a3 b3 )a2 b2 = pL (S(a3 ), a15 )pL (S(b5 ), b13 )ϕ (a2 , S(a4 )a14 , b14 )

ϕ −1 (S(a5 )a13 , b4 , S(b6 )b12 )σ (S(a6 )a12 , b3 )ϕ (b2 , S(a7 )a11 , S(b7 )b11 ) ϕ −1 (a1 , b1 , a9 b9 )(S(a8 )a10 )·(S(b8 )b10 )

(11.4.22)

for all a, b ∈ A. Now, the explicit formula for the multiplication · is the following: a·b = ϕ (S(a1 ), a10 , S(b1 )b12 ) f (b6 , a3 )σ (a8 , S(b3 ))ϕ −1 (S(a2 ), S(b5 ), a6 b9 )

11.4 The Function Algebra Braided Group

σ (a4 , b7 )ϕ −1 (a9 , S(b2 ), b11 )ϕ (S(b4 ), a7 , b10 )a5 b8 ,

427 (11.4.23)

for all a, b ∈ A. Indeed, it is easy to see that the multiplication · defined by (11.4.23) is a right A-colinear map. A straightforward but tedious computation ensures that · satisfies the relation (11.4.22); we leave all these details to the reader. It is not hard to see that the unit of A is 1A , the unit of A. Following the dual version of Proposition 11.2, the comultiplication of A is ob−1 (ξ ), where ξ is defined by the following composition: tained as Δ = θA⊗A μN

μN

aN,A,A

ξN : N −→ N ⊗ A −→ (N ⊗ A) ⊗ A −→ N ⊗ (A ⊗ A), for all N ∈ M A . Explicitly, for all n ∈ N,

ξN (n) = ϕ (n(1) , S(n(3) )n(5) , S(n(8) )n(10) )pL (S(n(7) ), n(11) ) pL (S(n(2) ), n(6) )n(0) ⊗ (n(4) ⊗ n(9) ).

(11.4.24)

The counit ε is obtained as ε (a) = θk−1 (l)(a), where l is the left unit constraint. Proposition 11.18 Let A be a dual quasi-Hopf algebra. Then the comultiplication of A is given for all a ∈ A by Δ(a) = ϕ −1 (S(a1 ), a5 , S(a7 ))β (a6 )ϕ (S(a2 )a4 , S(a8 ), a10 )a3 ⊗ a9 .

(11.4.25)

The counit of Δ is ε = α . Proof

Notice first that (3.7.3) and the definitions (11.4.13) of pL and qL imply: qL (a1 , b1 c1 )ϕ (a2 , b2 , c2 ) = α (a3 )ϕ −1 (S(a2 ), a4 , b1 )ϕ −1 (S(a1 ), a5 b2 , c),

ϕ

−1

(11.4.26)

(a, b1 , S(b3 )c1 )pL (S(b2 ), c2 ) = ϕ (a1 b1 , S(b5 ), c)ϕ −1 (a2 , b2 , S(b4 ))β (b3 ),

(11.4.27)

for all a, b, c ∈ A. On the other hand, from (11.4.24) we can easily see that

ξAR (a) = a0 ⊗ a1 = pL (S(a3 ), a7 )pL (S(a8 ), a12 ) ϕ (a2 , S(a4 )a6 , S(a9 )a11 )a1 ⊗ (a5 ⊗ a10 ). (11.4.28) Now, for all a ∈ A we compute: ΔA (a)

=

−1 θA⊗A (ξ )

=

qL (a1 , (a2 )1(1) )ε ((a2 )0 )(a2 )1(0)

(11.4.28)

=

qL (a1 , (a5 ⊗ a10 )(1) )pL (S(a8 ), a12 )pL (S(a3 ), a7 )

ϕ (a2 , S(a4 )a6 , S(a9 )a11 )(a5 ⊗ a10 )(0) (3.7.6)

=

qL (a1 , (S(a5 )a7 )(S(a12 )a14 ))pL (S(a10 ), a16 )pL (S(a3 ), a9 )

ϕ (a2 , S(a4 )a8 , S(a11 )a15 )a6 ⊗ a13 (11.4.26)

=

α (a3 )ϕ −1 (S(a2 ), a4 , S(a8 )a10 )ϕ −1 (S(a1 ), a5 (S(a7 )a11 ), S(a14 )a16 ) pL (S(a13 ), a17 )pL (S(a6 ), a12 )a9 ⊗ a15

428

Factorizable Quasi-Hopf Algebras (11.4.14)

=

α (a3 )ϕ −1 (S(a2 ), a4 , S(a6 )a8 )ϕ −1 (S(a1 ), a10 , S(a12 )a14 ) pL (S(a11 ), a15 )pL (S(a5 ), a9 )a7 ⊗ a13

(11.4.27)

=

α (a4 )ϕ −1 (S(a3 ), a5 , S(a7 )a9 )ϕ (S(a2 )a11 , S(a15 ), a17 ) ϕ −1 (S(a1 ), a12 , S(a14 ))β (a13 )pL (S(a6 ), a10 )a8 ⊗ a16

(11.4.13)

=

qL (a3 , S(a5 )a7 )ϕ (S(a2 )a9 , S(a13 ), a15 )ϕ −1 (S(a1 ), a10 , S(a12 ))

β (a11 )pL (S(a4 ), a8 )a6 ⊗ a14 (11.4.15)

=

ϕ −1 (S(a1 ), a5 , S(a7 ))β (a6 )ϕ (S(a2 )a4 , S(a8 ), a10 )a3 ⊗ a9 .

The counit of Δ is ε (a) = θk−1 (l)(a) = qL (a, 1) = α (a) for all a ∈ A, so ε = α . Let M be a finite dimensional right A-comodule and M ∗ its left dual object as in Proposition 3.52. By the dual version of Proposition 11.2, the reconstructed antipode S of A is characterized as being the unique morphism in C satisfying −1 ◦ (IdM⊗A ⊗ evM ) ◦ (aM,A,M∗ ⊗ IdM ) ◦ (a−1 (IdM ◦ S) ◦ μM = lM⊗A M,A,M ∗ ⊗ IdM )

◦ ((IdM ⊗ c−1 A,M ∗ ) ⊗ IdM ) ◦ ((IdM ⊗ μM ∗ ) ⊗ IdM ) ◦ (coevM ⊗ IdM ) ◦ rM , for any finite-dimensional object M of M A , where l, r, a, c, ev and coev are the left unit constraint, the right unit constraint, the associativity constraint, the braiding of M A , and the evaluation and coevaluation maps, respectively. This reads pL (S(m(1) ), m(3) )m(0) ⊗ S(m(2) ) = β (m(3) )pL (S2 (m(12) ), S(m(4) ))

σ −1 (S2 (m(11) )S(m(5) ), S(m(13) ))ϕ −1 (m(2) , S2 (m(10) )S(m(6) ), S(m(14) )) ϕ (m(1) [S2 (m(9) )S(m(7) )], S(m(15) ), m(17) )α (m(16) )m(0) ⊗ S(m(8) ), for any finite-dimensional right A-comodule M and m ∈ M. It follows that the above relation is equivalent to pL (S(a1 ), a3 )S(a2 ) =

β (a3 )pL (S2 (a12 ), S(a4 ))σ −1 (S2 (a11 )S(a5 ), S(a13 )) ϕ −1 (a2 , S2 (a10 )S(a6 ), S(a14 ))ϕ (a1 [S2 (a9 )S(a7 )], S(a15 ), a17 )α (a16 )S(a8 )

(3.7.3)

=

(3.7.8) (11.4.14)

=

(11.4.13)

β (a2 )pL (S2 (a11 ), S(a3 ))σ −1 (S2 (a10 )S(a4 ), S(a12 )) ϕ (S2 (a8 )S(a6 ), S(a14 ), a16 )ϕ (a1 , [S2 (a9 )S(a5 )]S(a13 ), a17 )α (a15 )S(a7 ) pL (S(a1 ), a13 )pL (S2 (a8 ), S(a2 ))σ −1 (S2 (a7 )S(a3 ), S(a9 ))

ϕ (S2 (a6 )S(a4 ), S(a10 ), a12 )α (a11 )S(a5 )

for all a ∈ A, and therefore S(a) = pL (S2 (a7 ), S(a1 ))σ −1 (S2 (a6 )S(a2 ), S(a8 ))

ϕ (S2 (a5 )S(a3 ), S(a9 ), a11 )α (a10 )S(a4 ), for all a ∈ A (it is not hard to see that S defined above is right A-colinear). We summarize all these facts in the following:

(11.4.29)

11.4 The Function Algebra Braided Group

429

Theorem 11.19 Let (A, σ ) be a CQT dual quasi-Hopf algebra. Then there is a braided Hopf algebra A in the category M A . A coincides with A as a k-linear space, and it is an object in M A by the right coadjoint coaction

ρA (a) = a2 ⊗ S(a1 )a3 . The algebra structure, the coalgebra structure and the antipode are a·b = ϕ (S(a1 ), a10 , S(b1 )b12 ) f (b6 , a3 )σ (a8 , S(b3 ))ϕ −1 (S(a2 ), S(b5 ), a6 b9 )

σ (a4 , b7 )ϕ −1 (a9 , S(b2 ), b11 )ϕ (S(b4 ), a7 , b10 )a5 b8 , Δ(a) = ϕ −1 (S(a1 ), a5 , S(a7 ))β (a6 )ϕ (S(a2 )a4 , S(a8 ), a10 )a3 ⊗ a9 , S(a) = pL (S2 (a7 ), S(a1 ))σ −1 (S2 (a6 )S(a2 ), S(a8 ))

ϕ (S2 (a5 )S(a3 ), S(a9 ), a11 )α (a10 )S(a4 ), for all a, b ∈ A. The unit element is 1A , the unit of A, and the counit is ε = α . We will call A the associated function algebra braided group of A. Remark 11.20 The braided group A is weakly braided commutative in the following sense. The multiplication mA satisfies the equality μM ⊗ μN

M ⊗ N −→

aM,A,N⊗A

−→

M ⊗ (A ⊗ (N ⊗ A))

−→

M ⊗ ((A ⊗ N) ⊗ A)

−→

M ⊗ ((N ⊗ A) ⊗ A)

−→

M ⊗ (N ⊗ (A ⊗ A))

−→

(M ⊗ N) ⊗ (A ⊗ A)

−→

(M ⊗ N) ⊗ A

IdM ⊗a−1 A,N,A IdM ⊗(cA,N ⊗IdA ) IdM ⊗aN,A,A a−1 M,N,A⊗A IdM⊗N ⊗mA

μM ⊗IdN

(M ⊗ A) ⊗ (N ⊗ A)

M ⊗ N −→

aM,A,N

M ⊗ (A ⊗ N)

−→

M ⊗ (N ⊗ A)

−→

M ⊗ ((N ⊗ A) ⊗ A)

−→

M ⊗ (N ⊗ (A ⊗ A))

−→

(M ⊗ N) ⊗ (A ⊗ A)

−→

(M ⊗ N) ⊗ A ,

IdM ⊗c−1 N,A

=

(M ⊗ A) ⊗ N

−→

IdM ⊗(μN ⊗IdA ) IdM ⊗aN,A,A a−1 M,N,A⊗A IdM⊗N ⊗mA

for all M, N ∈ M A . Note that by writing down explicitly the above equality we get a relation that is dual to the one in (11.2.5); the details are left to the reader. Suppose now that (H, R) is a finite-dimensional QT quasi-Hopf algebra. Then H ∗ , the linear dual of H, is in an obvious way a CQT dual quasi-Hopf algebra, so it makes sense to consider H ∗ , the function algebra braided group associated to H ∗ . It is a Hopf algebra in the category of right H ∗ -comodules, hence a Hopf algebra in the category of left H-modules. From (11.4.12), H ∗ is a left H-module via h  χ = h2  χ  S(h1 ),

(11.4.30)

for all h ∈ H and χ ∈ H ∗ . By Theorem 11.19, the structure of H ∗ as a Hopf algebra in H M is given by:

χ ·ψ = [x13Y 2 r1 y1 X 2  χ  S(x1 X 1 ) f 2 R1 ] [x23Y 3 y3 X23  ψ  S(x2Y 1 r2 y2 X13 ) f 1 R2 ], 1H ∗ = ε ,

(11.4.31) (11.4.32)

430

Factorizable Quasi-Hopf Algebras ΔH ∗ (χ ) = χ1  S(x1 ) ⊗ x23 X 3  χ2  x2 X 1 β S(x13 X 2 ),

(11.4.33)

ε H ∗ (χ ) = χ (α ),

(11.4.34)

1

2

1

S(χ ) = q12 R2 p˜2  χ S  q2 R S(q11 R1 p˜1 ),

(11.4.35)

for all χ , ψ ∈ H ∗ . Here pR = p1 ⊗ p2 and qR = q1 ⊗ q2 are the elements defined by 1 2 (3.2.19), f = f 1 ⊗ f 2 is the Drinfeld’s twist defined by (3.2.15), R−1 = R ⊗ R , and qL = q˜1 ⊗ q˜2 is the element given by (3.2.20). On the other hand, since (H, R) is finite dimensional, the categorical left dual of H has a braided Hopf algebra structure in H M . We have denoted H ∗ with this Hopf algebra structure by (H) . By the above, (H) is a left H-module via (h  χ )(h ) = χ (S(h)  h ), ∀ h, h ∈ H, χ ∈ H ∗ . By Proposition 2.54 the structure of (H) as a bialgebra in formulas

(11.4.36) HM

(χ ∗ ψ )(h) = χ , f 2  h2 ψ , f 1  h1 ,

is given by the (11.4.37)

1(H) = ε ,

(11.4.38)

Δ(H)∗ (χ ) = χ , (g  i e) • (g  j e) e ⊗ e,

(11.4.39)

ε (H) (χ ) = χ (β ),

(11.4.40)

1

2

j

i

where {i e}i=1,n and {i e}i=1,n are dual bases in H and H ∗ . Furthermore, by Proposition 2.66 we have that (H) is a Hopf algebra in H M with antipode given by S(H)∗ (χ ) = χ ◦ S.

(11.4.41)

We show that, up to a braided Hopf algebra isomorphism, H ∗ is nothing but (H) . Proposition 11.21 Let (H, R) be a finite-dimensional QT quasi-Hopf algebra, H the associated enveloping algebra braided group of H, (H) the left op-cop dual Hopf algebra structure of H in H M , and H ∗ the function algebra braided group associated to H ∗ . Then the map λ : (H) → H ∗ given for all χ ∈ H ∗ by

λ (χ ) = S−1 (g1 )  χ ◦ S  g2

(11.4.42)

is a braided Hopf algebra isomorphism. Here g1 ⊗ g2 is the inverse of the Drinfeld twist f ; see (3.2.16). Proof

The map λ is left H-linear since (h  λ (χ ))(h )

=

λ (χ ), S(h1 )h h2 

=

χ , g1 S(h2 )S(h )S(g2 S(h1 ))

(3.2.13),(8.7.1)

=

(11.4.36)

=



χ , S(h)  (g1 S(g2 h )) (h  χ ) ◦ S, g2 h S−1 (g1 ) = λ (h  χ )(h )

for all h, h ∈ H and χ ∈ H ∗ . Next, we show that λ is an algebra and coalgebra

11.4 The Function Algebra Braided Group

431

morphism, and that it is bijective. Firstly, for all χ , ψ ∈ H ∗ and h ∈ H we compute

λ (χ ∗ ψ )(h)

χ , f 2  (g1 S(g2 h))2 ψ , f 1  (g1 S(g2 h))1 

= (11.2.3),(8.7.1)

=

(3.2.13)

χ , f 2 x3 R1  y1 X 2 g12 G2 S(y2 X13 g21 h1 ) ψ , f11 x1 X 1 g11 G1 S( f21 x2 R2 y3 X23 g22 h2 )

(3.2.17),(3.1.7)

=

(3.2.13)

χ , f 2 x3 R1  G2(1,1) y1 g1 S(G2(1,2) y2 g21 G1 S(X21 )F 1 h1 X 2 ) ψ , f11 x1 G1 S( f21 x2 R2 G22 y3 g22 G2 S(X11 )F 2 h2 X 3 )

(3.2.17),(8.7.1)

=

(10.1.3)

χ , f 2 x3 G22 R1 G1  g1 S(g2 S(X21 y2 )F 1 h1 X 2 y3 ) ψ , f11 x1 G1 S( f21 x2 G21 R2 G2 S(X11 y1 )F 2 h2 X 3 )

(10.3.6),(3.2.17)

=

(8.7.1),(3.2.13)

χ , g1 S(g2 S(X21 y2 R11 x11 )F 1 h1 X 2 y3 R12 x21 ) ψ , G1 S(G2 S(X11 y1 R2 x2 )F 2 h2 X 3 x3 ),

and, on the other hand, by (11.4.31) we have (λ (χ )·λ (ψ ))(h)

=

χ , g1 S(g2 S(x1 X 1 ) f 2 R1 h1 x13Y 2 r1 y1 X 2 ) ψ , G1 S(G2 S(x2Y 1 r2 y2 X13 ) f 1 R2 h2 x23Y 3 y3 X23 )

(10.3.6),(3.1.9)

=

(10.1.3) (3.1.9),(10.1.1)

=

χ , g1 S(g2 S(Y21 R1 z1 x1 X 1 ) f 1 h1Y 2 z3 r1 x12 y1 X 2 ) ψ , G1 S(G2 S(Y11 R2 z2 r2 x22 y2 X13 ) f 2 h2Y 3 x3 y3 X23 ) χ , g1 S(g2 S(Y21 z2 R11 x11 ) f 1 h1Y 2 z3 R12 x21 ) ψ , G1 S(G2 S(Y11 z1 R2 y2 ) f 2 h2Y 3 y3 ),

as needed. It is not hard to see that λ (1(H) ) = 1H ∗ , so λ is an algebra morphism. Now, λ is a coalgebra morphism since (λ ⊗ λ )(Δ(H) (χ )) =

χ , (g1  i e) • (g2  j e)λ ( j e) ⊗ λ (i e)

(11.4.42),(4.1.9),(8.7.1)

χ , X 1 g11 G1 S(x1 X 2 g12 G2 i e)α x2 X13 g21 G1 S(x3 X23 g22 G2 j e) j e ⊗ i e

(3.2.17),(3.1.7),(3.2.1)

χ , G1 S(g1 S(X 2 x3 )i eX 3 )α g2 S(G2 S(X11 x1 ) j eX21 x2 ) j e ⊗ i e

= =

(8.7.7),(11.4.42)

λ (χ ), S(X11 x1 ) j eX21 x2 β S(X 2 x3 )i eX 3 ) j e ⊗ i e

(3.1.9),(11.4.33)

=

λ (χ )1  S(X11 x1 ) ⊗ X 3  λ (χ )2  X21 x2 β S(X 2 x3 )

=

ΔH ∗ (λ (χ )),

=

for all χ ∈ H ∗ , and since the definitions of counits imply ε H ∗ ◦ λ = ε (H) . It is easy to see that λ is bijective with inverse λ −1 (χ ) = S( f 2 )  χ ◦ S−1  f 1 , for all χ ∈ H ∗ . Thus, the proof is complete. We have seen two processes that associate to a finite-dimensional QT quasi-Hopf algebra H a braided Hopf algebra structure on H ∗ within H M . The first one associates what we called the function algebra braided group on H ∗ , which was denoted by H ∗ , while the second one associates H ∗ as in Proposition 10.30 (see also Remark 10.31(2). Actually, up to an isomorphism, these two processes coincide.

432

Factorizable Quasi-Hopf Algebras

Proposition 11.22 Let H be a finite-dimensional QT quasi-Hopf algebra, H ∗ the function algebra braided group on H ∗ and H ∗ the Hopf algebra in H M as in Proposition 10.30. Then S, the antipode of H ∗ , yields a braided Hopf algebra isomorphism between H ∗ and H ∗ . Proof

One can see easily that

(h  χ  h ) ◦ S = S−1 (h )  χ S  S−1 (h) , ∀ h, h ∈ H and χ ∈ H ∗ , where, as before,  and  are the left and right regular actions of the algebra H on its dual space H ∗ . The above formula together with (11.4.30) and (10.5.2) implies that S : H ∗ → H ∗ is an isomorphism in H M . Also, it can be easily checked that S behaves well with respect to the units and counits of H ∗ and H ∗ . It is also a multiplicative morphism since S(χ ·ψ ), h

(11.4.31)

=

(x13Y 2 r1 y1 X 2  χ  S(x1 X 1 ) f 2 R1 ) (x23Y 3 y3 X23  ψ  S(x2Y 1 r2 y2 X13 ) f 1 R2 ), S(h)

(3.2.13),(10.1.3)

=

χ , S(h1 x1 X 1 ) f 2 x23 R1Y 2 r1 y1 X 2  ψ , S(h2 x2Y 1 r2 y2 X13 ) f 1 x13 R2Y 3 y3 X23 

=

χ S, S−1 ( f 2 x23 R1Y 2 r1 y1 X 2 )h1 x1 X 1  ψ S, S−1 ( f 1 x13 R2Y 3 y3 X23 )h2 x2Y 1 r2 y2 X13 

(10.1.2)

=

(x1 X 1  χ S  S−1 ( f 2 x23Y 3 R1 X 2 )) (x2Y 1 R21 X13  ψ S  S−1 ( f 1 x13Y 2 R22 X23 )), h

(10.5.8)

=

(χ S) ◦ (ψ S), h,

for all h ∈ H, as required. Finally, S respects the comultiplications of H ∗ and H ∗ since ΔH ∗ (χ S) = χ2 S ⊗ χ1 S, for all χ ∈ H ∗ , and therefore ΔH ∗ (χ S)

(10.5.9)

=

X11 p1  χ1 S  S−1 (X21 p2 ) ⊗ X 2  χ2 S  S−1 (X 3 )

=

(X21 p2  χ1  S(X11 p1 )) ◦ S ⊗ (X 3  χ2  S(X 2 )) ◦ S

=

(χ1  S(X11 p1 )) ◦ S ⊗ (X 3  χ2  X21 p2 S(X 2 )) ◦ S

(3.2.20),(11.5.7)

(χ1  S(x1 )) ◦ S ⊗ (x23 p˜2  χ2  x2 S(x13 p˜1 )) ◦ S

(3.2.20),(11.4.33)

(S ⊗ S)ΔH ∗ (χ ),

= =

for all χ ∈ H ∗ , as desired. This finishes the proof. Consequently, in the QT case we have the following description for the quantum double quasi-Hopf algebra. Theorem 11.23 The quantum double D(H) of a finite-dimensional QT quasi-Hopf algebra H can be characterized as follows: (i) D(H) is a biproduct between H ∗ , the function algebra braided group, and H; (ii) D(H) is a biproduct between (H) , the categorical left op-cop dual of the associated enveloping algebra braided group of H, and H.

11.5 Factorizable QT Quasi-Hopf Algebras

433

Proof The assertion (i) follows from Remark 10.31(1) and Proposition 11.22, while (ii) is a consequence of (i) and Proposition 11.21.

11.5 Factorizable QT Quasi-Hopf Algebras In this section we will introduce the notion of factorizable quasi-Hopf algebra and we will show that the quantum double is an example of this type. If (H, R) is a QT quasi-Hopf algebra we consider the k-linear map Q : H ∗ → H, given for all χ ∈ H ∗ by Q(χ ) = χ , S(X22 p˜2 ) f 1 R2 r1U 1 X 3 X 1 S(X12 p˜1 ) f 2 R1 r2U 2 ,

(11.5.1)

where r1 ⊗ r2 is another copy of R, pL = p˜1 ⊗ p˜2 is the element considered in (3.2.20) and U = U 1 ⊗U 2 is the element defined in (7.3.1). Definition 11.24 A QT quasi-Hopf algebra (H, R) is called factorizable if the map Q defined by (11.5.1) is bijective. In the second part of this section we will uncover the monoidal categorical interpretation for the definition of Q. Example 11.25 For (H(2), R± ) with R± as in Example 10.7, the map Q from (11.5.1) has the following form, for all χ ∈ H(2)∗ : Q(χ ) = χ (1)p− + χ (g)p+ . Since {p− , p+ } and {1, g} are bases for H(2) it follows that Q is bijective, so (H(2), R± ) are factorizable quasi-Hopf algebras. Proof

For H(2) the element pL has the form

pL = X 1 X 2 ⊗ X 3 = 1 − 2p− ⊗ p− = 1 − (1 − g) ⊗ p− = 1 ⊗ p+ + g ⊗ p− = f . Also, one can easily see that X 1 X12 ⊗ X22 X 3 = 1 and since f = f −1 we conclude that X22 X 3 p˜2 f 1 ⊗ X 1 X12 p˜1 f 2 = 1. On the other hand, since ω 2 − 2ω = −2 it follows that R2 r1 ⊗ R1 r2 = (1 − ω p− ⊗ p− )2 = 1 − 2p− ⊗ p− . We have already seen that U = g ⊗ 1, and therefore S(X22 p˜2 ) f 1 R2 r1U 1 X 3 ⊗ X 1 S(X12 p˜1 ) f 2 R1 r2U 2 = (1 − 2p− ⊗ p− )(g ⊗ 1) = 1 ⊗ p− + g ⊗ p+ . It is now clear that Q(χ ) = χ (1)p− + χ (g)p+ , for all χ ∈ H(2)∗ . In what follows we will need a second formula for the map Q in (11.5.1). Also, another k-linear map Q : H ∗ → H is required.

434

Factorizable Quasi-Hopf Algebras

Proposition 11.26 Let (H, R) be a QT quasi-Hopf algebra. (i) The map Q defined by (11.5.1) has a second formula given for all χ ∈ H ∗ by Q(χ ) = χ , q˜1 X 1 R2 r1 p1 q˜21 X 2 R1 r2 p2 S(q˜22 X 3 ),

(11.5.2)

where qL = q˜1 ⊗ q˜2 and pR = p1 ⊗ p2 are the elements defined by (3.2.20) and (3.2.19), respectively. (ii) Let Q : H ∗ → H be the k-linear map defined for all χ ∈ H ∗ by Q(χ ) = χ , S−1 (X 3 )q2 R1 r2 X22 p˜2 q1 R2 r1 X12 p˜1 S−1 (X 1 ),

(11.5.3)

where qR = q1 ⊗ q2 and pL = p˜1 ⊗ p˜2 are the elements defined by (3.2.19) and (3.2.20), respectively. Then Q is bijective if and only if Q is bijective. Proof

(i) We claim that R1U 1 ⊗ R2U 2 = q˜12 R1 p1 ⊗ q˜11 R2 p2 S(q˜2 ).

(11.5.4)

Indeed, we calculate: q˜12 R1 p1 ⊗ q˜11 R2 p2 S(q˜2 )

(10.1.3)

R1 q˜11 p1 ⊗ R2 q˜12 p2 S(q˜2 )

(7.3.5)

R1 (q˜1 S( p˜1 ))1U 1 p˜2 ⊗ R2 (q˜1 S( p˜1 ))2U 2 S(q˜2 )

(7.3.2)

R1 (q˜1 S(q˜21 p˜1 ))1U 1 q˜22 p˜2 ⊗ R2 (q˜1 S(q˜21 p˜1 ))2U 2

(3.2.24)

R1U 1 ⊗ R2U 2 ,

= = = =

as needed. Now, if we denote by Q˜ 1 ⊗ Q˜ 2 another copy of qL we have Q(χ )

(11.5.1),(10.3.3)

χ , S(x2 p˜21 ) f 1 R2 r1U 1 x3 p˜22 S(x1 p˜1 ) f 2 R1 r2U 2

(11.5.4),(3.2.21)

χ , S(x2 p˜21 ) f 1 q˜11 (x3 p˜22 )(1,1) R2 r1 p1 

= =

(10.1.3)

S(x1 p˜1 ) f 2 q˜12 (x3 p˜22 )(1,2) R1 r2 p2 S(q˜2 (x3 p˜22 )2 ) χ , S( p˜21 )Q˜ 1 X 1 ( p˜22 )(1,1) R2 r1 p1 

(3.2.28)

=

(3.1.7),(3.2.22)

=

(3.2.24)

=

S( p˜1 )q˜1 Q˜ 21 X 2 ( p˜22 )(1,2) R1 r2 p2 S(q˜2 Q˜ 22 X 3 ( p˜22 )2 ) χ , Q˜ 1 X 1 R2 r1 p1 S( p˜1 )q˜1 p˜21 Q˜ 21 X 2 R1 r2 p2 S(q˜2 p˜22 Q˜ 22 X 3 ) χ , Q˜ 1 X 1 R2 r1 p1 Q˜ 21 X 2 R1 r2 p2 S(Q˜ 22 X 3 ),

for all χ ∈ H ∗ . So we have proved the relation (11.5.2). (ii) For all χ ∈ H ∗ we have Q(χ )

(11.5.1),(7.3.1)

=

(10.3.6)×2

=

(11.5.2)

=

χ , S(X22 p˜2 ) f 1 R2 r1 g1 S(q2 )X 3 X 1 S(X12 p˜1 ) f 2 R1 r2 g2 S(q1 ) χ , S(q2 r1 R2 X22 p˜2 )X 3 X 1 S(q1 r2 R1 X12 p˜1 ) S(Q(χ ◦ S)).

Since the antipode S is bijective we conclude that Q is bijective if and only if Q is bijective, so our proof is complete. We provide an important family of factorizable QT quasi-Hopf algebras.

11.5 Factorizable QT Quasi-Hopf Algebras

435

Proposition 11.27 Let H be a finite-dimensional quasi-Hopf algebra and D(H) its quantum double. Then D(H) is a factorizable quasi-Hopf algebra. Proof We will show that in the quantum double case the map Q defined by (11.5.3) is bijective, so by Proposition 11.26 it follows that D(H) is factorizable. For this we will compute first the element R 2 R1 ⊗ R 1 R2 , where we denote by R1 ⊗ R2 another copy of the R-matrix R of D(H). In fact, if we denote by P1 ⊗ P2 another copy of the element pR then we compute: R 2 R1 ⊗ R 1 R2 =

(i e  p12 )(ε  S−1 (P2 ) j eP11 ) ⊗ (ε  S−1 (p2 )i ep11 )( j e  P21 )

(8.5.3) i

=

e  p12 S−1 (P2 ) j eP11 ⊗ (S−1 (p2 )i ep11 )(1,1)  j e  S−1 ((S−1 (p2 )i ep11 )2 )  (S−1 (p2 )i ep11 )(1,2) P21

=

i

e  p12 S−1 ((S−1 (p2 )i ep11 )2 P2 ) j e(S−1 (p2 )i ep11 )(1,1) P11 ⊗ j e  (S−1 (p2 )i ep11 )(1,2) P21

(3.2.21) i

=

e  S−1 ((S−1 (p2 )i e)2 P2 ) j e(S−1 (p2 )i e)(1,1) P11 p11 ⊗ j e  (S−1 (p2 )i e)(1,2) P21 p12 .

Now, H is a quasi-Hopf subalgebra of D(H), so we have to calculate the element b1 ⊗ b2 := (ε  S−1 (X 3 )q2 )R 1 R2 (ε  X22 p˜2 ) ⊗ (ε  q1 )R 2 R1 (ε  X12 p˜1 S−1 (X 1 )). By dual bases and (8.5.3) we have b1 ⊗ b2

=

(ε  S−1 (X 3 ))( j e  (q2 S−1 (q12 p2 )i e)(1,2) ((q11 )(1,1) P1 )2 p12 ) (ε  X22 p˜2 ) ⊗ i e  S−1 ((q2 S−1 (q12 p2 )i e)2 (q11 )(1,2) P2 S((q11 )2 )) j e (q2 S−1 (q12 p2 )i e)(1,1) ((q11 )(1,1) P1 )1 p11 )(ε  X12 p˜1 S−1 (X 1 ))

(3.2.21)

=

(3.2.23) (8.5.3)

=

(10.3.3)

(ε  S−1 (X 3 ))( j e  (i e)(1,2) P21 )(ε  X22 p˜2 ) ⊗ (i e  S−1 ((i e)2 P2 ) j e(i e)(1,1) P11 )(ε  X12 p˜1 S−1 (X 1 )) (ε  S−1 (x3 p˜22 )i e)( j e  P21 x2 p˜21 ) ⊗ (i e  S−1 (P2 ) j eP11 x1 p˜1 ).

Now we want an explicit formula for the element SD (b1 ) ⊗ b2 . To this end we need the following relations: S(P21 x2 p˜21 )1 f11 p1 ⊗ S(P21 x2 p˜21 )2 f21 p2 S( f 2 )S2 (P11 x1 p˜1 ) = g1 S(P1 y3 x22 p˜(1,2) ) ⊗ g2 S(S(y1 x1 p˜1 )α y2 x12 p˜2(1,1) ), S(P )2U ⊗ S(P )1U P = g ⊗ g . 1

2

1

1 2

2

1

(11.5.5) (11.5.6)

The first one follows by applying (3.2.13), (3.2.17), (3.1.7), (3.2.1) and then the formula f 1 β S( f 2 ) = S(α ) and (3.1.7), (3.2.1). The second one can be proved more easily by using (7.3.1), (3.2.13) and (3.2.23); we leave the details to the reader. Therefore, if we denote by G1 ⊗ G2 another copy of f −1 then from the definition

436

Factorizable Quasi-Hopf Algebras

(8.5.14) of SD , (11.5.5), (11.5.6) and the axioms of a quasi-Hopf algebra we obtain SD (b1 ) ⊗ b2 (ε  S(P21 x2 p˜21 ))SD (e j  1)(ε  S(ei )x3 p˜22 )

=

⊗ (ei  S−1 (P2 )e j P11 x1 p˜1 ) (ε  S(P21 x2 p˜21 ) f 1 )(p11U 1  S

=

−1

(e j )  f 2 S−1 (p2 )

 p12U 2 S(ei )x3 p˜22 ) ⊗ (ei  S−1 (P2 )e j P11 x1 p˜1 ) (ε  S(P21 x2 p˜21 ) f 1 )(S

=

−1

(e j )  p12U 2 S(ei )x3 p˜22 )

⊗ (ei  S−1 (p11U 1 P2 )e j S−1 ( f 2 S−1 (p2 ))P11 x1 p˜1 ) (8.5.3)

=

(S

−1

1 (e j )  S(P21 x2 p˜21 )(1,2) f(1,2) p12U 2 S(ei )x3 p˜22 ) ⊗ (ei  S−1 (U 1 P2 )

1 S−1 (S(P21 x2 p˜21 )(1,1) f(1,1) p11 )e j S−1 ( f 2 S−1 (S(P21 x2 p˜21 )2 f21 p2 ))P11 x1 p˜1 ) (11.5.5)

=

(S

−1

(e j )  g12 S(P1 y3 x22 p˜2(1,2) )2U 2 S(ei )x3 p˜22 ) ⊗ (ei  S−1 (U 1 P2 )

S−1 (g11 S(P1 y3 x22 p˜2(1,2) )1 )e j S−2 (g2 S(S(y1 x1 p˜1 )α y2 x12 p˜2(1,1) ) (10.3.3),(11.5.6)

=

(3.2.20)

−1

2 (e j )  g12 S(q˜2 X(2,2) p˜22 )2 G2 S(ei )X 3 )

2 2 ⊗ (ei  S−1 (g11 S(q˜2 X(2,2) p˜22 )1 G1 )e j S−2 (g2 S(X 1 S(X12 p˜1 )q˜1 X(2,1) p˜21 ))

(3.2.22),(3.2.24)

=

(S

(S

−1

(e j )  g12 S(X 2 )2 G2 S(ei )X 3 )

⊗ (ei  S−1 (g11 S(X 2 )1 G1 )e j S−2 (g2 S(X 1 )) (3.2.13)

=

(S

−1 j

( e)  S(i e)) ⊗ (X12 S−1 (g12 G2 )  i e  S−1 (X 3 )

 X22 S−1 (g11 G1 ) j eS−2 (g2 S(X 1 ))). We prove that Q is injective. Let D ∈ (D(H))∗ be such that Q(D ◦ SD ) = 0. This means D(SD (b1 ))b2 = 0, which is equivalent to D(S

−1 j

( e)  S(i e))i e, S−1 (X 3 )hX12 S−1 (g12 G2 ) χ , X22 S−1 (g11 G1 ) j eS−2 (g2 S(X 1 )) = 0,

for all h ∈ H and χ ∈ H ∗ . In particular, D(S

−1 j

( e)  S(i e))i e, S−1 (X 3 )(S−1 (x3 )hS−1 (F 2 f21 )x12 )X12 S−1 (g12 G2 ) S−2 (S(x1 ) f 2 )  χ  S−1 (F 1 f11 )x22 , X22 S−1 (g11 G1 ) j eS−2 (g2 S(X 1 )) = 0,

for all h ∈ H and χ ∈ H ∗ , and therefore D(S

−1

(χ )  S(h)) = 0, ∀ χ ∈ H ∗ and h ∈ H.

Since the antipode S is bijective (H is finite dimensional) we conclude that D = 0 and by using the bijectivity of SD it follows that Q is injective. Finally, Q is bijective because D(H) is finite dimensional, so the proof is finished. We would like to stress that formula (11.5.1) was chosen in such a way that it provides a left H-module morphism from H ∗ , the function algebra braided group

11.5 Factorizable QT Quasi-Hopf Algebras

437

associated to H ∗ , to H, the enveloping braided group H of (H, R). Indeed, for all χ ∈ H ∗ and h ∈ H we have: (11.5.1)

h  Q(χ )

χ , S(X22 p˜2 ) f 1 R2 r1U 1 X 3 h1 X 1 S(X12 p˜1 ) f 2 R1 r2U 2 S(h2 )

=

(7.3.2),(10.1.3)

=

(3.2.13)

χ , S((h(2,1) X 2 )2 p˜2 ) f 1 R2 r1U 1 h(2,2) X 3  h1 X 1 S((h(2,1) X 2 )1 p˜1 ) f 2 R1 r2U 2

(3.1.7),(3.2.22)

=

(11.5.1),(11.4.30)

=

χ , S(X22 p˜2 h1 ) f 1 R2 r1U 1 X 3 h2 X 1 S(X12 p˜1 ) f 2 R1 r2U 2 Q(h2  χ  S(h1 )) = Q(h  χ ).

It is quite remarkable that (11.5.1) is a braided Hopf algebra morphism, too. Proposition 11.28 Let (H, R) be a finite-dimensional QT quasi-Hopf algebra, H the associated enveloping algebra braided group of H and H ∗ the function algebra braided group associated to H ∗ . Then the map Q defined by (11.5.1) is a braided Hopf algebra morphism in H M from H ∗ to H. Proof We have already seen that Q is a morphism in H M . Hence, it remains to show that Q is an algebra and a coalgebra morphism. To this end, we will use the second formula (11.5.2) for the map Q. We set R = R1 ⊗ R2 = r1 ⊗ r2 = R1 ⊗ R2 = R1 ⊗ R2 = r1 ⊗ r2 = R 1 ⊗ R 2 , qL = q˜1 ⊗ q˜2 = q˜1 ⊗ q˜2 and pR = p1 ⊗ p2 = P1 ⊗ P2 . Now, for all χ , ψ ∈ H ∗ we compute: Q(χ ·ψ ) =

χ , S(x1 X 1 ) f 2 R1 q˜11 Z11 R21 r11 p11 x13Y 2 r1 y1 X 2  ψ , S(x2Y 1 r2 y2 X13 ) f 1 R2 q˜12 Z21 R22 r12 p12 x23Y 3 y3 X23 q˜21 Z 2 R1 r2 p2 S(q˜22 Z 3 )

(10.1.3),(5.5.16)

=

3 χ , S(x1 X 1 ) f 2 q˜12 [Z 1 R2 r1 x(1,1) p1 ]2 R1Y 2 r1 y1 X 2  3 p1 ]1 R2Y 3 y3 X23  ψ , S(x2Y 1 r2 y2 X13 ) f 1 q˜11 [Z 1 R2 r1 x(1,1) 3 p2 S(q˜22 Z 3 x23 ) q˜21 Z 2 R1 r2 x(1,2)

(10.1.3),(3.1.7)

=

(3.2.28)

21 T 2 Z21 R22 r12 p12 R1Y 2 r1 y1 X 2  χ , S(X 1 )q˜1 Q ψ , S(Y 1 r2 y2 X13 )q˜1 T 1 Z11 R21 r11 p11 R2Y 3 y3 X23 

2 T13 Z 2 R1 r2 p2 S(q˜22 Q

(2,2) T23 Z 3 ) q˜21 Q (2,1)

(3.1.9),(3.1.7)

=

(5.2.7),(10.1.3)

21 x2 Z 2 R22 R1 r11 p11Y 2 r1 y1 X 2  χ , S(X 1 )q˜1V 1 Q (2,1) ψ , S(x1Y 1 r2 y2 X13 )q˜1 x12 Z 1 R21 R2 r12 p12Y 3 y3 X23 

22 x2 Z 3 R1 r2 p2 S(q˜22V 3 x3 ) q˜21V 2 Q (2,2)

(3.1.7),(10.1.3)

=

(5.5.16)

(3.1.9),(10.1.3)

=

(10.1.1),(10.1.2)

21 Z 2 R22 R1 r11 T12Y 2 r1 y1 p11 X 2  χ , S(X )q˜1V 1 Q 1

ψ , S(T 1Y 1 r2 y2 (p12 X 3 )1 )q˜1 Z 1 R21 R2 r12 T22Y 3 y3 (p12 X 3 )2 

22 Z 3 R1 r2 T 3 p2 S(q˜22V 3 ) q˜21V 2 Q

21 R2 Z 3 x3 R1W 2 r1 z1 T 2 r1Y11 y1 p11 X 2  χ , S(X 1 )q˜1V 1 Q ψ , S(T 1 r2Y21 y2 (p12 X 3 )1 )q˜1 Z 1 R2 x2 R2W 3 z3 R 1 T13Y 2 y3 (p12 X 3 )2 

438

Factorizable Quasi-Hopf Algebras

(3.1.9),(10.1.3)

=

(10.1.2)

22 R1 Z 2 R1 x1W 1 r2 z2 R 2 T23Y 3 p2 S(q˜22V 3 ) q˜21V 2 Q

21 R2 Z 3 x3W23 R1 T 2 r1 D1 z11Y11 y1 p11 X 2  χ , S(X 1 )q˜1V 1 Q ψ , S(W 1 T11 r12 D2 z12Y21 y2 (p12 X 3 )1 )q˜1 Z 1 R2 x2W13 R2 T 3 z3 R 1

22 R1 Z 2 R1 x1W 2 T21 r22 D3 z2 R 2Y 3 p2 S(q˜22V 3 ) Y 2 y3 (p12 X 3 )2 q˜21V 2 Q

(10.1.2),(3.1.9)

=

(3.1.7)

21 R2 Z 3 T 3 r1Y 1 p11 X 2  χ , S(X 1 )q˜1V 1 Q 2 R 1C2Y22 (p12 X 3 )2  ψ , S(T 1 r12C1Y12 (p12 X 3 )1 )q˜1 Z 1 R2 T22 r(2,2)

22 R1 Z 2 R1 T12 r2 R 2C3Y 3 p2 S(q˜22V 3 ) q˜21V 2 Q (2,1) (5.2.7),(10.1.3)

=

(3.1.7),(3.2.1)

χ , S(y1 X 1 )q˜1 R2 r1 y21Y 1 p11 X 2  ψ , S(x1C1 (Y 2 p12 )1 X13 )α x2 R2 R 1C2 (Y 2 p12 )2 X23  q˜2 R1 r2 y22 x3 R1 R 2C3Y 3 p2 S(y3 )

(5.5.16),(10.1.3)

=

(3.1.7),(3.2.1)

χ , S(y1 X 1 )q˜1 R2 r1 y21 z1 X 2  ψ , S(C1 p11 X13 )q˜1 R2 R 1C2 p12 X23 q˜2 R1 r2 y22 z2 q˜2 R1 R 2C3 p2 S(y3 z3 ).

On the other hand, if we denote by P1 ⊗ P2 another copy of pR then by (4.1.9), (3.2.19), (11.5.2) we have: Q(χ ) ◦ Q(ψ ) χ , q˜1Y 1 R2 r1 P1 ψ , q˜1 Z 1 R2 r1 p1 

21 Z 2 R1 r2 p2 S(y3 Q

22 Z 3 ) q1 y11 q˜21Y 2 R1 r2 P2 S(q2 y12 q˜22Y 3 )y2 Q

= (5.2.7),(10.1.3)

=

(5.5.16) (3.2.22),(10.1.3)

=

(3.1.7),(3.2.21)

χ , S(X 1 P11 )q˜1 R2 r1 X 2 P21 ψ , q˜1 Z 1 R2 r1 p1 

22 Z 3 ) q1 y11 q˜2 R1 r2 X 3 P2 S(q2 y12 )y2 Q˜ 21 Z 2 R1 r2 p2 S(y3 Q χ , S(X 1 (q11 P1 )1 y11 )q˜1 R2 r1 X 2 (q11 P1 )2 y12 

21 Z 2 ψ , q˜1 Z 1 R2 r1 p1 q˜2 R1 r2 X 3 q12 P2 S(q2 )y2 Q

22 Z 3 ) R1 r2 p2 S(y3 Q

(3.2.23),(5.2.7),(3.1.9)

=

(10.1.3),(5.5.16) (3.2.22),(10.1.3)

=

(3.1.7),(3.2.21)

χ , S(y1 X 1 )q˜1 R2 r1 y21 x1 X 2 ψ , S(Y 1 p11 )q˜1 R2 r1Y 2 p12  q˜2 R1 r2 y22 x2 X13 q˜2 R1 r2Y 3 p2 S(y3 x3 X23 ) χ , S(y1 X 1 )q˜1 R2 r1 y21 x1 X 2  ψ , S(Y 1 p11 X13 )q˜1 R2 r1Y 2 p12 X23  q˜2 R1 r2 y22 x2 q˜2 R1 r2Y 3 p2 S(y3 x3 ).

By the above it follows that Q is multiplicative. Since Q(1H ∗ ) = Q(ε ) = β = 1H , we conclude that Q is an algebra map. Thus, one has only to show that Q is a coalgebra map. To this end, observe first that (3.1.9), (3.2.1) imply X11 p1 ⊗ X21 p2 S(X 2 ) ⊗ X 3 = x1 ⊗ x2 S(x13 p˜1 ) ⊗ x23 p˜2 .

(11.5.7)

Also, it is not hard to see that (11.2.3), (8.7.1), (10.1.1), (10.1.3) and (10.3.6) imply ΔH (h) = x1 X 1 h1 r2 g2 S(x2Y 1 R2 y2 X13 ) ⊗ x13Y 2 R1 y1 X 2 h2 r1 g1 S(x23Y 3 y3 X23 ). (11.5.8)

11.5 Factorizable QT Quasi-Hopf Algebras

439

Therefore, by (11.5.8) and (11.5.2), for any χ ∈ H ∗ we have ΔH (Q(χ )) =

χ , q˜1 Z 1 R2 r1 p1 x1 X 1 q˜2(1,1) Z12 R11 r21 p21 S(q˜22 Z 3 )1 r2 g2 S(x2Y 1 R2 y2 X13 ) ⊗ x13Y 2 R1 y1 X 2 q˜2(1,2) Z22 R12 r22 p22 S(q˜22 Z 3 )2 r1 g1 S(x23Y 3 y3 X23 )

(10.1.3),(3.2.13)

=

(3.2.25),(3.1.7)

χ , q˜1 Z 1 R2 r1V 1 (T11 p1 )1 P1  x1 X 1 (q˜21 Z 2 )1 R11 r2 r22V 3 T21 p2 S(x2Y 1 R2 y2 (X 3 q˜22 )1 Z13 T 2 ) ⊗ x13Y 2 R1 y1 X 2 (q˜21 Z 2 )2 R12 r1 r21V 2 (T11 p1 )2 P2 S(x23Y 3 y3 (X 3 q˜22 )2 Z23 T 3 )

(11.5.7),(10.1.2)

=

(10.1.1),(5.2.7)

χ , S(v1 )q˜1 v21 R2t 3 r12 R1 z11 P1  x1 X 1 q˜21 v2(2,1) R11t 1 r2 z2 S(x2Y 1 R2 y2 X13 v31 z31 p˜1 ) ⊗ x13Y 2 R1 y1 X 2 q˜22 v2(2,2) R12t 2 r11 R2 z12 P2 S(x23Y 3 y3 X23 v32 z32 p˜2 )

(10.1.1),(10.1.3)

=

(10.1.1)

χ , S(v1 )q˜1 v21 T 1 R2 r1t 1V 1 R 2 R1 z11 P1  x1 X 1 q˜21 v2(2,1) T 2 R1 r2t 2 r2V 3 z2 S(x2Y 1 R2 y2 X13 v31 z31 p˜1 ) ⊗ x13Y 2 R1 y1 X 2 q˜22 v2(2,2) T 3t 3 r1V 2 R 1 R2 z12 P2 S(x23Y 3 y3 X23 v32 z32 p˜2 )

(3.1.7),(5.2.7)

=

(3.2.22),(3.1.9)

χ , S(v1 )q˜1 v21 X11 R2 r1t 1V 1 R 2 R1 z11 P1  x1 q˜2 v22 X21 R1 r2t 2 r2V 3 z2 S(x2Y 1 R2 y2 v3(2,1) X13 z31 p˜1 ) ⊗ x13Y 2 R1 y1 v31 X 2t 3 r1V 2 R 1 R2 z12 P2 S(x23Y 3 y3 v3(2,2) X23 z32 p˜2 )

(10.1.3),(3.1.9)

=

(3.2.22)

χ , S(v1 )q˜1 v21 R2 r1t 1 Z 1 R 2 R1 P1  x1 q˜2 v22 R1 r2t 2 X 1 r2 z2 S(x2Y 1 R2 y2 (v3t 3 )(2,1) X13 z31 p˜1 ) ⊗ x13Y 2 R1 y1 (v3t 3 )1 X 2 r1 z1 Z 2 R 1 R2 P2 S(x23Y 3 y3 (v3t 3 )(2,2) X23 z32 p˜2 Z 3 )

(3.1.7),(10.1.3)

=

(3.1.9),(10.1.2)

χ , S(v1 )q˜1 v21 R2 r1t 1 Z 1 R 2 R1 P1 x1 q˜2 v22 R1 r2t 2 T 1 X 1 R21V 2 y12 z2 S(x2 (v3t 3 )1Y 1 T12 X 2 R22V 3 y2 z31 p˜1 ) ⊗ x13 (v3t 3 )(2,1) Y 2 T22 X 3 R1V 1 y11 z1 Z 2 R 1 R2 P2 S(x23 (v3t 3 )(2,2)Y 3 T 3 y3 z32 p˜2 Z 3 )

(3.1.9),(3.2.20)

=

(3.2.1) (3.1.9),(3.2.1)

=

(3.2.22),(10.1.3)

χ , S(v1 )q˜1 v21 R2 r1t 1 Z 1 R 2 R1 P1 x1 q˜2 v22 R1 r2t 2 X 1 β S(x2 (v3t 3 )1 X 2 ) ⊗ x13 (v3t 3 )(2,1) X13 Z 2 R 1 R2 P2 S(x23 (v3t 3 )(2,2) X23 Z 3 ) 2 2 χ , S(t 1 x1 )q˜1 R2 r1t12 x(1,1) z1 Z 1 R 2 R1 P1 q˜2 R1 r2t22 x(1,2) z2 β S(t 3 x22 z3 )

⊗ x13 Z 2 R 1 R2 P2 S(x23 Z 3 ) (3.1.7),(3.2.1)

=

(10.1.3),(11.5.2)

Q(χ1  S(x1 )) ⊗ χ2 (x2 Z 1 R 2 R1 P1 )x13 Z 2 R 1 R2 P2 S(x23 Z 3 ).

On the other hand, by (11.4.33) we have (Q ⊗ Q)(ΔH ∗ (χ )) = (11.5.2),(3.2.21)

=

(10.1.3) (3.1.7),(3.2.22)

=

(3.2.20)

Q(χ1  S(x1 )) ⊗ Q(x23 X 3  χ2  x2 X 1 β S(x13 X 2 )) Q(χ1  S(x1 )) ⊗ χ2 , x2 X 1 β S(x13 X 2 )q˜1 Z 1 (x23 X 3 )(1,1) R 2 R1 P1 

22 Z 3 (x23 X 3 )2 )

21 Z 2 (x23 X 3 )(1,2) R 1 R2 P2 S(Q Q Q(χ1  S(x1 )) ⊗ χ2 , x2 S( p˜1 )q˜1 p˜21 Z 1 R 2 R1 P1 

440

Factorizable Quasi-Hopf Algebras

2 p˜22 )1 Z 2 R 1 R2 P2 S(x23 (Q

2 p˜22 )2 Z 3 ) x13 (Q (3.2.24)

=

Q(χ1  S(x1 )) ⊗ χ2 , x2 Z 1 R 2 R1 P1 x13 Z 2 R 1 R2 P2 S(x23 Z 3 ).

So Q is a coalgebra map since (ε H ◦ Q)(χ ) = χ (α ) = ε H ∗ . This ends the proof. Summarizing, we can now present the true meaning of the map Q : H ∗ → H defined in (11.5.1). It is a morphism of braided groups from H ∗ , the function algebra braided group associated to H ∗ , to H, the associated enveloping algebra braided group of H. When H is factorizable in the sense that the map Q is bijective then Q : H∗ ∼ = H as braided Hopf algebras. In other words, the function algebra braided group associated to H ∗ and the associated enveloping algebra braided group of H are categorical self dual, cf. Proposition 11.21.

11.6 Factorizable Implies Unimodular We show that any finite-dimensional factorizable QT quasi-Hopf algebra is unimodular. In particular, we obtain that for any finite-dimensional quasi-Hopf algebra H its quantum double D(H) is always a unimodular quasi-Hopf algebra. Throughout this section, H is a finite-dimensional quasi-Hopf algebra, t ∈ H is a non-zero left integral in H and μ is the modular element of H ∗ ; see (7.2.6). We also consider λ a non-zero left cointegral on H, r a non-zero right integral in H such that λ (r) = 1 and g the modular element of H with inverse g−1 as in (7.6.2). Remark 11.29 Let H be a finite-dimensional quasi-Hopf algebra, t a non-zero left integral in H and μ the modular element of H ∗ . Then the relation (7.5.16) can be rewritten in the form t1 p1 ⊗ t2 p2 S(h  μ ) = t1 p1 h ⊗ t2 p2 ,

(11.6.1)

for all h ∈ H, where for all χ ∈ H ∗ we denote h  χ = χ (h1 )h2 . We can now prove the main result of this section. Theorem 11.30 Let (H, R) be a finite-dimensional QT quasi-Hopf algebra and μ the modular element of H ∗ . Then the following assertions hold: (i) If qR = q1 ⊗ q2 = Q1 ⊗ Q2 and pR = p1 ⊗ p2 = P1 ⊗ P2 are the elements defined by (3.2.19) then

μ (Q1 )q2t2 p2 S(Q2 (R2 P2  μ ))R1 P1 ⊗ q1t1 p1 = S(u)q1t1 p1 ⊗ q2t2 p2 ,

(11.6.2)

where R = R1 ⊗ R2 is the R-matrix of H and u is the element defined in (10.3.4). (ii) If (H, R) is factorizable then H is unimodular. Proof

(i) Note that g1 S(g2 α ) = β , (10.3.6), (10.3.8) and (10.3.10) imply R1 β S(R2 ) = S(β u).

(11.6.3)

11.6 Factorizable Implies Unimodular

441

Now, from (11.6.1) we have

μ (Q1 )q2t2 p2 S(Q2 (R2 P2  μ ))R1 P1 ⊗ q1t1 p1 =

μ (Q1 )q2t2 p2 S(Q2 )R1 P1 ⊗ q1t1 p1 R2 P2

(7.2.6)

q2t2 Q12 p2 S(Q2 )R1 P1 ⊗ q1t1 Q11 p1 R2 P2

(3.2.23)

=

q2t2 R1 P1 ⊗ q1t1 R2 P2

(10.1.3)

q2 R1t1 P1 ⊗ q1 R2t2 P2

(7.2.3)

q2 R1 β Q1t1 P1 ⊗ q1 R2 Q2t2 P2

(7.2.2)

q2 R1 β S(q1 R2 )Q1t1 P1 ⊗ Q2t2 P2

(11.6.3)

S(q1 β uS−1 (q2 ))Q1t1 P1 ⊗ Q2t2 P2

= = = = =

(10.3.10),(3.2.19),(3.2.2)

=

S(u)Q1t1 P1 ⊗ Q2t2 P2 ,

and this proves the first assertion.  (ii) Let (λ ,t) ∈ L × lH be as in Proposition 7.59, that is, λ is a non-zero left cointegral on H and t is a non-zero left integral in H such that λ (S−1 (t)) = 1 and λ (q2t2 p2 )q1t1 p1 = 1H . So the definition (7.6.2) applies. By applying IdH ⊗ λ to the equality (11.6.2) we obtain

μ (Q1 )S−1 (g−1 )S(Q2 (R2 P2  μ ))R1 P1 = S(u), and since S−1 (g)S(u) = S(uS−2 (g)) = S(gu), it follows that the above relation is equivalent to

μ (Q1 )S(Q2 (R2 P2  μ ))R1 P1 = S(u)S(g).

(11.6.4)

On the other hand, if we denote by r1 ⊗ r2 another copy of R, we have

μ (Q1 )S(Q2 (R2 P2  μ ))R1 P1 (3.2.19)

μ (X 1 R21 P12 )S(X 2 R22 P22 )α X 3 R1 P1

(10.1.2)

μ (X 1 R2 y2 P12 )S(r2 X 3 y3 P22 )α r1 X 2 R1 y1 P1

= =

(10.3.8),(10.3.10),(3.2.19)

=

(3.2.25),(10.1.3)

=

μ (q1 R2 y2 P12 )S(S(q2 )y3 P22 )uR1 y1 P1 1 μ (q1 X(1,1) p11 R2 P2 S(X 3 ) f 1 ) 1 S(S(q2 )X21 p2 S(X 2 ) f 2 )uX(1,2) p12 R1 P1

(10.3.10),(3.2.21)

μ (X 1 q1 p11 R2 P2 S(X 3 ) f 1 )S(S(q2 p12 )p2 S(X 2 ) f 2 )uR1 P1

(3.2.23),(10.3.10)

μ (X 1 R2 P2 S(X 3 ) f 1 )uS−1 (S(X 2 ) f 2 )R1 P1 .

= =

From the above computation and (11.6.4) we obtain

μ (X 1 R2 P2 S(X 3 ) f 1 )S−1 (S(X 2 ) f 2 )R1 P1 = u−1 S(u)S(g).

(11.6.5)

2 1 But, as we know, if (H, R) is QT then R˜ = R−1 21 = R ⊗ R is another R-matrix for H. ˜ instead of (H, R), we find Repeating the above computations for (H, R)

μ (X 1 r1 P2 S(X 3 ) f 1 )S−1 (S(X 2 ) f 2 )r2 P1 = u˜−1 S(u)S(g), ˜

(11.6.6)

442

Factorizable Quasi-Hopf Algebras

˜ instead of (H, R), where we denote by u˜ the element defined as in (10.3.4) for (H, R) 1 2 −1 and where r ⊗ r is another copy of R . More precisely, we have that u˜ = S(u−1 ).

(11.6.7)

Indeed, one can easily see that (11.6.3) and (10.3.10) imply r2 β S(r1 ) = S−1 (β )u−1 = u−1 S(β ).

(11.6.8)

Now, we compute: =

S(r1 x2 β S(x3 ))α r2 x1

(∗)

S(β f 1 r1 x2 β S(x3 )) f 2 r2 x1

u˜

=

(10.3.8),(3.2.19)

=

S(r2 β S(r1 ) f 2 p2 ) f 1 p1

(11.6.8)

S(S−1 ( f 1 p1 )u−1 S(β ) f 2 p2 )

(10.3.10),(∗),(3.2.19),(3.2.2)

S(u−1 S(p1 )α p2 ) = S(u−1 ),

= =

where (*) means that we use the relation S(β f 1 ) f 2 = α . Now, since S2 (u) = u the relation (11.6.6) becomes

μ (X 1 r1 P2 S(X 3 ) f 1 )S−1 (S(X 2 ) f 2 )r2 P1 = S(u)u−1 S(g).

(11.6.9)

From Corollary 10.18 we know that uS(u) = S(u)u, so u−1 S(u) = S(u)u−1 . Hence, by (11.6.5) and (11.6.9) we obtain

μ (X 1 R2 P2 S(X 3 ) f 1 )S−1 (S(X 2 ) f 2 )R1 P1 = μ (X 1 r1 P2 S(X 3 ) f 1 )S−1 (S(X 2 ) f 2 )r2 P1 . This comes out explicitly as μ (R2 P2 )R1 P1 = μ (r1 P2 )r2 P1 , and implies

μ (Q11 R2 P2 S(Q2 ))Q12 R1 P1 = μ (Q11 r1 P2 S(Q2 ))Q12 r2 P1 . From (10.1.3) and (3.2.23) we deduce that

μ (R2 )R1 = μ (r1 )r2 ⇔ μ (R2 r1 )R1 r2 = 1H .

(11.6.10)

Finally, the above relation allows to compute: Q(μ )

= (11.6.10),(3.2.19)

=

(3.2.1),(3.2.20)

=

μ (q˜1 X 1 R2 r1 p1 )q˜21 X 2 R1 r2 p2 S(q˜22 X 3 ) μ (q˜1 X 1 x1 )q˜21 X 2 x2 β S(q˜22 X 3 x3 ) μ (α )β = Q(μ (α )ε ).

If (H, R) is factorizable then Q is bijective, so μ = μ (α )ε . In particular, 1 = μ (1H ) = μ (α )ε (1H ) = μ (α ). Hence μ = ε , and this means that H is unimodular. Theorem 11.31 Let H be a finite-dimensional quasi-Hopf algebra. Then the quantum double D(H) of H is a unimodular quasi-Hopf algebra. Proof

It is a consequence of Proposition 11.27 and Theorem 11.30.

11.7 The Quantum Double of a Factorizable Quasi-Hopf Algebra

443

Remark 11.32 Let (H, R) be a finite-dimensional QT quasi-Hopf algebra. By the proof of Theorem 11.30 we can derive a nicer formula for the antipode S of the function algebra braided group associated to H ∗ ; see (11.4.35). In fact, since (H cop , R−1 ) is a QT quasi-Hopf algebra as well, by (10.1.1) applied to (H cop , R−1 ) (which is actually equivalent to (10.1.1) for (H, R)) we have that 1

2

1

q12 R2 p˜2 ⊗ q2 R S(q11 R1 p˜1 ) 1

2

=

q12 x2 R X 3 y2 p˜2 ⊗ q2 x3 R X 2 r2 y1 S(q11 x1 X 1 r1 y2 p˜1 )

(3.2.20),(5.5.17)

q1Y12 R X 3 ⊗ S−1 (Y 3 )q2Y22 R X 2 r2 β S(Y 1 x1 X 1 r1 )

(10.1.3),(11.6.8)

q1 R Y22 X 3 ⊗ S−1 (Y 3 )q2 R Y12 X 2 u−1 S(Y 1 X 1 β )

= =

1

1

2

2

(10.3.10),(3.2.20) 1 1 2 2 2 = q R Y2 p˜ ⊗ u−1 S(Y 1 S(q2 R Y12 p˜1 )Y 3 ) (10.3.3) 1 2 = q1 R x2 p˜21 ⊗ u−1 S(S(q2 R x1 p˜1 )x3 p˜22 ) (10.3.1) 1 −1 2

=

R ⊗ u S(R ).

Thus, by (11.4.35) we get that S(χ ) = R1  χ S  u−1 S(R2 ), for all χ ∈ H ∗ .

11.7 The Quantum Double of a Factorizable Quasi-Hopf Algebra Throughout this section, (H, R) will be a finite-dimensional QT quasi-Hopf algebra and D(H) its quantum double. Thus D(H) is a biproduct quasi-Hopf algebra. The goal of this section is to show that when H is, moreover, factorizable then D(H) is nothing but a twist deformation of H ⊗ H. When there is no danger of confusion, the elements χ  h of D(H) will be simply denoted by χ h. Since H ∗ can be viewed only as a k-linear subspace of D(H), we will denote by

χ(1) ⊗ χ(2) = ΔD (χ  1H ) (8.5.9)

= (ε  X 1Y 1 )(p11 x1  χ2  Y 2 S−1 (p2 )  p12 x2 ) ⊗ X12  χ1  S−1 (X 3 )  X22Y 3 x3 .

In this notation, for all χ ∈ H ∗ and h ∈ H, the comultiplication ΔD of D(H) is ΔD (χ h) = χ(1) h1 ⊗ χ(2) h2 . By Lemma 10.26, there exists a quasi-Hopf algebra projection π : D(H) → H covering the canonical inclusion iD : H → D(H); see (10.5.1). Recall that π is a quasi-Hopf algebra morphism and π ◦ iD = IdH . 2 1 As we have seen before, R˜ := R−1 21 = R ⊗ R is another R-matrix for H. So there is always a second projection π : D(H) → H covering the canonical inclusion iD . Explicitly, the morphism π is given, for all χ ∈ H ∗ and h ∈ H, by 2 1 π (χ  h) = χ (q2 R )q1 R h.

(11.7.1)

444

Factorizable Quasi-Hopf Algebras

For H a quasi-Hopf algebra, A a quasi-bialgebra and ν : H → A a quasi-bialgebra morphism, we denote by H co(ν ) = {h ∈ H | h1 ⊗ ν (h2 ) = x1 hS(x23 X 3 ) f 1 ⊗ ν (x2 X 1 β S(x13 X 2 ) f 2 )}, (11.7.2) the set of alternative coinvariants of H relative to its structure in H MHH and the morphism ν . Lemma 11.33 Let (H, R) be a finite-dimensional QT quasi-Hopf algebra, and π and π the quasi-Hopf algebra morphisms defined by (10.5.1) and (11.7.1), respectively. Let j : D(H)co(π ) → D(H) be the inclusion map and Ψ : H ∗ → D(H)co(π ) defined by Ψ(χ ) = χ(1) β S(π (χ(2) ))

(11.7.3)

for all χ ∈ H ∗ . Then the following assertions hold: (1) Ψ is well defined and bijective. (2) If Q is the map defined by (11.5.3) then S ◦ Q = π ◦ j ◦ Ψ. In particular, Q is bijective if and only if π |D(H)co(π ) is bijective. Proof

(1) For all χ ∈ H ∗ we have (Id ⊗ π )ΔD (Ψ(χ )) = χ((1),(1)) β1 S(π (χ(2) ))1 ⊗ χ((1),(2)) β2 S(π (χ(2) ))2 ,

where we use the Sweedler type notation (ΔD ⊗ Id)(ΔD (χ )) = χ((1),(1)) ⊗ χ((1),(2)) ⊗ χ(2) , (Id ⊗ ΔD )(ΔD (χ )) = χ(1) ⊗ χ((2),(1)) ⊗ χ((2),(2)) . Now, since H is a quasi-Hopf subalgebra of D(H) and π is a quasi-Hopf algebra morphism such that π (h) = h for any h ∈ H, by a direct computation one can show that Ψ(χ ) ∈ D(H)co(π ) , so Ψ is well defined. We claim that the inverse of Ψ, Ψ−1 : D(H)co(π ) → H ∗ , is given for all D ∈ D(H)co(π ) by the formula Ψ−1 (D) = (Id ⊗ ε )(D). Indeed, Ψ−1 is a left inverse since (Ψ−1 ◦ Ψ)(χ ) = Id ⊗ ε , χ(1) β S(π (χ(2) )) = (Id ⊗ ε )(χ(1) )εD (χ(2) ) = (Id ⊗ ε )(χ ⊗ 1) = χ , for all χ ∈ H ∗ . It is also a right inverse. If D = i χ i h ∈ D(H)co(π ) then i χ(1) i h1 ⊗ π (i χ(2) )i h2

= x1 [i χ i h]S(x23 X 3 ) f 1 ⊗ x2 X 1 β S(x13 X 2 ) f 2

in D(H) ⊗ H. Therefore, (Ψ ◦ Ψ−1 )(D) = ε (i h)Ψ(i χ ) = ε (i h)i χ(1) β S(π (i χ(2) )) = i χ(1) i h1 β S(π (i χ(2) )i h2 )

11.7 The Quantum Double of a Factorizable Quasi-Hopf Algebra

445

= x1 [i χ i h]S(x23 X 3 ) f 1 β S(x2 X 1 β S(x13 X 2 ) f 2 ) = i χ i h = D, because of f 1 β S( f 2 ) = S(α ), and (3.2.1) and (3.2.2). (2) By (11.7.3), (8.5.9) and (10.5.1), for any χ ∈ H ∗ we find that Ψ(χ ) = (X 1Y 1 )(1,1) p11 x1  χ  S−1 (X 3 )q2 R1 X12Y 2 S−1 ((X 1Y 1 )2 p2 )  (X 1Y 1 )(1,2) p12 x2 β S(q1 R2 X22Y 3 x3 ) and, if we denote by Q1 ⊗ Q2 another copy of qR , and by r1 ⊗ r2 another copy of R, then by (11.7.1) we compute that (π ◦ j ◦ Ψ)(χ ) =

2

χ , S−1 (X 3 )q2 R1 X12Y 2 S−1 ((X 1Y 1 )2 p2 )Q2 R (X 1Y 1 )(1,1) p11 x1  1

Q1 R (X 1Y 1 )(1,2) p12 x2 β S(q1 R2 X22Y 3 x3 ) (10.1.3),(3.2.21)

=

2

χ , S−1 (X 3 )q2 R1 X12Y 2 S−1 (p2 )Q2 R p11 x1  1

X 1Y 1 Q1 R p12 x2 β S(q1 R2 X22Y 3 x3 ) (10.1.3),(3.2.23)

=

χ , S−1 (X 3 )q2 R1 X12Y 2 R x1 X 1Y 1 R x2 β S(q1 R2 X22Y 3 x3 )

(10.1.3),(10.1.1)

χ , S−1 (X 3 )q2 X22 R1 r2Y 3 R X 1Y 1 R1 β S(q1 X12 R2 r1Y 2 R2 )

=

(3.2.1),(10.1.5),(10.1.3)

=

(3.2.20),(11.5.3)

=

2

1

2

1

1

χ , S−1 (X 3 )q2 R1 r2 X22Y 3 S(q1 R2 r1 X12Y 2 S−1 (X 1Y 1 β )) (S ◦ Q)(χ ),

as needed. Since H is finite dimensional the antipode S is bijective, so Q is bijective if and only if π ◦ j is bijective. Thus, the proof is complete. The notion of right quasi-Hopf bimodule was introduced in Definition 6.1. For M a right quasi-Hopf bimodule we have denoted by M co(H) the set of alternative coinvariants of M, see Remark 6.17 for the explicit definition of M co(H) . Lemma 11.34 Let D, A and B be quasi-bialgebras and ϑ , υ , κ quasi-bialgebra morphisms as in the diagram below: ϑ

D υ

κ

B

A

ζ

A ⊗ B.

Suppose υ ◦ κ = IdB and let ζ := (ϑ ⊗ υ ) ◦ ΔD . The following assertions hold: (i) D and A ⊗ B are right quasi-Hopf B-bimodules via the following structures: b · d · b = κ (b)d κ (b ), D ∈ B MBB : ρD (d) = d1 ⊗ υ (d2 ),  b · (a ⊗ b) · b = ϑ (κ (b1 ))aϑ (κ (b 1 )) ⊗ b2 bb 2 , B A ⊗ B ∈ B MB : ρA⊗B (a ⊗ b) = ϑ (κ (x1 ))aϑ (κ (X 1 )) ⊗ x2 b1 X 2 ⊗ x3 b2 X 3 ,

446

Factorizable Quasi-Hopf Algebras

for a ∈ A, b, b , b ∈ B, d ∈ D, and ζ becomes a quasi-Hopf B-bimodule morphism. (ii) If D, A and B are, moreover, quasi-Hopf algebras and ϑ , υ and κ are, moreover, quasi-Hopf algebra maps then Dco(B) = Dco(υ ) and (A ⊗ B)co(B) = {ϑ (κ (x1 ))aϑ (κ (S(x23 X 3 ) f 1 )) ⊗ x2 X 1 β S(x13 X 2 ) f 2 | a ∈ A}. Proof Since no confusion is possible we will write without subscripts D, A or B in the tensor components of the reassociators of D, A or B, respectively. The same thing will be done when we write their inverses. (i) This follows from the first part of the proof of Proposition 9.24. Also, one can check directly that ζ becomes a morphism in B MBB ; details are left to the reader. (ii) By definitions we have Dco(B) = {d ∈ D | ρD (d) = x1 · d · S(x23 X 3 ) f 1 ⊗ x2 X 1 β S(x13 X 2 ) f 2 } = {d ∈ D | d1 ⊗ υ (d2 ) = κ (x1 )d κ (S(x23 X 3 ) f 1 ) ⊗ υ (κ (x2 X 1 β S(x13 X 2 ) f 2 ))} = Dco(υ ) . Recall that Dco(B) = Im(E A⊗B ), where E A⊗B is the projection on the space of alternative coinvariants of A ⊗ B defined in (6.3.7). Having in mind the structure of A ⊗ B as a right quasi-Hopf B-bimodule, we have that E A⊗B (a ⊗ b)

(ϑ (κ (x1 ))aϑ (κ (X 1 )) ⊗ x2 b1 X 2 ) · β S(x3 b2 X 3 )

= (3.2.13)

=

(3.2.14)

ϑ (κ (x1 ))aϑ κ (X 1 δ 1 S(x23 b(2,2) X23 ) f 1 ) ⊗ x2 b1 X 2 δ 2 S(x13 b(2,1) X13 ) f 2

(3.2.6)

ϑ (κ (x1 ))aϑ κ (β S(x23 b(2,2) X 3 ) f 1 ) ⊗ x2 b1 X 1 β S(x13 b(2,1) X 2 ) f 2

(3.1.7)

ϑ (κ (x1 ))aϑ κ (β S(b))ϑ κ (S(x23 X 3 ) f 1 ) ⊗ x2 X 1 β S(x13 X 2 ) f 2 ,

= =

(3.2.1)

for all a ∈ A, b ∈ B. It follows that Dco(B) = {ϑ (κ (x1 ))aϑ (κ (S(x23 X 3 ) f 1 )) ⊗ x2 X 1 β S(x13 X 2 ) f 2 | a ∈ A}, since A = {aϑ κ (β S(b)) | a ∈ A, b ∈ B}. This finishes our proof. Proposition 11.35 Let D be a quasi-Hopf algebra, A and B two quasi-bialgebras and ϑ : D → A, υ : D → B two quasi-bialgebra maps. Consider ζ : D → A ⊗ B given by ζ (d) = ϑ (d1 ) ⊗ υ (d2 ), for all d ∈ D. (1) Suppose that (D, R) is QT and define F = F1 ⊗ F2 ∈ (A ⊗ B)⊗2 by 1

2

F = ϑ (Y11 x1 X 1 y11 ) ⊗ υ (Y21 x2 R X 3 y2 ) ⊗ ϑ (Y 2 x3 R X 2 y12 ) ⊗ υ (Y 3 y3 ), 1

2

(11.7.4)

where, as usual, R ⊗ R is the inverse of the R-matrix R of D. Then F is a twist on A ⊗ B (here A ⊗ B has the componentwise quasi-bialgebra structure) and ζ : D → (A ⊗ B)F is a quasi-bialgebra morphism. Moreover, if A and B are quasi-Hopf algebras and ϑ and υ are quasi-Hopf algebra morphisms, then ζ : D → (A ⊗ B)U F is a quasi-Hopf algebra morphism, where U = ϑ (R2 g2 ) ⊗ υ (R1 g1 ).

11.7 The Quantum Double of a Factorizable Quasi-Hopf Algebra

447

(2) Suppose that A and B are quasi-Hopf algebras, ϑ and υ are quasi-Hopf algebra morphisms, and that there exists a quasi-Hopf algebra map κ : B → D such that υ ◦ κ = IdB . Then ζ is a bijective map if and only if the restriction of ϑ provides a bijection from Dco(ϑ ) to A. Proof (1) We have that ζ = (ϑ ⊗ υ ) ◦ ΔD , so clearly ζ is an algebra map. It also respects the comultiplications. Indeed, by applying (3.1.3), (3.1.7) twice, (10.1.3), and then again (3.1.7) twice, it is not hard to see that, for all d ∈ D, (Δ(A⊗B)F ◦ ζ )(d) = ((ζ ⊗ ζ ) ◦ ΔD )(d). Obviously, εA⊗B ◦ ζ = εD , so ζ respects the counits. It remains to show that (ζ ⊗ ζ ⊗ ζ )(ΦD ) = Φ(A⊗B)F . This follows from a long, technical but straightforward computation; we leave the details to the reader. Suppose that A, B are quasi-Hopf algebras and that ϑ and υ are quasi-Hopf algebra morphisms. In this case, ζ : D → (A ⊗ B)U F is also a quasi-bialgebra morphism since = (A ⊗ B) as quasi-bialgebras. Thus, we are left to show that (A ⊗ B)U F F

ζ (α ) = Uα(A⊗B)F ,

ζ (β ) = β(A⊗B)F U−1 ,

(ζ ◦ SD )(d) = USA⊗B (ζ (d))U−1

for all d ∈ D. Take F−1 = G1 ⊗ G2 as being the inverse of the twist F. By (3.2.4) and (11.7.4) we compute:

α(A⊗B)F

=

SA⊗B (G1 )αA⊗B G2

=

ϑ (S(Y11 x1 X 1 y11 )α Y21 x2 R2 X 3 y2 ) ⊗ υ (S(Y 2 x3 R1 X 2 y12 )α Y 3 y3 )

(3.2.1),(10.1.2)

2

(3.2.1),(10.1.5),(10.1.3)

=

(3.2.5),(3.2.14)

1

ϑ (S(R21 X 2 R y11 )α R22 X 3 y2 ) ⊗ υ (S(R1 X 1 R y12 )α y3 )

=

2

1

ϑ (S(X 2 y12 R )α X 3 y2 ) ⊗ υ (S(X 1 y11 R )α y3 ) 2

1

=

ϑ (S(R )γ 1 ) ⊗ υ (S(R )γ 2 )

=

ϑ (S(R ) f 1 α1 ) ⊗ υ (S(R ) f 2 α2 )

(10.3.6)

=

2

1

ϑ ( f 2 R α1 ) ⊗ υ ( f 1 R α2 ) = U−1 ζ (α ), 2

1

as needed. In a similar manner one can prove that β(A⊗B)F = ζ (β )U; the details are left to the reader. Finally, for all d ∈ D we have USA⊗B (ζ (d))U−1

= (3.2.13),(10.1.3)

=

2

1

ϑ (R2 g2 S(d1 ) f 2 R ) ⊗ υ (R1 g1 S(d2 ) f 1 R ) ϑ (S(d)1 ) ⊗ υ (S(d)2 ) = ζ (S(d)).

(2) We are in the same hypothesis as in Lemma 11.34, so ζ : D → A ⊗ B is a right quasi-Hopf B-bimodule morphism. As we explained before Lemma 11.34, the morphism ζ is bijective if and only if ζ0 , the restriction of ζ , defines an isomorphism between Dco(B) and (A ⊗ B)co(B) . But Dco(B) = Dco(υ ) , so if d ∈ Dco(B) then

ζ (d) = ϑ (d1 ) ⊗ υ (d2 )

448

Factorizable Quasi-Hopf Algebras = ϑ (κ (x1 ))ϑ (d)ϑ (κ (S(x23 X 3 ) f 1 )) ⊗ x2 X 1 β S(x13 X 2 ) f 2 ,

because of υ ◦ κ = IdB . Hence, by Lemma 11.34, ζ is bijective if and only if the map

ζ0 : Dco(υ ) → {ϑ (κ (x1 ))aϑ (κ (S(x23 X 3 ) f 1 )) ⊗ x2 X 1 β S(x13 X 2 ) f 2 | a ∈ A}, ζ0 (d) = ϑ (κ (x1 ))ϑ (d)ϑ (κ (S(x23 X 3 ) f 1 )) ⊗ x2 X 1 β S(x13 X 2 ) f 2 is bijective. Now, it follows that ζ is bijective if and only if the restriction of ϑ defines a bijection between Dco(υ ) and A. We can now state the structure theorem of D(H) when H is factorizable. Theorem 11.36 Let (H, R) be a finite-dimensional QT quasi-Hopf algebra and π , π : D(H) → H the quasi-Hopf algebra morphisms given by (10.5.1) and (11.7.1). Define ζ : D(H) → H ⊗ H by ζ (D) = π (D1 ) ⊗ π (D2 ), for all D ∈ D(H), and F = Y11 x1 X 1 y11 ⊗Y21 x2 R2 X 3 y2 ⊗Y 2 x3 R1 X 2 y12 ⊗Y 3 y3 ,

(11.7.5)

where R1 ⊗ R2 is the R-matrix R of H. Then the following assertions hold: (1) ζ : D(H) → (H ⊗ H)U F is a quasi-Hopf algebra morphism, where we define 1 2 U := R g2 ⊗ R g1 . (2) ζ is bijective if and only if (H, R) is factorizable. Proof We consider in Proposition 11.35 D = D(H), A = B = H, ϑ = π , υ = π and κ = iD . So the map ζ in the statement is the map ζ in Proposition 11.35 specialized for our case. Moreover, from definition (10.4.1) of the R-matrix R of D(H) we have

π (R 1 ) ⊗ π (R 2 )

2

1

=

S−1 (p2 )i ep11 ⊗ i e, q2 R q1 R p12

=

S−1 (p2 )q2 R p11 ⊗ q1 R p12

2

1

(10.1.3),(3.2.23) −1 2 2 1 2 1 = S (p )q p2 R ⊗ q1 p11 R 1

2

1

= R ⊗R .

2

Since π and π are algebra maps we obtain that π (R ) ⊗ π (R ) = R2 ⊗ R1 , so the twist (11.7.5) is the twist F defined in (11.7.4) specialized for our situation. Also, the element U is the element U defined in Proposition 11.35 specialized for our context and this proves the first assertion. By applying again Proposition 11.35 we have that ζ is bijective if and only if the restriction of π provides a bijection from D(H)co(π ) to H. By Lemma 11.33 this is equivalent to Q bijective. Finally, by Proposition 11.26 we obtain that ζ is bijective if and only if (H, R) is factorizable, and this finishes our proof. We end this section with an application. At first sight there is no relationship between D(H(2)) described in Proposition 10.21 and H(2) ⊗ H(2), so it comes as a surprise that these two quasi-Hopf algebras are twist equivalent. To show this, we will use Theorem 11.36. Example 11.37 Let X,Y be the algebra generators of D(H(2)) defined in Proposition 10.21, and x, y the generators of H(2) ⊗ H(2) ∼ = k[C2 × C2 ], the tensor product

11.7 The Quantum Double of a Factorizable Quasi-Hopf Algebra

449

quasi-Hopf algebra. Let ω± = 1 ± i and consider the elements U± = p+ ⊗ 1 + p− ⊗ g + ω± p− ⊗ p− , F± = 1 − 2px− py− ⊗ px− py+ − 2px+ py− ⊗ px− py− − ω± px− ⊗ py− , where px± = 12 (1 ± x) and py± = 12 (1 ± y). Then the maps ζ± : D(H(2)) → (H(2) ⊗ U H(2))F±± , given by 1 ζ± (Y ) = −1 + ω± px− + ω∓ py− = − (ω± x + ω∓ y), 2

ζ± (X) = xy,

are quasi-Hopf algebra isomorphisms. Proof By Example 11.25, H(2) is a factorizable quasi-Hopf algebra. Then everything will follow from the general isomorphism presented in Theorem 11.36. For H(2) we have qR = 1 ⊗ p+ − g ⊗ p− . Also, it is easy to see that the inverse of R± = 1 − ω± p− ⊗ p− is R∓ = 1 − ω∓ p− ⊗ p− , and therefore q2 R1± ⊗ q1 R2± = p+ ⊗ 1 − p− ⊗ g − ω± p− ⊗ p− , 2

1

q2 R± ⊗ q1 R± = q2 R1∓ ⊗ q1 R2∓ = p+ ⊗ 1 − p− ⊗ g − ω∓ p− ⊗ p− . From the structure of D(H(2)) in Proposition 10.21 we see that π (X) = π˜ (X) = g,

π (Y ) = π (μ  1) = μ (p+ )1 − μ (p− )g − ω± μ (p− )p− = −g − ω± p− , and, in a similar way, π˜ (Y ) = −g − ω∓ p− . We get that π (XY ) = −1 + ω± p− and π˜ (XY ) = −1 + ω∓ p− , so ζ± (X) = g ⊗ g = xy and 1 ζ± (Y ) = − (π ⊗ π˜ ) (Y ⊗Y + XY ⊗Y +Y ⊗ XY − XY ⊗ XY ) . 2 After some straightforward computations we obtain

π (Y ) ⊗ π˜ (Y ) = xy + 2px− py− + ω± xpy− + ω∓ ypx− , π (XY ) ⊗ π˜ (Y ) = x − 2px− py− − ω± xpy− + ω∓ px− , π (Y ) ⊗ π˜ (XY ) = y − 2px− py− + ω± py− − ω∓ ypx− , π (XY ) ⊗ π˜ (XY ) = 1 + 2px− py− − ω± py− − ω∓ px− . Thus, we can compute: 1 ζ± (Y ) = − (xy + x + y − 1 − 4px− py− + 2ω± py− + 2ω∓ px− ) 2 = 1 − x − y − ω± py− − ω∓ px− = −1 + (2 − ω∓ )px− + (2 − ω ±)py− = −1 + ω± px− + ω∓ py− , and this is exactly what we need. Finally, one can easily see that the corresponding elements U± and F± for (H(2), R± ) are exactly the ones defined in the statement; we leave the details to the reader.

450

Factorizable Quasi-Hopf Algebras

Remarks 11.38

(1) Keeping the notation used in Example 11.37, we have that

xy xy (ζ± ⊗ ζ± ⊗ ζ± )(ΦX ) = Φxy := 1 − 2pxy − ⊗ p− ⊗ p− , 1 where ΦX is the reassociator of D(H(2)) and pxy − := 2 (1 − xy). Thus, Φxy is a 3cocycle for k[C2 ×C2 ] and Φxy = (Φx,y )F = (Φx Φy )F , because of (3.1.5). Here Φy = 1 − 2py− ⊗ py− ⊗ py− is the 3-cocycle on k[C2 ×C2 ] corresponding to y. In other words we have proved that the 3-cocycles Φxy and Φx Φy are equivalent. (2) It follows from Example 11.37 that k[C4 ] and k[C2 × C2 ] are isomorphic as algebras if char(k) = 2 and k contains a primitive fourth root of unity. This is well known and can be easily seen directly: k[C4 ] and k[C2 ×C2 ] are isomorphic (even as Hopf algebras) to their duals and the two duals are both isomorphic to k4 as algebras. More explicitly, we have the following: ζ+ = γ ◦ β ◦ α , where α , β , γ are the following three algebra isomorphisms ({e1 , e2 , e3 , e4 } is the standard basis of k4 ):

α : k[C4 ] = k[Y ]/(Y 4 − 1) → k4 , α (Y ) = e1 + ie2 − e3 − ie4 ; β : k4 → k[C2 ×C2 ] = k[x, y], β (e1 ) = px+ py+ , β (e2 ) = px− py+ , β (e3 ) = px+ py− , β (e4 ) = px− py− ; γ : k[C2 ×C2 ] → k[C2 ×C2 ], γ (x) = xy, γ (y) = −x, γ (xy) = −y.

11.8 Notes Factorizable (quasi-)Hopf algebras provide a special class of QT (quasi-)Hopf algebras, and in their theory an important role is played by the quantum double D(H). Factorizable Hopf algebras were introduced and studied by Reshetikhin and Semenov-Tian-Shansky [190]. They are important in Hennings’ investigation of 3manifold invariants [110]. Afterwards, Kauffman reworked Hennings’ construction; see [129] or [188] for more details. Factorizable quasi-Hopf algebras were introduced in [61] by using a categorical point of view due to Majid [148]. An alternative definition was recently introduced in [98]. The process that associates to a QT quasi-Hopf algebra a braided Hopf algebra was introduced in [54], and the dual case in [61]. The bosonisation for quasi-Hopf algebras was also taken from [54]. That D(H) is always factorizable, and therefore unimodular, was taken from [61], as well as the structure of D(H) when H itself is factorizable. Example 11.37 is from [52].

12 The Quantum Dimension and Involutory Quasi-Hopf Algebras

We compute the quantum dimension of a finite dimensional quasi-Hopf algebra H and of its quantum double D(H), within the rigid braided category of finite dimensional left D(H)modules. As we will see, this involves the semisimplicity of D(H) and leads to the notion of involutory quasi-Hopf algebra, a concept that will be studied at the end of this chapter.

12.1 The Integrals of a Quantum Double We provide explicit formulas for the integrals in the quantum double D(H) of a finite-dimensional quasi-Hopf algebra H. Our aim is to apply Theorem 7.28 in order to see when D(H) is a semisimple algebra, as this is important in computing the quantum dimension of H and D(H). In what follows, {ei }i is a basis of H and {ei }i is the corresponding dual basis of ∗ H . Ω is the element of H ⊗5 defined in (8.5.1), λ is a non-zero left cointegral on H, r is a non-zero right integral in H and μ is the modular element of H ∗ . If δ is the element defined in (3.2.6) and T := μ −1 (δ 2 )δ 1  λ ∈ H ∗ , we claim that T  r is a non-zero left and right integral in D(H). That it is non-zero follows easily from the fact that r is non-zero and T (r) = μ −1 (δ 2 )λ (rδ 1 ) = μ −1 (ε (δ 1 )δ 2 )λ (r) = μ −1 (β )λ (r) = 0,



as L × rH (λ  , r ) → λ  (r ) ∈ k is non-degenerate. The difficult part is to show that T  r is a left and right integral in D(H). To prove that it is a left integral we need the formula Y 1 δ 1 S(Y23 ) ⊗Y 2 δ 2 S(Y13 ) = β S( p˜2 ) ⊗ S( p˜1 ),

(12.1.1)

which can be deduced from the definitions of δ and pL , and (3.1.9) and (3.2.1). Proposition 12.1

With the above notation, T  r is a left integral in D(H).

We check this assertion by direct computation. If ϕ ∈ H ∗ and h ∈ H then

Proof

(ϕ  h)(T  r) =

μ −1 (δ 2 )(Ω1  ϕ  Ω5 )(Ω2 h(1,1) δ 1  λ  S−1 (h2 )Ω4 )  Ω3 h(1,2) r

452

The Quantum Dimension and Involutory Quasi-Hopf Algebras (7.2.7)

μ −1 (Ω3 h(1,2) δ 2 )(Ω1  ϕ  Ω5 )(Ω2 h(1,1) δ 1  λ  S−1 (h2 )Ω4 )  r

(7.5.11)

μ −1 (Ω3 h(1,2) δ 2 S((h2 )1 ))(Ω2 h(1,1) δ 1 S((h2 )2 )  λ  Ω4 )  Ω3 h(1,2) r

(3.2.8)

ε (h)μ −1 (Ω3 δ 2 )(Ω1  ϕ  Ω5 )(Ω2 δ 1  λ  Ω4 )  r.

= = =

Therefore it suffices to show that

μ −1 (Ω3 δ 2 )λ (Ω4 h2 Ω2 δ 1 )Ω5 h1 Ω1 = μ −1 (δ 2 )λ (hδ 1 )S−1 (α ), for all h ∈ H. To this end, we compute

μ −1 (Ω3 δ 2 )λ (Ω4 h2 Ω2 δ 1 )Ω5 h1 Ω1 (8.5.1)

=

1 μ −1 (X21 y3 x22 δ 2 )λ (S−1 ( f 1 X 2 x3 )h2 X(1,2) y2 x12 δ 1 ) 1 S−1 ( f 2 X 3 )h1 X(1,1) y1 x1

(3.1.9)

=

(12.1.1) (7.5.11)

=

(3.2.22) (3.2.19)

=

(7.3.5) (7.5.11)

=

(7.3.2) (7.55)

=

1 μ −1 (X21 y2 x13 S( p˜1 ))λ (S−1 ( f 1 X 2 y3 x23 )h2 X(1,2) y12 x2 β S( p˜2 )) 1 S−1 ( f 2 X 3 )h1 X(1,1) y11 x1 1 μ −1 (X21 y2 S( p˜1 ))λ (S−1 ( f 1 X 2 y3 )h2 X(1,2) y12 x2 β S( p˜2 x3 )) 1 S−1 ( f 2 X 3 )h1 X(1,1) y11 x1

μ −1 (X21 y2 S( p˜1 ))λ (S−1 ( f 1 X 2 y3 )(hX11 y1 S(P˜ 1 ))2U 2 S( p˜2 )) S−1 ( f 2 X 3 )(hX11 y1 S(P˜ 1 ))1U 1 P˜ 2 2 μ −1 (X21 y2 S( p˜1 ))μ (X12 y31 )λ , S−1 ( f 1 )(hX11 y1 S(X(2,1) y3(2,1) p˜21 P˜ 1 ))2U 2  2 2 y3(2,1) p˜21 P˜ 1 ))1U 1 X(2,2) y3(2,2) p˜22 P˜ 2 S−1 ( f 2 X 3 )(hX11 y1 S(X(2,1) 2 μ −1 (X21 y2 S( p˜1 ))μ (X12 y31 )μ (q11 z1 )λ , hX11 y1 S(q12 z2 X(2,1) y3(2,1) p˜21 P˜ 1 ) 2 y3(2,2) p˜22 P˜ 2 S−1 (X 3 )q2 z3 X(2,2)

(3.1.7)

=

2 2 μ −1 (X21 y2 )μ (q11 X(1,1) y3(1,1) z1 p˜1 )λ , hX11 y1 S(q12 X(1,2) y3(1,2) z2 p˜21 P˜ 1 

S−1 (X 3 )q2 X22 y32 z3 p˜22 P˜ 2 (10.3.3)

=

(5.5.17)

μ ((q12 )1 (x2 y31Y 2 )1 Z 2 S−1 (q1(1,2) x21 y2Y 1 Z 1 β )) λ , hq1(1,1) x11 y1 S((q12 )2 (x2 y31Y 2 )2 Z 3 P˜ 1 )q2 x3 y32Y 3 P˜ 2

(3.1.9),(3.1.7)

=

(3.2.1)

μ ((q12 z3 )1 Z 2 S−1 (q1(1,2) z2 Z 1 β )) λ , hq1(1,1) z1 y1 S((q12 z3 )2 Z 3 y2 P˜ 1 )q2 y3 P˜ 2

(3.1.7)

=

μ −1 (z2 (q12 )1 Z 1 β S(z31 (q12 )(2,1) Z 2 )) λ , hz1 q11 y1 S(z32 (q12 )(2,2) Z 3 y2 P˜ 1 )q2 y3 P˜ 2

(3.1.7),(3.2.1)

μ −1 (z2 Z 1 β S(z31 Z 2 ))λ , hz1 q11 y1 S(z32 Z 3 q12 y2 P˜ 1 )q2 y3 P˜ 2

(3.2.20),(3.2.1)

μ −1 (z2 Z 1 β S(z31 Z 2 ))λ , hz1 β S(z32 Z 3 )S−1 (α )

= =

(3.2.6)

=

μ −1 (δ 2 )λ (hδ 1 )S−1 (α ),

for all h ∈ H, and this finishes the proof.

12.1 The Integrals of a Quantum Double

453

Corollary 12.2 The quantum double D(H) is a semisimple algebra if and only if H is semisimple and admits a normalized left cointegral, that is, a left cointegral λ satisfying λ (S−1 (α )β ) = 0. Proof This is an immediate consequence of Theorem 7.28. Note that, for the nonzero left integral T = μ −1 (δ 2 )δ 1  λ  r in D(H), we have

εD (T) = ε (r)μ −1 (δ 2 )λ (S−1 (α )δ 1 ), and so εD (T) = 0 if and only if ε (r) = 0 and μ −1 (δ 2 )λ (S−1 (α )δ 1 ) = 0. But ε (r) = 0 implies H semisimple, and therefore unimodular, in which case μ −1 (δ 2 )δ 1 = β . Examples 12.3 (1) D(H(2)) is semisimple because H(2) is semisimple and the left cointegral Pg on H(2) found in Example 7.53 satisfies Pg (S−1 (g)) = Pg (g) = 1. (2) D(H± (8)) is not semisimple as H± (8) is not semisimple by Examples 7.31 (2). From Theorem 11.31 we know that T = T  r is a right integral in D(H), too. We will present a direct proof now, and for this we need first some technical results. For Hopf algebras we have g−1 = S(g). Indeed, in this case L is an ideal of ∗ H and since dimk L = 1 it follows that for any h∗ ∈ H ∗ there is a ch∗ ∈ k such that λ h∗ = ch∗ λ . Evaluating in t we obtain ch∗ = h∗ (S−1 (g−1 )), and so λ (h1 )h2 = λ (h)S−1 (g−1 ), for all h ∈ H. From here we get that Δ(S−1 (g−1 )) = S−1 (g−1 ) ⊗ S−1 (g−1 ), that is, S−1 (g−1 ) is a grouplike element of H. In particular this implies that S−1 (g−1 )−1 = S(S−1 (g−1 )), which is clearly equivalent to g−1 = S−1 (g), and so to g−1 = S(g), too. As a consequence, λ (h1 )h2 = λ (h)g, for all h ∈ H. For quasi-Hopf algebras the above argument does not apply. Nevertheless, in this case we can prove the following. Proposition 12.4 bra H then

If λ is a left cointegral on a finite-dimensional quasi-Hopf alge-

λ (S−1 ( f 2 )h1 g1 S(h ))S−1 ( f 1 )h2 g2 = μ ( f 1 )μ −1 (U22 U2 α )μ (β ) μ (U 1 y12 x2 )λ (hS(y3 x23 h2 p˜2 )) S−1 (g−1 y11 x1 )S(S(U12 U1 y2 x13 h1 p˜1 ) f 2 ) (12.1.2) for all h, h ∈ H, where U = U 1 ⊗U 2 = U1 ⊗ U2 is the element defined in (7.3.1) and μ is the modular element of H ∗ . Proof

The Nakayama isomorphism ξcop for H cop is given by

ξcop : H ∗ → H,

ξcop (h∗ ) = h∗ (S−1 (q˜1t1 p˜1 ))q˜2t2 p˜2 ,

−1 (h) = h  Λ ◦ S, where and is an isomorphism of left H-modules with inverse ξcop Λ is a right cointegral on H satisfying Λ(S(t)) = 1; see Remark 7.60. −1 (h). If q∗ := h∗ ◦ S−1 then Now take h ∈ H and h∗ = ξcop (7.6.4)

h = q∗ (q˜1t1 p˜1 )q˜2t2 p˜2 = q∗ (q1t1 p1 )q2t2 p2 S−1 (u), where u is the element introduced in Proposition 7.61. Set pR = p1 ⊗ p2 = P1 ⊗ P2

454

The Quantum Dimension and Involutory Quasi-Hopf Algebras

and qR = q1 ⊗ q2 = Q1 ⊗ Q2 and compute Δ(hS−1 (u−1 )) q∗ (q1t1 p1 )q21t(2,1) p21 ⊗ q22t(2,2) p22

= (3.2.26)

=

(3.2.25),(7.2.6)

=

q∗ (q1 Q11 x1t1 p1 )S−1 (g2 )q2 Q12 x2t(2,1) p21 ⊗ S−1 (g1 )Q2 x3t(2,2) p22

μ (X 1 )q∗ (q1 (Q1t1 P1 )1 p1 )S−1 (g2 )q2 (Q1t1 P1 )2 p2 S(X 3 ) f 1 ⊗ S−1 (g1 )Q2t2 P2 S(X 2 ) f 2 .

This equality is equivalent to (S−1 ( f 2 ) ⊗ S−1 ( f 1 ))Δ(hS−1 (u−1 ))(g1 ⊗ g2 ) = μ (X 1 )q∗ (q1 (Q1t1 P1 )1 p1 )q2 (Q1t1 P1 )2 p2 S(X 3 ) ⊗ Q2t2 P2 S(X 2 ). By applying λ ⊗ IdH to this formula, we find

λ (S−1 ( f 2 )(hS−1 (u−1 ))1 g1 S(h ))S−1 ( f 1 )(hS−1 (u−1 ))2 g2 = (7.5.15)

=

μ (X 1 )q∗ (q1 (Q1t1 P1 )1 p1 )λ (q2 (Q1t1 P1 )2 p2 S(h X 3 ))Q2t2 P2 S(X 2 ) μ (x1 X 1 )q∗ (q˜2 x3 h2 X23 p˜2 )λ (S−1 (q˜1 )Q1t1 P1 S(x2 h1 X13 p˜1 ))Q2t2 P2 S(X 2 ).

−1 (h) = h  Λ◦S, hence q∗ = (h  Λ◦S)◦S−1 = Λ  S(h). By We know that h∗ = ξcop Proposition 7.61 we have that Λ = (λ ◦S−1 )  u−1 , hence q∗ = (λ ◦S−1 )  u−1 S(h), and this implies that

λ (S−1 ( f 2 )(hS−1 (u−1 ))1 g1 S(h ))S−1 ( f 1 )(hS−1 (u−1 ))2 g2 = μ (x1 X 1 ) λ (S−1 (q˜2 x3 h2 X23 p˜2 )hS−1 (u−1 ))λ (S−1 (q˜1 )Q1t1 p1 S(x2 h1 X13 p˜1 ))Q2t2 p2 S(X 2 ). Since u is invertible it follows that

λ (S−1 ( f 2 )h1 g1 S(h ))S−1 ( f 1 )h2 g2 = μ (x1 X 1 )λ (S−1 (q˜2 x3 h2 X23 p˜2 )h) λ (S−1 (q˜1 )Q1t1 p1 S(x2 h1 X13 p˜1 ))Q2t2 p2 S(X 2 ) (7.5.16),(3.2.13)

=

3 μ (x1 X 1 g1 S(x22 h(1,2) X(1,2) p˜12 ) f 1 )λ (S−1 (q˜2 x3 h2 X23 p˜2 )h) 3 λ (S−1 (q˜1 )Q1t1 p1 )Q2t2 p2 S(X 2 g2 S(x12 h(1,1) X(1,1) p˜11 ) f 2 )

(3.2.27),(7.5.11)

=

(3.1.7),(7.5.16) (7.6.2),(3.2.28)

=

(6.5.1) (3.2.13),(3.2.28)

=

μ (q˜11 x1 )μ (S((q˜12 x2 )2 y2 (h2 p˜2 )1 P˜ 1 ) f 1 )λ (S−1 (q˜2 x3 y3 (h2 p˜2 )2 P˜ 2 )h) λ (Q1t1 p1 )Q2t2 p2 S(S((q˜12 x2 )1 y1 h1 p˜1 ) f 2 ) λ (S−1 (q˜2 y3 (x23 h2 p˜2 )2 P˜ 2 )h)μ (S(G22 S(q1 x11 )2 q˜12 y2 (x23 h2 p˜2 )1 P˜ 1 ) f 1 ) μ (G1 S(q2 x21 )x2 )S−1 (g−1 )S(S(G21 S(q1 x11 )1 q˜11 y1 x13 h1 p˜1 ) f 2 ) 1 μ (G1 S(q2 x21 )x2 )μ (S(G22 g2 S(X 1 q11 x(1,1) )q˜1 (Q˜ 2 X23 x23 h2 p˜2 )1 P˜ 1 ) f 1 )

λ (S−1 (q˜2 (Q˜ 2 X23 x23 h2 p˜2 )2 P˜ 2 )h)S−1 (g−1 ) 1 S(S(G21 g1 S(X 2 q12 x(1,2) )Q˜ 1 X13 x13 h1 p˜1 ) f 2 ) (6.5.1),(7.6.7) −1 = μ (α )μ (β )μ (G1 S(q2 (y11 x1 )2 )y12 x2 )μ (S(G22 g2 S(Q1 q11 (y11 x1 )(1,1) )) f 1 ) (3.2.21) λ (hS(y3 x23 h2 p˜2 ))S−1 (g−1 )S(S(G21 g1 S(Q2 q12 (y11 x1 )(1,2) )y2 x13 h1 p˜1 ) f 2 )

12.1 The Integrals of a Quantum Double (3.2.26),(3.2.17)

=

(3.1.7),(3.1.9) (3.2.13),(3.1.7)

=

(7.3.1) (7.6.13)

=

455

μ −1 (α )μ (β )μ (Y 1 g11 G1 S((q2 y12 )2 )F 1 y21 x1 )μ (S(Y 3 g2 S(q1 y11 )) f 1 ) λ (hS(y3 x3 h2 p˜2 ))S−1 (g−1 )S(S(Y 2 g12 G2 S((q2 y12 )1 )F 2 y22 x2 h1 p˜1 ) f 2 ) μ −1 (α )μ (β )μ ( f 1 )μ (S(y1 )1Y 1U11 y21 x1 )μ −1 (S(y1 )(2,2)Y 3U 2 ) λ (hS(y3 x3 h2 p˜2 ))S−1 (g−1 )S(S(S(y1 )(2,1)Y 2U21 y22 x2 h1 p˜1 ) f 2 ) μ (β f 1 )μ −1 (Y 3U 2 α )μ (Y 1U11 y21 x1 )λ (hS(y3 x3 h2 p˜2 )) S−1 (g−1 y1 )S(S(Y 2U21 y22 x2 h1 p˜1 ) f 2 ).

Thus we have shown that

λ (S−1 ( f 2 )h1 g1 S(h ))S−1 ( f 1 )h2 g2 = μ (β f 1 )μ −1 (Y 3U 2 α ) μ (Y 1U11 y21 x1 )λ (hS(y3 x3 h2 p˜2 ))S−1 (g−1 y1 )S(S(Y 2U21 y22 x2 h1 p˜1 ) f 2 ), (12.1.3) for all h, h ∈ H. Finally, substituting (7.5.1) in (12.1.3) and applying (7.6.13), (3.1.9), we easily obtain (12.1.2), as desired. To prove that T  r is a right integral in D(H) we need the following formulas. Lemma 12.5 Let H be a finite-dimensional quasi-Hopf algebra, h ∈ H and r ∈ Then the following relations hold:

H r

.

X11 x1 δ 1 S(X23 ) ⊗ X21 x2 δ12 S(X13 )1 ⊗ X 2 x3 δ22 S(X13 )2 = (β S(X 3 ))1 g1 S(x3 ) ⊗ (β S(X 3 ))2 g2 S(x2 ) f 1 ⊗ x2 X 1 β S(x1 X 2 ) f 2 , (12.1.4) f 2V 1 S−1 ( f 1 )1 ⊗V 2 S−1 ( f 1 )2 = qL ,

(12.1.5)

S(p )F 2 f22 X 3 ⊗ S(p2 f 1 X 1 )F 1 f12 X 2 = 1H ⊗ α , V 1 r1 ⊗ g−1V 2 r2 = V 2 r2 p2 ⊗ S2 (V 1 r1 p1 )α ,

(12.1.6)

1

V r1 ⊗ S 1

−1

S( p˜2 ) f 1 r

(h)V r2 = μ (h1 )h2V r1 ⊗V r2 , 2

−1

1 ⊗g

1

S( p˜1 ) f 2 r

2

Proof

1

1

(12.1.8)

2

= μ (S(p ) f )S(p ) f V r2 P2 ⊗ S2 (V 1 r1 P1 )α . 2

(12.1.7)

2 2

(12.1.9)

We have

X11 x1 δ 1 S(X23 ) ⊗ X21 x2 δ12 S(X13 )1 ⊗ X 2 x3 δ22 S(X13 )2 (3.2.14),(3.2.13)

=

(3.1.7)

3 (X 1 β1 )1 x1 g1 S(X23 ) ⊗ (X 1 β1 )2 x2 g21 G1 S(X(1,2) )f1 3 ⊗ X 2 β2 x3 g22 G2 S(X(1,1) )f2

(3.2.17),(3.1.7)

=

3 3 (X 1 β1 g1 )1 G1 S(x3 X(2,2) ) ⊗ (X 1 β1 g1 )2 G2 S(x2 X(2,1) )f1

⊗ X 2 β2 g2 S(x2 X13 ) f 2 (3.2.14),(3.2.13)

=

(X 1 δ 1 S(X23 ))1 G1 S(x3 ) ⊗ (X 1 δ 1 S(X23 ))2 G2 S(x2 ) f 1 ⊗ X 2 δ 2 S(x1 X13 ) f 2

(3.2.6)

=

(β S(X 3 ))1 g1 S(x3 ) ⊗ (β S(X 3 ))2 g2 S(x2 ) f 1 ⊗ x2 X 1 β S(x1 X 2 ) f 2 ,

and this proves (12.1.4). The equalities (12.1.5) and (12.1.6) follow from the defini-

456

The Quantum Dimension and Involutory Quasi-Hopf Algebras

tions of V and pR , and from (3.2.17), (7.5.12). (12.1.8) can be proved with the help of (7.3.3). The verification of all these details is left to the reader. In order to show (12.1.7), notice that (7.6.8) and (7.6.1) imply that V 1 r1U 1 ⊗ g−1V 2 r2U 2 = V 2 r2U 2 ⊗ S2 (V 1 r1U 1 ). As we have already observed, (7.6.10) guarantees that Δ(r)U = Δ(r)pR , and so V 1 r1 ⊗ g−1V 2 r2

(3.2.23)

= V 1 r1 q11 p1 ⊗ g−1V 2 r2 q12 p2 S(q2 )

= V 1 r1 p1 ⊗ g−1V 2 r2 p2 α = V 2 r2 p2 ⊗ S2 (V 1 r1 p1 )α , as required. Finally, we have S( p˜2 ) f 1 r1 ⊗ g−1 S( p˜1 ) f 2 r2 (7.3.8)

μ −1 (g1 )q1 r1 ⊗ S−1 (g2 )q2 r2

(7.3.4)

μ −1 (g1 )μ (q˜1 )q˜2V 1 r1 ⊗ S−1 (g2 )V 2 r2

(12.1.8)

μ (S(g1 )q˜1 g21 )q˜2 g22V 1 r1 ⊗ g−1V 2 r2

= = =

(12.1.7),(7.6.12)

=

μ (S(p2 ) f 1 )S(p1 ) f 2V 2 r2 P2 ⊗ S2 (V 1 r1 P1 )α ,

proving (12.1.9). This makes the proof complete. We can now (re)prove that a quantum double quasi-Hopf algebra is unimodular. 

Theorem 12.6 If H is a finite-dimensional quasi-Hopf algebra, 0 = r ∈ rH and 0 = λ ∈ L then T = μ −1 (δ 2 )δ 1  λ  r is a non-zero right integral in D(H). Consequently, D(H) is a unimodular quasi-Hopf algebra. Proof

For ϕ ∈ H ∗ and h ∈ H, we compute that

T(ϕ  h) (8.5.2),(8.5.1)

=

(7.5.11),(7.2.7) (12.1.4),(3.1.7)

=

(3.2.13),(3.2.17)

(y1 X11 x1 δ 1 S(X23 )  λ  S−1 ( f 2 ))(y2 (X21 x2 δ12 S(X13 )1 )1 r(1,1)  ϕ  S−1 ( f 1 X 2 x3 δ22 S(X13 )2 ))  y3 (X21 x2 δ12 S(X13 )1 )2 r(1,2) h (β S(X 3 ))(1,1) g11 G1 S(x3 y3 )  λ  S−1 (F2 ))((β S(X 3 ))(1,2) g12 G2 S(x22 y2 )F 1 f11 r(1,1)  ϕ  S−1 (F1 X 1 β S(x1 X 2 ) f 2 r2 ))  (β S(X 3 ))2 g2 S(x12 y1 )F 2 f21 r(1,2) h

(3.2.13)

=

(3.2.14)

(12.1.2)

=

(3.1.9)

=

(3.2.22)

((δ 1 S(X23 ))1 G1 S(x3 y3 )  λ  S−1 (F2 ))((δ 1 S(X23 ))2 G2 S(x22 y2 )F 1 f11 r(1,1)  ϕ  S−1 (F1 X 1 β S(x1 X 2 ) f 2 r2 ))  δ 2 S(x12 y1 X13 )F 2 f21 r(1,2) h

ϕ S−1 (g−1 z11t 1 X 1 β S(x1 X 2 ) f 2 r2 )S(x22 y2 S(U12 U1 z2t13 x13 y31 p˜1 )F2 )  F 1 f11 r(1,1) μ (β F1 )μ −1 (U22 U2 α )μ (U 1 z12t 2 )

δ 1 S(z3t23 x23 y32 p˜2 X23 )  λ  δ 2 S(x12 y1 X13 )F 2 f21 r(1,2) h

ϕ S−1 (g−1 z11t 1 X 1 β S(Y 1 y11 x1 X 2 ) f 2 r2 )S(Y 3 y2 S(U12 U1 z2t13 y31 p˜1 )F2 )  F 1 f11 r(1,1) μ (β F1 )μ −1 (U22 U2 α )μ (U 1 z12t 2 )

12.2 The Cointegrals of a Quantum Double (11.5.7),(3.1.7)

=

(7.2.7),(3.1.9)

(7.2.7),(3.2.13)

=

(3.1.7)

(3.2.21),(3.1.9)

=

(3.1.7),(5.5.16)

=

(3.2.21)

(3.1.9),(3.2.20)

=

(3.2.13)

δ 1 S(z3t23 y32 p˜2 x3 X23 )  λ  δ 2 S(Y 2 y12 x2 X13 )F 2 f21 r(1,2) h

ϕ S−1 (g−1 z1t 1 X 1 β S(Y 1 p11 x1 X 2 ) f 2 r2 )S(Y 3 p2 S(U12 U1 z3t22 )F2 )  F 1 f11 r(1,1) μ (β F1 )μ −1 (U22 U2 α )μ (U 1 z2t12 ) δ 1 S(t 3 x3 X23 )  λ  δ 2 S(Y 2 p12 x2 X13 )F 2 f21 r(1,2) h  2 ϕ S−1 (g−1 z1t 1 X 1 β S(Y 1 (z21t(1,1) p1 )1 x1 X 2 ) f 2 r2 )μ (β F1 )μ (U 1 )  2 p2 S(U12 U1 z3t22 )F2 )F 1 f11 r(1,1) μ −1 (U22 U2 α ) S(Y 3 z22t(1,2) 2 δ 1 S(t 3 x3 X23 )  λ  δ 2 S(Y 2 (z21t(1,1) p1 )2 x2 X13 )F 2 f21 r(1,2) h

ϕ S−1 (g−1 z1 X 1 β S(Y 1 (z21 p1 )1 X 2 ) f 2 r2 )S(Y 3 z22 p2 S(U12 U1 z3 )F2 )  F 1 f11 r(1,1) μ (β F1 )μ −1 (U22 U2 α )μ (U 1 ) δ 1  λ

 δ 2 S(Y 2 (z21 p1 )2 X 3 )F 2 f21 r(1,2) h

ϕ S−1 (g−1 z1 X 1 β S(z21t 1 X 2 ) f 2 r2 )S((z22t 2 X13 )2 p2 S(U12 U1 z3t 3 X23 )F2 )  F 1 f11 r(1,1) μ (β F1 )μ −1 (U22 U2 α )μ (U 1 ) δ 1  λ  δ 2 S((z22t 2 X13 )1 p1 )F 2 f21 r(1,2) h

 ϕ S−1 (g−1 S( p˜1 ) f 2 r2 )S(p2 S(U12 U1 )F2 )F 1 (S( p˜2 ) f 1 r1 )1 μ (β F1 )

μ −1 (U22 U2 α )μ (U 1 ) δ 1  λ  δ 2 S(p1 )F 2 (S( p˜2 ) f 1 r1 )2 h  (12.1.9),(3.2.13) −1 = ϕ S (α )S(p2 S(U1 )F2 S(U 2 )2V 1 r1 P1 )F 1 (S(P1 ) f 2V 2 r2 P2 )1 μ (S(P2 ) f 1 )μ (β F1 )μ −1 (U 2 α )μ (S(U 2 )1 ) δ 1  λ (7.2.7),(12.1.8)

=

(7.3.1),(3.2.19) (7.2.7),(12.1.5)

=

(7.2.8) (7.2.7),(10.4.6)

=

(3.2.25),(3.1.7) (12.1.6)

=

(3.2.1),(3.2.2)

457

 δ 2 S(p1 )F 2 (S(P1 ) f 2V 2 r2 P2 )2 h

 μ −1 (β )μ (α )ϕ S−1 (α )S(p2 S(U1 )F2V 1 r1 P1 )F 1 (V 2 r2 P2 )1

μ (β F1 )μ −1 (U2 α ) δ 1  λ  δ 2 S(p1 )F 2 (V 2 r2 P2 )2 h

 μ −1 (U2 )ϕ S−1 (α )S(p2 S(U1 )q˜1 r1 P1 )F 1 (q˜2 r2 P2 )2 δ 1  λ  δ 2 S(p1 )F 2 (q˜2 r2 P2 )2 h

 ϕ S−1 (α )S(p2 f 1 X 1 (r1 p1 )1 P1 )F 1 f12 X 2 (r1 p1 )2 P2 δ 1  λ  δ 2 S(p1 )F 2 f22 X 3 r2 p2 h

 ϕ S−1 (α )S((r1 p1 )1 P1 )α (r1 p1 )2 P2 δ 1  λ  δ 2 r2 p2 h

=

ϕ (S−1 (α )) δ 1  λ  δ 2 rβ h

=

ϕ (S−1 (α ))ε (h)μ −1 (δ 2 )  λ  r = εD (ϕ  h)T  r,

as required. This finishes the proof.

12.2 The Cointegrals of a Quantum Double In what follows, we will identify D(H)∗ ∼ = H ⊗ H ∗.

458

The Quantum Dimension and Involutory Quasi-Hopf Algebras

Proposition 12.7

Take non-zero elements λ ∈ L and r ∈

H r

. Then

Γ = r  μ ( p˜1 )S( p˜2 )  λ  μ −1 ( f 1 )S−1 ( f 2 ) ∈ D(H)∗

(12.2.1)

is a non-zero left cointegral on D(H). Proof It is clear that Γ = 0. Since H is a quasi-Hopf subalgebra of D(H) via iD it follows that the elements U and V for D(H) are UD = ε  U 1 ⊗ ε  U 2 and VD = ε  V 1 ⊗ ε  V 2 . Identifying D(H)∗ ∼ = H ⊗ H ∗ , we compute Γ((ε  V 2 )(X12  ϕ1  S−1 (X 3 )  X22Y 3 x3 h2 )(ε  U 2 )) (ε  V 1 )(ε  X 1Y 1 )(p11 x1  ϕ2  Y 2 S−1 (p2 )  p12 x2 h1 )(ε  U 1 ) (8.5.2)

=

2 2 ϕ1 (S−1 (V22 X 3 )rV(1,1) X12 )λ (S−1 ( f 2 )V(1,2) X22Y 3 x3 h2U 2 S( p˜2 ))μ ( p˜1 )

μ −1 ( f 1 )(ε  V 1 X 1Y 1 )(p11 x1  ϕ2  Y 2 S−1 (p2 )  p12 x2 h1U 1 ) ϕ (r)μ (S−1 ( f 1 )V22 X 3 )μ ( p˜1 )λ (S−1 ( f 2 )V12 X 2 h2U 2 S( p˜2 ))

=

ε  V 1 X 1 h1U 1 (7.3.1),(3.2.13)

=

ϕ (r)μ −1 (S(X 3 )F 1 f11 p11 )μ ( p˜1 )λ (S−1 (S(X 2 )F 2 f21 p12 )h2U 2 S( p˜2 )) ε  S−1 (S(X 1 ) f 2 p2 )h1U 1

(3.2.17),(5.5.16)

=

ϕ (r)μ −1 ( f 1 x1 )μ ( p˜1 )λ (S−1 (F 1 f12 x12 p1 )h2U 2 S( p˜2 )) ε  S−1 (F 2 f22 x22 p2 S(x3 ))h1U 1

(3.2.13),(7.3.1)

=

(7.3.2)

ϕ (r)μ −1 ( f 1 x1 )μ ( p˜1 )λ (V 2 (S−1 ( f 2 x2 )hS( p˜21 ))2U 2 ) ε  x3V 1 (S−1 ( f 2 x2 )hS( p˜21 ))1U 1 p˜22

(7.5.13),(7.5.11)

ϕ (r)μ −1 ( f 1 x1 )μ (x12 y1 p˜1 )λ (S−1 ( f 2 )hS(x22 y2 p˜21 ))ε  x3 y3 p˜22

(3.2.20),(3.1.9)

ϕ (r)μ ( p˜1 )μ −1 ( f 1 )λ (S−1 ( f 2 )hS( p˜2 ))ε  1 = Γ(ϕ  h)ε  1H .

= =

As D(H) is unimodular, it follows that Γ is a left cointegral on D(H). Corollary 12.8 D(H) admits a normalized left cointegral if and only if D(H) is a semisimple algebra. Proof

For the non-zero left cointegral Γ defined in (12.2.1) we have −1 Γ(SD (ε  α )(ε  β )) = Γ(ε  S−1 (α )β )

= ε (r)μ ( p˜1 )μ −1 ( f 1 )λ (S−1 (α f 2 )β p˜2 ). This is non-zero if and only if ε (r) = 0 and μ ( p˜1 )μ −1 ( f 1 )λ (S−1 (α f 2 )β p˜2 ) = 0. But, as we have already mentioned, ε (r) = 0 implies H unimodular, and in this case

μ ( p˜1 )μ −1 ( f 1 )λ (S−1 (α f 2 )β p˜2 ) = λ (S−1 (α )β ). Then the result follows from Corollary 12.2. Now we describe the space of right cointegrals on D(H). Proposition 12.9 If t ∈ right cointegral on D(H).

H l

and λ ∈ L are non-zero then t  λ ◦ S is a non-zero

12.2 The Cointegrals of a Quantum Double

459

Proof Since D(H) is unimodular, Γ ◦ SD is a non-zero right cointegral on D(H); see Corollary 7.69. So it suffices to show that Γ ◦ SD = S(r)  λ ◦ S. By applying μ to both sides of (12.1.2) we obtain after a straightforward computation that

μ (S−1 ( f 1 )h2 g2 )λ (S−1 ( f 2 )h1 g1 S(h )) = μ −1 (α g−1 )μ (q˜1 )μ (q˜21 h1 p˜1 )λ (hS(q˜22 h2 p˜2 )), for all h, h ∈ H. Consequently, by (3.2.24) we obtain that

μ −1 (q˜1 )μ (S−1 ( f 1 )S(h)2 g2 )λ (S−1 ( f 1 )S(h)1 g1 S(h q˜2 )) = μ −1 (α g−1 )μ (q˜1 )μ (q˜21 h1 )λ (S(q˜22 h2 h)),

(12.2.2)

for all h, h ∈ H. Then we compute: Γ ◦ SD (ϕ  h) =

Γ((ε  S(h) f 1 )(p11U 1  ϕ ◦ S−1  f 2 S−1 (p2 )  p12U 2 ))

=

1 ϕ ◦ S−1 ( f 2 S−1 (S(h)2 f21 p2 )rS(h)(1,1) f(1,1) p11U 1 ) 1 μ ( p˜1 )μ −1 (F 1 )λ (S−1 (F 2 )S(h)(1,2) f(1,2) p12U 2 S( p˜2 ))

(7.6.12)

ϕ (S−1 (r))μ (S−1 (F 1 )S(h)2 g2 S(q˜1 ))μ ( p˜1 )λ (S−1 (F 2 )S(h)1 g1 S( p˜2 q˜2 ))

(12.2.2)

μ −1 (α g−1 )ϕ (S−1 (r))μ ((q˜2 p˜2 )1 )μ (q˜1 p˜1 )λ (S((q˜2 p˜2 )2 h))

(3.2.20)

μ −1 (g−1 )ϕ (S−1 (r))μ −1 (α )μ (α )μ −1 (β )λ (S(h))

(7.2.8)

ϕ ((μ −1 (g)μ (β ))−1 S−1 (r))λ (S(h)) = (S(r)  λ ◦ S)(ϕ  h),

= = = =

for all ϕ ∈ H ∗ and h ∈ H, where in the last equality we used Corollary 7.73. The modular element of D(H)∗ is μD = εD . Our next aim is to compute the modular element gD of D(H). To this end, we will need an explicit formula for the inverse of the antipode SD of D(H) and a lemma. −1 of the antipode of D(H) is given Proposition 12.10 The composition inverse SD ∗ by the following formula, for all ϕ ∈ H and h ∈ H: −1 (ϕ  h) = (ε  S−1 ( f 2 h))(p11 S−1 (q2 g2 )  ϕ ◦ S  S−1 (p2 f 1 )  p12 S−1 (q1 g1 )). SD

Proof

We first observe that (3.2.21) and (3.2.23) imply that (ε  q1 h1 )(p11  ϕ  q2 h2 S−1 (p2 )  p12 ) = h1  ϕ  h2 ,

for all ϕ ∈ H ∗ and h ∈ H. Consequently, (ε  q1 S(P1 )1 )(p11U 1 P2 f 1  ϕ  q2 S(P1 )2 S−1 (p2 )  p12U 2 )(ε  f 2 ) = (7.3.1),(3.2.13)

=

(S(P1 )1U 1 P2 f 1  ϕ  S(P1 )2U 2 )(ε  f 2 ) (3.2.23)

(g1 S(q2 P21 )P2 f 1  ϕ  g2 S(q1 p11 ))(ε  f 2 ) = ϕ  1H ,

for all ϕ ∈ H ∗ . By the definition of SD we have SD (S−1 (g2 )  ϕ  S−1 (g1 )) = p11U 1  ϕ ◦ S−1  S−1 (p2 )  p12U 2 ,

460

The Quantum Dimension and Involutory Quasi-Hopf Algebras

and combining these two relations we find for all ϕ ∈ H ∗ that ϕ  1H equals (ε  q1 S(P1 )1 )SD (S−1 (g2 )  (P2 f 1  ϕ  q2 S(P1 )2 ) ◦ S  S−1 (g1 ))(ε  f 2 ). −1 As H is a quasi-Hopf subalgebra of D(H), SD (ε  h) = ε  S−1 (h), for all h ∈ H. −1 This and the fact that SD is an anti-algebra morphism imply −1 (ϕ  h) SD

−1 −1 SD (ε  h)SD (ϕ  1H )

−1 2 (ε  S ( f h)) S−1 (q2 S(P1 )2 g2 )  ϕ ◦ S

= =

 S−1 (P2 f 1 )  S−1 (q1 S(P1 )1 g1 ) (3.2.13)

=



(ε  S−1 ( f 2 h))(P11 S−1 (q2 g2 )  ϕ ◦ S  S−1 (P2 f 1 )  P21 S−1 (q1 g1 )),

for all ϕ ∈ H ∗ and h ∈ H. Lemma 12.11 Let H be a finite-dimensional quasi-Hopf algebra. Then for all r ∈ H r and h ∈ H the following equalities hold: hr1 ⊗ r2 = μ −1 (h1 p1 )q1 r1 ⊗ S−1 (h2 p2 )q2 r2 ,

(12.2.3)

r1U ⊗ r2U S(h) = r1U h ⊗ r2U .

(12.2.4)

1

Proof

2

1

2

(12.2.3) follows since (3.2.23)

hr1 ⊗ r2 = hq1 p11 r1 ⊗ S−1 (p2 )q2 p12 r2 (3.2.21) 1

=

q (h1 p1 r)1 ⊗ S−1 (h2 p2 )q2 (h1 p1 r)2

=

μ −1 (h1 p1 )q1 r1 ⊗ S−1 (h2 p2 )q2 r2 .

(12.2.4) is a direct consequence of (7.3.2). In order to compute the modular element gD of D(H) we need a left integral T −1 (T)) = 1. Since μD = εD in D(H) and a left cointegral Γ on D(H) such that Γ(SD it turns out that this is equivalent to Γ(T) = 1. Also note that the unimodularity of −1 . D(H) implies that Γ ◦ SD = Γ ◦ SD  −1 2  Now take T = μ (δ )  λ  r for some 0 = λ  ∈ L and 0 = r ∈ rH , and let Γ be defined as in (12.2.1). A simple inspection ensures that −1 Γ ◦ SD (T) = μ (δ 1 )μ −1 (δ 2 )λ  (S(r))λ (S(r ))

and since S(δ 1 )αδ 2 = S(β ) and ε (g) = μ (β ), by Remark 7.60 we conclude that −1 (T) = μ −1 (α )−1 λ  (S(r))λ (r ). Thus we have to consider λ , λ  and r, r such Γ ◦ SD  that λ (S(r))λ (r ) = μ −1 (α ). Proposition 12.12

The modular element gD of D(H) is given by

−1 (μ  g12 S−2 (g−1 )) gD = μ (g11 )μ −1 (g2 )SD

= μ (q˜1 g1 )μ −1 ( p˜1 )(ε  S−3 (g−1 ))(μ −1  (S−1 (q˜2 g2 )  μ −1 ) p˜2 ).

12.2 The Cointegrals of a Quantum Double

461

H

Proof Let λ , λ  ∈ L and r, r ∈ r be such that λ  (S(r))λ (r ) = μ −1 (α ). By the above comments, (7.6.2) and (8.5.9), the modular element gD can be computed as: gD

−1 μ −1 (δ 2 )Γ ◦ SD ((q21 X 2 )1  (δ 1  λ  )1  S−1 (X 3 )  (q21 X 2 )2Y 3 x3 r2 P2 )

=

−1 ((ε  q1 X11Y 1 )(p11 x1  (δ 1  λ  )2  Y 2 S−1 (p2 )  p12 x2 r1 P1 )) SD

μ (q21 )λ (S(q22Y 3 x3 r2 P2 ))λ  (S(r))μ (Y 2 S−1 (p2 ))μ (p11 x1 )

=

−1 μ −1 (δ 2 )μ (δ 1 )SD ((ε  q1Y 1 )(μ  p12 x2 r1 P1 )) (12.2.3)

=

μ −1 (p1(2,1) x12 P1 )λ  (S(r))μ (q21 )λ (S(q22Y 3 x3 S−1 (p1(2,2) x22 P2 )Q2 r2 P2 )) −1 μ −1 (α )−1 μ (β )μ (Y 2 S−1 (p2 ))μ (p11 x1 )SD ((ε  q1Y 1 )(μ  Q1 r1 P1 ))

(5.5.16)

=

μ −1 (X 2 S−1 (X 1 β ))μ −1 (α )−1 λ  (S(r))μ (Y 2 S−1 (p2 ))μ (q21 )

(3.1.7)

−1 λ (S(q22Y 3 S−1 (X 3 p1 β )Q2 r2 P2 ))SD ((ε  q1Y 1 )(μ  Q1 r1 P1 ))

(7.2.7)

μ −1 (α )−1 λ  (S(r))μ (q21 )μ (Y 2 S−1 (p2 ))λ (S(q22Y 3 S−1 (p1 β )V 2 r2 U 2 ))

=

(7.6.10)

−1 ((ε  q1Y 1 )(μ  V 1 r1 U 1 )) SD

(12.1.8)

μ −1 (q12Y21 p2 S(Y 2 ))λ  (S(r))μ (q21 )λ (S(q22Y 3 S−1 (q11Y11 p1 β )V 2 r2 U 2 ))

=

−1 μ −1 (α )−1 SD (μ  V 1 r1 U 1 ) (3.2.6)

μ −1 (q12 δ 2 S(q21 ))λ  (S(r))λ ◦ S(S−1 (q11 δ 1 S(q22 ))V 2 r2 U 2 )

=

−1 μ −1 (α )−1 SD (μ  V 1 r1 U 1 ) (3.2.14)

=

(3.2.13)

μ −1 ((q1 β S(q2 ))2 g2 )λ  (S(r))λ ◦ S(S−1 ((q1 β S(q2 ))1 g1 )V 2 r2 U 2 ) −1 μ −1 (α )−1 SD (μ  V 1 r1 U 1 )

(12.1.8)

−1 μ (g11 )μ −1 (g2 )μ −1 (α )−1 λ  (S(r))λ ◦ S(V 2 r2 U 2 )SD (μ  g12V 1 r1 U 1 )

(7.5.13)

−1 μ (g11 )μ −1 (g2 )SD (μ  g12 S−2 (g−1 )).

= =

−1 = Γ ◦ SD , In the second equality we used Proposition 12.9 and the fact that Γ ◦ SD H in the third one we used the properties S(r) ∈ l and μ is an algebra map, and in the last equality Remark 7.60 and (12.2.4). We have also denoted by P1 ⊗ P2 another copy of pR . This proves the first formula for gD . For the second one we use the form −1 found above to compute of SD

gD

=

μ (g11 )μ −1 (g2 )(ε  S−3 (g−1 )S−1 ( f 2 g12 )) (p11 S−1 (q2 G2 )  μ −1  S−1 (p2 f 1 )  p12 S−1 (q1 G1 ))

(8.5.2)

=

μ −1 ((S−1 ( f 2 g12 )1 p1 )1 S−1 (q2 G2 ))μ (S−1 ( f 2 g12 )2 p2 f 1 )μ (g11 ) μ −1 (g2 )(ε  S−3 (g−1 ))(μ −1  (S−1 (( f 2 g12 )1 p1 )2 S−1 (q1 G1 ))

(3.2.13),(3.2.17)

=

μ −1 (S−1 ( f 2 x3 g1(2,2) G2 )1 S−1 (q2 G2 ))μ −1 (g2 ) μ (S−1 (F 2 f21 x2 g1(2,1) G1 )β F 1 f11 x1 g11 ) (ε  S−3 (g−1 ))(μ −1  S−1 ( f 2 x3 g1(2,2) G2 )2 S−1 (q1 G1 ))

(7.5.12),(3.1.7)

=

(3.2.1)

μ −1 (S−1 (α x2 G1 )x1 )μ −1 (S−1 (g1 x3 G2 )1 S−1 (q2 G2 ))μ −1 (g2 ) (ε  S−3 (g−1 ))(μ −1  S−1 (g1 x3 G2 )2 S−1 (q1 G1 ))

462

The Quantum Dimension and Involutory Quasi-Hopf Algebras (3.2.20),(3.2.13)

=

μ (q˜1 G1 )μ (S( p˜1 ) f 2 q˜22 G22 G2 )

(7.3.8)

(ε  S−3 (g−1 ))(μ −1  S−1 (S( p˜2 ) f 1 q˜21 G21 G1 ))

(3.2.13)

μ (q˜1 G1 )μ −1 ( p˜1 )(ε  S−3 (g−1 ))(μ −1  (S−1 (q˜2 G2 )  μ −1 ) p˜2 ),

=

and this completes the proof.

12.3 The Quantum Dimension Let C be a braided category which is left rigid. If V is an object of C , and evV and coevV are the evaluation and coevaluation morphisms associated to V, we define the quantum dimension (or representation-theoretic rank) of V as follows: dim(V ) = evV ◦ cV,V ∗ ◦ coevV . If H is a quasi-Hopf algebra then the category H M fd of finite-dimensional modules over H is left rigid. Therefore, if H is a QT quasi-Hopf algebra and V a finitedimensional left H-module it makes sense to consider the representation-theoretic rank of V . If R = R1 ⊗ R2 is an R-matrix for H then n

dim(V ) = ∑ vi (S(R2 )α R1 β · vi ) = Tr(η ),

(12.3.1)

i=1

where η := S(R2 )α R1 β . Here Tr(η ) is the trace of the linear endomorphism of V defined by v → η · v, and {vi , vi }i are dual bases in V and V ∗ . Let u be the element defined in (10.3.4). By (10.3.8) we have that S(R2 )α R1 = S(α )u, so by (10.3.7) we obtain

η = S(S(β )α )u = uS−1 (α )β .

(12.3.2)

The aim of this section is to compute the quantum dimension of H and D(H) within the braided rigid monoidal category D(H) M fd . If {ei , ei }i are dual bases in H and H ∗ , by (10.4.2) and the fact that H is a quasi-Hopf subalgebra of D(H) we get that ηD , the corresponding element η for D(H), is given by n

−1

ηD = ∑ β  S (ei )  ei S−1 (α )β . i=1

12.3.1 The Quantum Dimension of H We first compute the quantum dimension (or representation-theoretic rank) of H within the braided rigid category D(H) M fd . Recall that H is a left D(H)-module via the action → defined in (8.7.11). It can be rewritten as follows: (ϕ  h) → h

(3.1.9)

=

(3.2.1)

ϕ , S−1 (Y 3 )q2Y22 y32 S−1 (q˜1 y21 (h  h )1 g1 )y1 

Y 1 q˜2 y22 (h  h )2 g2 S(q1Y12 y31 )

 (3.2.13) = ϕ , S−1 q˜1 (y2 (h  h )S(Y 2 y3 ))1U 1Y 3 y1  (7.3.1)

12.3 The Quantum Dimension

463

Y 1 q˜2 (y2 (h  h )S(Y 2 y3 ))2U 2

 (3.2.22) = ϕ , S−1 q˜1 (Y21 y2 (h  h )S(Y 2 y3 ))1U 1Y 3 Y11 y1  q˜2 (Y21 y2 (h  h )S(Y 2 y3 ))2U 2 . Hence we have shown that for all ϕ ∈ H ∗ and h, h ∈ H we have

 (ϕ  h) → h = ϕ , S−1 q˜1 (Y21 y2 (h  h )S(Y 2 y3 ))1U 1Y 3 Y11 y1  q˜2 (Y21 y2 (h  h )S(Y 2 y3 ))2U 2 .

(12.3.3)

So this action defines on H a left D(H)-module structure, and on H0 a left D(H)module algebra structure; see Proposition 8.42. For the computation of dim(H) we need the following formulas. Lemma 12.13 Let H be a finite-dimensional quasi-Hopf algebra and {ei }i a basis in H with dual basis {ei }i . Then for all h, h , h ∈ H the following relations hold: n

∑ ei , S−1 (β )S−2 (Q˜ 1 (ei )1 h )hq˜2 Q˜ 22 (ei )(2,2) h S−1 (q˜1 Q˜ 21 (ei )(2,1) )

i=1

n

= ∑ ei , S−1 (β )S−2 (Q˜ 1 (ei )1 h )q˜2 Q˜ 22 (ei )(2,2) h S−1 (q˜1 Q˜ 21 (ei )(2,1) )h, (12.3.4) i=1

n

∑ ei , S−1 (β )S−2 (Q˜ 1 (ei )1 X 1 p11 h )h1 q˜2 Q˜ 22 (ei )(2,2) X 3 p2 S(h2 )h

i=1

n

S−1 (q˜1 Q˜ 21 (ei )(2,1) X 2 p12 ) = ∑ ei , S−1 (β )S−2 (Q˜ 1 (ei )1 X 1 p11 h1 h ) i=1 2 2 ˜ q˜ Q2 (ei )(2,2) X 3 p2 h S−1 (q˜1 Q˜ 21 (ei )(2,1) X 2 p12 h2 ),

(12.3.5)

where we denoted qL = q˜1 ⊗ q˜2 = Q˜ 1 ⊗ Q˜ 2 and pR = p1 ⊗ p2 . Proof In order to prove (12.3.4) we will apply (3.2.22) twice, and then the properties of dual bases and (3.2.1). Explicitly, n

∑ ei , S−1 (β )S−2 (Q˜ 1 (ei )1 h )hq˜2 Q˜ 22 (ei )(2,2) h S−1 (q˜1 Q˜ 21 (ei )(2,1) )

i=1

n

= ∑ ei , S−1 (β )S−2 (Q˜ 1 (ei )1 h )q˜2 (h2 Q˜ 2 )2 (ei )(2,2) h i=1

S−1 (q˜1 (h2 Q˜ 2 )1 (ei )(2,1) )h1  n

= ∑ ei , S−1 (β )S−2 (S(h(2,1) )Q˜ 1 (h(2,2) ei )1 h )q˜2 Q˜ 22 (h(2,2) ei )(2,2) h i=1

S−1 (q˜1 Q˜ 21 (h(2,2) ei )(2,1) )h1  n

= ∑ ei , h(2,2) S−1 (h(2,1) β )S−2 (Q˜ 1 (ei )1 h )q˜2 Q˜ 22 (ei )(2,2) h i=1

S−1 (q˜1 Q˜ 21 (ei )(2,1) )h1 

464

The Quantum Dimension and Involutory Quasi-Hopf Algebras n

= ∑ ei , S−1 (β )S−2 (Q˜ 1 (ei )1 h )q˜2 Q˜ 22 (ei )(2,2) h S−1 (q˜1 Q˜ 21 (ei )(2,1) )h. i=1

In a similar manner one can prove (12.3.5). It follows by applying (12.3.4), dual bases, (3.1.7) and (3.2.21); we leave the details to the reader. We are now able to compute dim(H). Proposition 12.14 Let H be a finite-dimensional quasi-Hopf algebra. Then the quantum dimension of H within D(H) M fd is

 dim(H) = Tr h → S−2 (S(β )α hβ S(α )) . (12.3.6) Proof We set pR = p1 ⊗ p2 = P1 ⊗ P2 , qL = q˜1 ⊗ q˜2 = Q˜ 1 ⊗ Q˜ 2 and f = f 1 ⊗ f 2 = F 1 ⊗ F 2 . Then by (12.3.1) and the above expression of ηD we have: dim(H) n

= (12.3.3)

=

∑ e j ,

i, j=1 n

 ei  S−1 (α ei β )β → e j 



∑ ei , S−1

1 1 2 −1  q˜ (Y2 y (S (α ei β )β  e j )S(Y 2 y3 ))1U 1Y 3 Y11 y1 

i, j=1

e j , q˜2 (Y21 y2 (S−1 (α ei β )β  e j )S(Y 2 y3 ))2U 2  n

=

∑

ek ,Y21 y2 (S−1 (α ei β )β  e j )S(Y 2 y3 )

i, j,k=1

=

Y11 y1  ei , S−1 (q˜1 (ek )1U 1Y 3 )e j , q˜2 (ek )2U 2  n

 ∑ ek ,Y21 y2 S−1 (α eiY11 y1 β )β  q˜2 (ek )2U 2 S(Y 2 y3 )

i,k=1

ei , S−1 (q˜1 (ek )1U 1Y 3 ) (8.7.1),(3.2.13)

=

(3.2.14)

n

1 y12 δ 2 ) ∑ ek ,Y21 y2 S−1 ( f 2Y(1,2)



−1 S (α ei )β  q˜2 (ek )2U 2

i,k=1

1 y11 δ 1 S(Y 2 y3 )ei , S−1 (q˜1 (ek )1U 1Y 3 ) f 1Y(1,1) (3.2.6),(3.1.9)

=

(3.2.19)

n

1 p2 ) ∑ ek ,Y21 S−1 ( f 2Y(1,2)



S−1 (α ei )β  q˜2 (ek )2U 2



i,k=1

1 p1 β S(Y 2 )ei , S−1 (q˜1 (ek )1U 1Y 3 ) f 1Y(1,1) (3.2.21),(3.2.20)

=

n

∑ S( p˜1 )  ek , S−1 ( f 2 p2 )



−1 S (α ei )β  q˜2 (ek )2U 2 f 1 p1 

i,k=1

=

ei , S−1 (q˜1 (ek )1U 1 p˜2 ) n 

∑ ek , S−1 ( f 2 p2 ) S−1 (α ei )β  q˜2 (ek )2 S( p˜1 )2U 2 f 1 p1 

i,k=1

ei , S−1 (q˜1 (ek )1 S( p˜1 )1U 1 p˜2 )

12.3 The Quantum Dimension (7.3.5)

=

(8.7.1),(3.2.13)

=

(3.2.19)

=

n

∑ ek , S−1 ( f 2 p2 )

(3.2.20),(3.2.13)

=

(3.2.14)



−1 S (α S−1 (q˜1 (ek )1 P1 ))β  q˜2 (ek )2 P2 f 1 p1 

k=1 n

∑ ek  x3 , S−1 ( f 2 S−1 (F 1 q˜11 (ek )(1,1) p11 g1 )x2 β )

k=1

−1  S (α )β  q˜2 (ek )2 p2 f 1 S−1 (F 2 q˜12 (ek )(1,2) p12 g2 )x1  n

3 (ek )(1,1) p11 g1 )x2 β ) S−1 (α )β ∑ ek , S−1 ( f 2 S−1 (F 1 q˜11 x(1,1)

k=1

 3 q˜2 x23 (ek )2 p2 f 1 S−1 (F 2 q˜12 x(1,2) (ek )(1,2) p12 g2 )x1 

n

∑ ek , S−1 (γ 2 S−1 (Q˜ 1 X 1 (ek )(1,1) p11 g1 )β )

k=1

β1 q˜2 Q˜ 22 X 3 (ek )2 p2 S(β2 )γ 1 S−1 (q˜1 Q˜ 21 X 2 (ek )(1,2) p12 g2 ) (3.1.7),(12.3.4)

=

n

∑ ek , S−1 (γ 2 S−1 (Q˜ 1 (ek )1 X 1 p11 g1 )β )q˜2 Q˜ 22

k=1

(ek )(2,2) X 3 p2 S(β2 )γ 1 S−1 (q˜1 Q˜ 21 (ek )(2,1) X 2 p12 g2 )β1  (12.3.5),(3.2.14)

=

n

∑ ek , S−1 (γ 2 S−1 (Q˜ 1 (ek )1 X 1 p11 δ 1 )β )

k=1

q˜2 Q˜ 22 (ek )(2,2) X 3 p2 γ 1 S−1 (q˜1 Q˜ 21 (ek )(2,1) X 2 p12 δ 2 ) (3.2.5),(3.1.5)

=

n

∑ ek , S−1 (β )S−2 (Q˜ 1 (ek )1 X 1 p11Y11 x1 β S(S(Z 1 )α y3 Z23Y 3 ))

k=1

q˜2 Q˜ 22 (ek )(2,2) X 3 p2 S−1 (q˜1 Q˜ 21 (ek )(2,1) X 2 p12Y21 x2 β S(S(y1 Z 2 )α y2 Z13Y 2 x3 ) (3.2.21),(3.1.7)

=

n

∑ ek , S−1 (β )S−2 (Q˜ 1 (ek )1 X 1 p11 x1 β S(S(Z 1 )α y3 Z23Y 3 ))

k=1

q˜2 Q˜ 22 (ek )(2,2) X 3 p2 S−1 (q˜1 Q˜ 21 (ek )(2,1) X 2 p12 x2 β S(S(y1 Z 2Y21 )α y2 Z13Y 2 x3 )Y11  (12.3.4),(3.1.9)

=

(3.2.1)

n

∑ ek , S−1 (β )S−2 (Q˜ 1 (ek )1 X 1 p11 x1 β S(α ))Y 1 q˜2 Q˜ 22 (ek )(2,2)

k=1

X 3 p2 S−1 (q˜1 Q˜ 21 (ek )(2,1) X 2 p12 x2 β S(S(Y 2 )α Y 3 x3 )) (3.2.19),(12.3.4)

=

n

∑ q1  ek , S−1 (β )S−2 (Q˜ 1 (ek )1 X 1 p11 P1 β S(α ))q˜2 Q˜ 22

k=1

(ek )(2,2) X 3 p2 S(q2 )S−1 (q˜1 Q˜ 21 (ek )(2,1) X 2 p12 P2 ) (3.1.7)

=

465

n

∑ ek , S−1 (β )S−2 (Q˜ 1 (ek )1 X 1 (q11 p1 )1 P1 β S(α ))q˜2 Q˜ 22

k=1

(ek )(2,2) X 3 q12 p2 S(q2 )S−1 (q˜1 Q˜ 21 (ek )(2,1) X 2 (q11 p1 )2 P2 )

466

The Quantum Dimension and Involutory Quasi-Hopf Algebras (3.2.23),(3.2.19)

=

n

∑ ek , S−1 (β )S−2 (Q˜ 1 (ek )1 β S(α ))q˜2

k=1

Q˜ 22 (ek )(2,2) S−1 (q˜1 Q˜ 21 (ek )(2,1) β ) (3.2.1),(3.2.20)

=

(3.2.20),(3.2.2)

=

n

∑ ek , S−1 (β )S−2 (α ek β S(α ))q˜2 S−1 (q˜1 β )

k=1 n

∑ ek , S−2 (S(β )α ek β S(α ))

k=1



Tr h → S−2 (S(β )α hβ S(α )) ,

=

so the proof is finished. Further on in this chapter we shall see that the quantum dimension of H is closely connected with what will be called the trace formula for quasi-Hopf algebras.

12.3.2 The Quantum Dimension of D(H) We show that dim(D(H)) = dim(H) within D(H) M fd . Lemma 12.15 In a quasi-Hopf algebra H the following relations hold: Ω11 δ 1 S2 (Ω4 ) ⊗ Ω1(2,1) δ12 g1 S(Ω3 ) ⊗ Ω1(2,2) δ22 g2 S(Ω2 ) ⊗ Ω5 = X 1 p11 P1 S( f 1 p˜1 ) ⊗ X 2 p12 P2 ⊗ X 3 p2 ⊗ S−1 ( f 2 p˜2 ),

γ X ⊗ 1

1

f 1 γ12 X 2 ⊗ f 2 γ22 X 3

= S(X

(12.3.7)

) f 1 γ11 ⊗ S(X 2 ) f 2 γ21 ⊗ S(X 1 )γ 2 .

Proof

⊗ · · · ⊗ Ω5 ,

⊗ δ 2,

γ1

⊗ γ 2,

(12.3.8)

f −1

Here Ω = δ= γ= f = ⊗ = g1 ⊗ g2 , qR = q1 ⊗ q2 , pR = p1 ⊗ p2 = P1 ⊗ P2 and qL = q˜1 ⊗ q˜2 are the elements defined in (8.5.1), (3.2.5), (3.2.15), (3.2.16), (3.2.19) and (3.2.20), respectively. Ω1

δ1

3

f1

f 2,

Using the definition of δ and Ω we compute:

Ω11 δ 1 S2 (Ω4 ) ⊗ Ω1(2,1) δ12 g1 S(Ω3 ) ⊗ Ω1(2,2) δ22 g2 S(Ω2 ) ⊗ Ω5 (3.2.13) 1 1 = X(1,1) y1 p1 β S( f 1 X 2 ) ⊗ X(1,1) y1 p2 g1 S(X21 y3 ) 1 1 (2,1) (2,1) 1 (3.2.19),(3.1.7)

=

1 1 y1 p2 g2 S(X(1,2) y2 ) ⊗ S−1 ( f 2 X 3 ) ⊗ X(1,1) (2,2) (2,2) 2



 Y 1 (X11 )(1,1) p1 1 P1 β S( f 1 X 2 ) ⊗Y 2 (X11 )(1,1) p1 2 P2 S(X21 )

⊗Y 3 (X11 )(1,2) p2 S((X11 )2 ) ⊗ S−1 ( f 2 X 3 ) (3.2.21),(3.2.20)

=

Y 1 p11 P1 S( f 1 p˜1 ) ⊗Y 2 p12 P2 ⊗Y 3 p2 ⊗ S−1 ( f 2 p˜2 ),

so (12.3.7) is proved. The relation in (12.3.8) follows more easily since

γ 1 X 1 ⊗ f 1 γ12 X 2 ⊗ f 2 γ22 X 3 (3.2.14)

=

(3.1.7),(3.2.17)

=

(3.2.14)

=

F 1 α1 X 1 ⊗ f 1 F12 α(2,1) X 2 ⊗ f 2 F22 α(2,2) X 3 S(X 3 ) f 1 F11 α(1,1) ⊗ S(X 2 ) f 2 F21 α(1,2) ⊗ S(X 1 )F 2 α2 S(X 3 ) f 1 γ11 ⊗ S(X 2 ) f 2 γ21 ⊗ S(X 1 )γ 2 ,

12.3 The Quantum Dimension

467

where we denoted by F 1 ⊗ F 2 another copy of f . In (7.2.1) we have constructed a projection onto the space of left integrals of a finite-dimensional quasi-Hopf algebra H. Replacing the quasi-Hopf algebra H by H cop we obtain a second projection onto the space of left integrals, denoted in what ˜ Since in H cop we have (qR )cop = q˜2 ⊗ q˜1 we obtain follows by P. n

 H

i=1

l

˜ P(h) = ∑ ei , S−1 (β )S−2 (q˜1 (ei )1 )hq˜2 (ei )2 ∈

, ∀ h ∈ H.

(12.3.9)

We can now compute the representation-theoretic rank of D(H). Proposition 12.16 Let H be a finite-dimensional quasi-Hopf algebra and D(H) its quantum double. Then

 dim(D(H)) = dim(H) = Tr h → S−2 (S(β )α hβ S(α )) . Proof We set pR = p1 ⊗ p2 = P1 ⊗ P2 , qR = q1 ⊗ q2 = Q1 ⊗ Q2 and f = f 1 ⊗ f 2 = F 1 ⊗ F 2 = F 1 ⊗ F 2 . The expression of ηD allows to compute: dim(D(H)) n

=

∑ ei  e j , ηD (ei  e j )

i, j=1 n

= (8.5.2)

=

∑

ei  e j , (β  S

∑

S

i, j,k=1 n

−1

−1

(ek )  S(β )α  ek )(ei  e j )

(ek ), S(β )α Ω5 (ei )1 Ω1 β 

i, j,k=1

(3.2.13),(3.2.14)

=

n

∑

ei , S−1 ((ek )2 )Ω4 (ei )2 Ω2 (ek )(1,1) e j , Ω3 (ek )(1,2) e j    e j , Ω3 S−1 S(β1 )γ 2 Ω52 (ei )(1,2) Ω12 δ 2 e j  2

i, j=1

ei , S−2 (S(β2 )γ 1 Ω51 (ei )(1,1) Ω11 δ 1 )Ω4 (ei )2 Ω2   S−1 S(β1 )γ 2 Ω52 (ei )(1,2) Ω12 δ 2  (3.2.13),(12.3.7)

=

n

∑ e , S i

−2

1



S(β2

)γ 1 S−1 (F 2 p˜2 )

1 ((ei )1 )1Y

1 1 1 p1 P S(F 1 p˜1 )



i, j=1

(3.1.7),(3.2.21)

=

n

∑

(ei )2

 S−1 f 2 S(β1 )2 γ22 S−1 (F 2 p˜2 )(2,2) ((ei )1 )(2,2)Y 3 p2 

 e j , S−1 f 1 S(β1 )1 γ12 S−1 (F 2 p˜2 )(2,1) ((ei )1 )(2,1)Y 2 p12 P2 e j 

 ei , S−2 S(β2 )γ 1Y 1 (S−1 (F 2 p˜2 )1 p1 )1 (ei )1 P1 S(F 1 p˜1 )

i, j=1

 S−1 f 2 S(β1 )2 γ22Y 3 S−1 (F 2 p˜2 )2 p2 

 e j , S−1 f 1 S(β1 )1 γ12Y 2 (S−1 (F 2 p˜2 )1 p1 )2 (ei )2 P2 e j 

468

The Quantum Dimension and Involutory Quasi-Hopf Algebras

(3.2.13),(3.2.19)

=

(3.2.17)

(7.5.12),(3.2.1)

=

(3.2.20)

n

∑ ei , S−2

i, j=1



S(β2 )γ 1Y 1 S−1 (F 2 x3 p˜22 g2 )1 (ei )1 P1 S(F 1 F11 x1 p˜1 )



 S−1 f 2 S(β1 )2 γ22Y 3 S−1 (F 2 F21 x2 p˜21 g1 )β 

 e j , S−1 f 1 S(β1 )1 γ12Y 2 S−1 (F 2 x3 p˜22 g2 )2 (ei )2 P2 e j 

n

∑ ei , S−2 (S(β2 )γ 1Y 1 S−1 (q˜2 p˜22 g2 )1 (ei )1 P1 S( p˜1 )q˜1 p˜21 g1

i, j=1

S( f 2 S(β1 )2 γ22Y 3 ))e j , S−1 ( f 1 S(β1 )1 γ12Y 2 (3.2.24),(3.2.13)

=

n

∑

S−1 (q˜2 p˜22 g2 )2 (ei )2 P2 )e j 

 ei , S−2 S(β2 g2 )γ 1Y 1 (ei )1 P1 g1 S( f 2 γ22Y 3 ) β(1,1) 

i, j=1

(12.3.8)

=

n

∑

e j , S−1 ( f 1 γ12Y 2 (ei )2 P2 )β(1,2) e j 

 ei , S−2 S(Y 3 β2 g2 ) f 1 γ11 (ei )1 (β1 )(1,1) P1 g1 S(S(Y 1 )γ 2 ) 

i, j=1

(3.2.21),(3.2.14)

=

n

∑

e j , (β1 )2 S−1 (S(Y 2 ) f 2 γ21 (ei )2 (β1 )(1,2) P2 )e j 

 ei , γ 1 S−2 S(Y 3 δ 2 ) f 1 (ei )1 P1 δ 1 S(S(Y 1 )γ 2 ) 

i, j=1

e j , S−1 (S(Y 2 ) f 2 (ei )2 P2 )e j  (3.2.5),(3.2.6)

=

n

∑ S

−2

(ei ), S(X 1 x11Y 1 )α x3 y32 Z 3 S−1 ( f 1 P1 y1 β )(ei )2Y 3 y2 Z 1 β

i, j=1

S(S(X 2 )α X 3 x2 y31 Z 2 )S2 (x21 )e j , S−1 ( f 2 P2 )(ei )1Y 2 e j  (3.2.19)

=

n

∑ S

−2

1 (ei ), S(q1 x11Y 1 )α x3 y32 Z 3 S−1 ( f 1 P1 y1 β )(ei )2 x(2,2) Y3

i, j=1

1 y2 Z 1 β S(S(q2 )x2 y31 Z 2 )e j , S−1 ( f 2 P2 )(ei )1 x(2,1) Y 2e j  (3.1.7),(3.2.13)

=

n

∑ S

−2

(ei ), S(q1Y 1 )α x3 y32 Z 3 S−1 ( f 1 (x11 )(1,1) P1 y1 β )(ei )2Y 3 x21

i, j=1

y2 Z 1 β S(S(q2 )x2 y31 Z 2 ) e j , (x11 )2 S−1 ( f 2 (x11 )(1,2) P2 )(ei )1Y 2 e j  (3.2.21),(3.1.9)

=

(3.2.1)

n

∑ S

−2

(ei ), S(q1Y 1 )α S−1 ( f 1 P1 p1 β )(ei )2Y 3 p2 S2 (q2 )

i, j=1

e j , S−1 ( f 2 P2 )(ei )1Y 2 e j  (3.2.26)

=

n

∑ S

−2

1 (ei ), S(q1 Q11 x(1,1) )α S−1 ( f 1 P1 p1 β )(ei )2 S−1 (x2 g1 )

i, j=1

1 e j Q2 x21 p2 e j , S−1 ( f 2 P2 )(ei )1 S−1 (x3 g2 )q2 Q12 x(1,2)

12.4 The Trace Formula for Quasi-Hopf Algebras (3.2.13)

=

n

∑ S

−1

469

1 (ei ), S(q1 Q11 x(1,1) )α S−1 (x2 (ei )1 P1 p1 β )Q2 x21 p2 

i, j=1

1 e j e j , S−1 (x3 (ei )2 P2 )q2 Q12 x(1,2) n

∑ S

=

i, j=1

(3.2.21),(3.2.23)

=

(6.5.1)

=

(3.2.19),(12.3.9)

=

−1

 (ei ), α S−1 x2 (ei )1 q11 (Q1 x11 )(1,1) P1 p1 β Q2 x21 p2 

 e j , q2 (Q1 x11 )2 S−1 x3 (ei )2 q12 (Q1 x11 )(1,2) P2 e j 

n

∑ S

−1

(ei ), α S−1 (x2 (ei )1 Q1 x11 p1 β )Q2 x21 p2 e j , S−1 (x3 (ei )2 )e j 

∑ S

−1

(ei ), α S−1 (q˜1 (ei )1 X 1 p1 β )X 2 p2 S(X 3 )e j , S−1 (q˜2 (ei )2 )e j 

i, j=1 n i, j=1 n

∑ e j , S−1

 ˜ −2 (β S(α ))) e j  P(S

j=1

 ˜ −2 (β S(α ))) ε P(S

(∗)

=

(12.3.9)

=

n

∑ ei , S−2 (S(β )α ei β S(α ))

i=1

 Tr h → S−2 (S(β )α hβ S(α )) ,

  ˜ −2 (β S(α ))) ∈ H . where in (*) we used the fact that S−1 P(S r =

12.4 The Trace Formula for Quasi-Hopf Algebras We will show that the quantum dimension dim(H) = dim(D(H)) computed in the previous section is zero unless D(H) is a semisimple algebra. To this end, we need a trace formula for quasi-Hopf algebras. As before, by Tr(χ ) we denote the trace of an endomorphism χ of a finite-dimensional k-vector space V . This means Tr(χ ) = ∑ni=1 vi (χ (vi )), where {vi , vi }i are dual bases in V and V ∗ . Recall also that, by the proof of Theorem 7.48, we have

ν : L ⊗ H λ ⊗ h → λ · h = S(h)  λ ∈ H ∗ , an isomorphism of right quasi-Hopf H-bimodules. The structure of H ∗ in H MHH is the one in Proposition 7.46, while L ⊗ H is a right quasi-Hopf H-bimodules via the structure given, for all λ ∈ L and h, h , h ∈ H, by h · (λ ⊗ h) · h = μ (h1 )λ ⊗ h2 hh ,

(12.4.1)

λ ⊗ h → μ (x )λ ⊗ x h1 ⊗ x h2 .

(12.4.2)

1

2

3

Theorem 12.17 Let H be a finite-dimensional quasi-Hopf algebra, μ the modular element of H ∗ , λ a non-zero left cointegral on H and r a right integral in H such that λ (S(r)) = 1. Then:

470

The Quantum Dimension and Involutory Quasi-Hopf Algebras

(i) For any linear map χ : H → H we have that

 Tr(χ ) = μ (q11 x1 )λ χ (q2 x3 r2 p2 )S(q12 x2 r1 p1 ) ; 

(ii) Tr h → β S(α )S2 (h)S(β )α = ε (r)λ (S−1 (α )β ). In particular, H is semisimple and admits a normalized left cointegral if and only if 

Tr h → β S(α )S2 (h)S(β )α = 0. Proof For any linear map χ : H → H we denote by χ ∗ : H ∗ → H ∗ the transpose of χ . We also denote by ϑ : H ∗ ⊗H → End(H ∗ ) the linear map defined for all ϕ , ψ ∈ H ∗ and h ∈ H by ϑ (ϕ ⊗ h)(ψ ) = ψ (h)ϕ . Then one can see that

ϑ (ϕ ⊗ h) ◦ χ ∗ = ϑ (ϕ ⊗ χ (h)),

(12.4.3)

Tr(ϑ (ϕ ⊗ h)) = ϕ (h),

(12.4.4)

for all ϕ ∈ H ∗ , h ∈ H and χ ∈ End(H). (i) The fact that ν is right H-colinear shows, by using (12.4.2) and (7.5.5), that

ϕ (V 1 h1U 1 )λ (V 2 h2U 2 S(h )) = μ (x1 )ϕ (x3 h2 )λ (hS(x2 h1 )), for all ϕ ∈ H ∗ and h, h ∈ H. If we write the above equation for h = r and use the  fact that S(r) ∈ l such that λ (S(r)) = 1, we obtain

ϕ (S−1 (β )hα ) = μ (x1 )ϕ (x3 r2 )λ (hS(x2 r1 )), for all ϕ ∈ H ∗ and h ∈ H. In particular, we have that p2  ϕ  q2 , S−1 (β )S−1 (q1 )hS(p1 )α  = μ (x1 )p2  ϕ  q2 , x3 r2 λ (S−1 (q1 )hS(p1 )S(x2 r1 )), and this comes out explicitly as ϕ (h) = μ (q11 x1 )ϕ (q2 x3 r2 p2 )λ (hS(q12 x2 r1 p1 )), for all ϕ ∈ H ∗ and h ∈ H, where we used (7.5.10). In other words we obtained

ϑ (λ q12 x2 r1 p1 ⊗ μ (q11 x1 )q2 x3 r2 p2 ) = IdH ∗ ,

(12.4.5)

where for h∗ ∈ H ∗ and h ∈ H we denote h∗ h = S(h)  h∗ . Now, by using (12.4.3), (12.4.4) and the fact that Tr(χ ) = Tr(χ ∗ ) we conclude that Tr(χ ) = Tr(χ ∗ ) = Tr(IdH ∗ ◦ χ ∗ ) 

= Tr η (λ q12 x2 r1 p1 ⊗ μ (q11 x1 )q2 x3 r2 p2 ) ◦ χ ∗

 = Tr η (λ q12 x2 r1 p1 ⊗ μ (q11 x1 )χ (q2 x3 r2 p2 )

 = μ (q11 x1 )λ χ (q2 x3 r2 p2 )S(q12 x2 r1 p1 ) . (ii) Combining (3.2.23) and (3.2.21) we obtain r1 ⊗ r2 = r1 p1 ⊗ r2 p2 α = r1 p1 S−1 (α ) ⊗ r2 p2 . Now, by part (i) we have 

Tr h → β S(α )S2 (h)S(β )α

(12.4.6)

12.4 The Trace Formula for Quasi-Hopf Algebras

 = μ (q11 x1 )λ β S(α )S2 (q2 x3 r2 p2 )S(β )α S(q12 x2 r1 p1 )

 (12.4.6) = μ (q11 x1 )λ β S(α )S(q12 x2 r1 β S(q2 x3 r2 ))

 (3.2.1),(3.2.19) = ε (r)μ (q11 p1 )λ β S(α )S(q12 p2 S(q2 )) (3.2.23)

=

471

ε (r)λ (β S(α )).

Next, we claim that ε (r)λ (β S(α )) = ε (r)λ (S−1 (α )β ). Indeed, if H is not semisimple then by Theorem 7.28 we have ε (r) = 0, and therefore

ε (r)λ (β S(α )) = ε (r)λ (S−1 (α )β ) = 0. 

On the other hand, if H is semisimple then by the same theorem we have that ε ( l ) =  ε ( r ) = 0. In this situation H is unimodular, so μ = ε . Finally, by (7.5.10) we get

λ (S−1 (α )β ) = μ (α1 )λ (β S(α2 )) = ε (α1 )λ (β S(α2 )) = λ (β S(α )), as claimed. Thus the proof is finished. As a consequence of Proposition 12.16 and Theorem 12.17 we obtain the following formula for the representation-theoretic ranks of H and D(H). Theorem 12.18 Let H be a finite-dimensional quasi-Hopf algebra, λ a left cointegral on H and r a right integral in H such that λ (r) = 1. Then dim(H) = dim(D(H)) = ε (r)λ (S−1 (α )β ) = εD (β  λ  r). In particular, if H is not semisimple or it does not admit a normalized left cointegral, then dim(H) = dim(D(H)) = 0. Proof By λop we denote a left cointegral on H op . It is straightforward to check that in H op we have μop = μ −1 := μ ◦ S, and that the roles of U and V interchange. So λop is an element of H ∗ satisfying

λop (V 2 h2U 2 )V 1 h1U 1 = μ −1 (X 1 )λop (S−1 (X 2 )h)X 3 , ∀ h ∈ H. Note that, if H is unimodular, then μ = ε and therefore a left cointegral on H op is nothing else than a left cointegral on H. By applying Theorem 12.17 to the quasi-Hopf algebra H op we obtain

 Tr h → S−2 (S(β )α hβ S(α ) = ε (t)λop (S−1 (α )β ), where t is a left integral in H such that λop (S−1 (t)) = 1. If we denote r = S−1 (t) we get that r is a right integral in H such that λop (r) = 1. It follows that ε (t) = ε (r), and

 dim(H) = dim(D(H)) = Tr h → S−2 (S(β )α hβ S(α ) = ε (r)λop (S−1 (α )β ). Finally, we apply the same trick as in the proof of the previous theorem. Namely, if H is not semisimple then ε (r) = 0 and we are done. If H is semisimple then it is unimodular. In this case we have seen that λop is a left cointegral on H and since λop (r) = 1 the above equality finishes the proof.

472

The Quantum Dimension and Involutory Quasi-Hopf Algebras

12.5 Involutory Quasi-Hopf Algebras The aim of this section is to introduce and study involutory quasi-Hopf algebras. Definition 12.19

A quasi-Hopf algebra is called involutory if S2 (h) = S(β )α hβ S(α ), ∀ h ∈ H.

(12.5.1)

The definition of an involutory quasi-Hopf algebra occurs as a consequence of the formula for the quantum dimension of H and D(H), the quantum double of H, and this is h → S−2 (S(β )α hβ S(α )) = IdH ; clearly, it is equivalent to the relation (12.5.1). A second way to introduce this notion is by using the trace formula for quasi-Hopf algebras, namely, h → β S(α )S2 (h)S(β )α = IdH . The result below says that the two ways above are equivalent. Lemma 12.20 Let H be an involutory quasi-Hopf algebra. Then S(β )α is an invertible element and (S(β )α )−1 = β S(α ). In particular, S2 is inner and therefore S is bijective. Moreover, α and β are invertible elements and

α −1 = S−1 (αβ )β = β S(β α ), β

−1

= S(β α )α =

α S−1 (αβ ).

(12.5.2) (12.5.3)

Proof For simplicity denote U = S(β )α and V = β S(α ). Then S2 (h) = UhV, for all h ∈ H. Since S2 is an algebra map we get that H h → UhV ∈ H is an algebra endomorphism of H. Hence U is invertible and U−1 = V. In other words we have proved that S(β )α is invertible and (S(β )α )−1 = β S(α ), as claimed. The relation VU = 1H comes out as β S(β α )α = 1H . Thus β has a right inverse, namely S(β α )α . Similarly, from UV = 1H we obtain that S(β )αβ S(α ) = 1H . Since S is bijective this is equivalent to α S−1 (αβ )β = 1H , so β has also a left inverse, namely α S−1 (αβ ). Thus β is invertible and β −1 = S(β α )α = α S−1 (αβ ). The relations in (12.5.2) follow from the fact that if in an algebra A two elements a and b are such that ab and a are invertible, then b is also invertible and b−1 = (ab)−1 a. In fact, we apply this elementary result to A = H, a = S(β ) and b = α . One can easily verify that H is involutory if and only if H op is involutory, if and only if H cop is involutory, if and only if H op,cop is involutory. Proposition 12.21 If H is a finite-dimensional involutory quasi-Hopf algebra and dim(H) = 0 in k then H is semisimple and admits a normalized left cointegral. Consequently, the quantum double of H is a semisimple quasi-Hopf algebra admitting a normalized left cointegral. Proof By the comments made before Lemma 12.20, if H is as in the statement then, on the one hand, dim(H) = Tr(IdH ) = dim(H) = 0; see (12.3.6). On the other hand, dim(H) = ε (r)λ (S−1 (α )β ); see Theorem 12.18, where r is a non-zero right integral in H and λ is a non-zero left cointegral on H such that λ (r) = 1. Thus ε (r)λ (S−1 (α )β ) = 0, and this is equivalent to the fact that H is semisimple and

12.5 Involutory Quasi-Hopf Algebras

473

admits a normalized left cointegral. The last assertion in the statement follows from Corollary 12.2 and Corollary 12.8. For an involutory quasi-Hopf algebra we have a kind of skew-antipode property, in the following sense. Proposition 12.22 Let H be an involutory quasi-Hopf algebra. Then for all h ∈ H the following relations hold: S(h2 )β −1 h1 = ε (h)β −1 and h2 α −1 S(h1 ) = ε (h)α −1 .

(12.5.4)

Proof As we have seen in Remark 3.16(2), if U ∈ H is invertible then we can define a new quasi-Hopf algebra H U = (H, Δ, ε , Φ, SU , αU , βU ), where αU := Uα , βU := β U−1 and SU (h) := US(h)U−1 . Now, consider U = α . We know from Lemma 12.20 that U is invertible, so it makes sense to consider the quasi-Hopf algebra H U . In this particular case we have that αU = 1H , βU = β α and SU (h) = α −1 S(h)α = β S(β α )S(h)α = β S(α )S(β −1 hβ )S(β )α = S−1 (β −1 hβ ). Since SU (h1 )αU h2 = ε (h)αU for all h ∈ H, we get that S−1 (β −1 h1 β )h2 = ε (h)1H , and this is equivalent to S(h2 )β −1 h1 β = ε (h)1H , for all h ∈ H. It follows now that S(h2 )β −1 h1 = ε (h)β −1 for all h ∈ H, as needed. Similarly, by using the fact that h1 βU S(h2 ) = ε (h)βU for all h ∈ H, one can prove that h2 α −1 S(h1 ) = ε (h)α −1 for all h ∈ H; the details are left to the reader. We end this section by presenting examples of involutory quasi-Hopf algebras. Examples 12.23 (1) The two-dimensional quasi-Hopf algebra H(2) described in Example 3.26 is an involutory quasi-Hopf algebra since α = g has order 2, β = 1 and S is the identity map. (2) The quasi-Hopf algebra Dω (H) considered in Section 8.6 is an involutory quasi-Hopf algebra since for Dω (H) we have αDω (H) = 1, the unit of Dω (H), −1 2 ω βD−1 ω (H) = s(βDω (H) ) and s (ϕ #h) = βDω (H) (ϕ #h)βDω (H) , for all ϕ #h ∈ D (H). (3) For the quasi-Hopf algebra bos(H(2)+ ) = bos(H(2)− ) in Example 11.13 we have that the distinguished elements α and β are given by α = x and β = 1, and the antipode is the identity map. Thus bos(H(2)+ ) = bos(H(2)− ) is an involutory quasi-Hopf algebra. (4) Let D(H(2)) be the quantum double of H(2) with the quasi-Hopf algebra structure computed in Proposition 10.21. We have that the antipode for D(H(2)) is the identity map, α = X and β = 1, so D(H(2)) is an involutory quasi-Hopf algebra. The involutory property on H transfers on D(H) if a certain condition is fulfilled. Proposition 12.24

Let H be an involutory quasi-Hopf algebra such that

Δ(S(β )α ) = f −1 (S ⊗ S)( f21 )(S(β )α ⊗ S(β )α ), where f21 = f 2 ⊗ f 1 . Then D(H) is an involutory quasi-Hopf algebra.

(12.5.5)

474

The Quantum Dimension and Involutory Quasi-Hopf Algebras

Proof

From (10.4.7) we know that

2 SD (ϕ  h) = g11 G1 S( f 2 F22 )  S

−2

(ϕ )  F 1 S−1 (g2 )  g12 G2 S( f 1 F12 )S2 (h),

for all ϕ ∈ H ∗ and h ∈ H. By using (12.5.5) twice and (3.2.13) we obtain (Δ ⊗ IdH )(Δ(S(β )α )) = g11 G1 S( f 2 F22 )S(β )α ⊗ g12 G2 S( f 1 F12 )S(β )α ⊗ g2 S(F 1 )S(β )α . By (12.5.1) and (12.5.2) we have S(β S(α )) = S2 (α )S(β ) = S(β )α 2 β S(β α ) = S(β )α 2 α −1 = S(β )α , or, equivalently, S−1 (S(β )α ) = β S(α ). Then we have, for all ϕ ∈ H ∗ and h ∈ H, (ε  S(β )α )(ϕ  h)(ε  α S(β )) (8.5.2) =

g11 G1 S( f 2 F22 )S(β )α  ϕ  β S(α )S−1 (g2 S(F 1 ))  g12 G2 S(F 1 f12 )S2 (h)

= g11 G1 S( f 2 F22 )  S

−2

(ϕ )  F 1 S−1 (g2 )  g12 G2 S( f 1 F12 )S2 (h)

2 (ϕ  h). = SD

This means that D(H) is an involutory quasi-Hopf algebra. Notice that for H(2) the condition in (12.5.5) is Δ(g) = g ⊗ g, which is just part of the definition of H(2). So Proposition 12.24 gives us a direct argument for the fact that D(H(2)) is an involutory quasi-Hopf algebra. It is still an open problem if (12.5.5) is automatic for an involutory quasi-Hopf algebra H, and so to have H involutory if and only if D(H) is involutory.

12.6 Representations of Involutory Quasi-Hopf Algebras The goal of this section is to study the representations of an involutory quasi-Hopf algebra H over a field k. We will prove that if H is semisimple then the characteristic of k does not divide the dimension of any finite-dimensional absolutely simple Hmodule. Recall that a left H-module V is absolutely simple if for every field extension k ⊆ K, K ⊗V is a simple K ⊗ H-module or, equivalently, if every H-endomorphism of V is of the form cIdV for some scalar c ∈ k. The case when H is not semisimple is treated as well; in this case the characteristic of k divides the dimension of any finite-dimensional projective H-module. Now, in order to prove these results for involutory quasi-Hopf algebras we need some preliminary results. Let V and W be two left H-modules. Then one can easily see that the set of k-linear maps from V to W , Homk (V,W ), has a left H-module structure defined by (h · ψ )(v) = h1 · ψ (S(h2 ) · v), ∀ h ∈ H, ψ ∈ Homk (V,W ) and v ∈ V. Consequently, if V = W then Endk (V ) := Homk (V,V ) is a left H-module.

12.6 Representations of Involutory Quasi-Hopf Algebras

475

For any left H-module V we define V H , the set of H-invariants of V , as follows: V H = {v ∈ V | h · v = ε (h)v, ∀ h ∈ H}. In particular, we have that the set of H-invariants of Homk (V,W ) is Homk (V,W )H = {ψ ∈ Homk (V,W ) | h1 · ψ (S(h2 ) · v) = ε (h)ψ (v), ∀ h ∈ H, v ∈ V }. (12.6.1) H We can characterize Homk (V,W ) more precisely. Lemma 12.25 Let H be a quasi-Hopf algebra with bijective antipode and V , W two left H-modules. If we denote by HomH (V,W ) the set of left H-linear maps from V to W then

ν : Homk (V,W )H → HomH (V,W ),

ν (ψ )(v) = q1 · ψ (S(q2 ) · v),

for all ψ ∈ Homk (V,W ) and v ∈ V , is bijective. Its inverse is

ν −1 : HomH (V,W ) → Homk (V,W )H ,

ν −1 (χ )(v) = χ (β · v),

for all χ ∈ HomH (V,W ) and v ∈ V , where qR = q1 ⊗ q2 is as defined in (3.2.19). Proof We first show that ν is well defined, that is, ν (ψ ) is H-linear for any ψ ∈ Homk (V,W )H . Indeed, for any ψ ∈ Homk (V,W )H , h ∈ H and v ∈ V , we have h · ν (ψ )(v)

hq1 · ψ (S(q2 ) · v)

= (3.2.21)

=

q1 h(1,1) · ψ (S(h(1,2) )S(q2 )h2 · v)

(12.6.1),(3.1.8) 1

=

q · ψ (S(q2 )h · v) = ν (ψ )(h · v),

as needed. Now, since for all χ ∈ HomH (V,W ), h ∈ H and v ∈ V , we have h1 · ν −1 (χ )(S(h2 ) · v) = h1 · χ (β S(h2 ) · v) = χ (h1 β S(h2 ) · v) (3.2.1)

= ε (h)χ (β · v) = ε (h)ν −1 (χ )(v),

from (12.6.1) we deduce that ν −1 is well defined, too. So it remains to show that ν and ν −1 are inverses. Indeed, for all χ ∈ HomH (V,W ) and v ∈ V we have (ν ◦ ν −1 )(χ )(v) = q1 · ν −1 (χ )(S(q2 ) · v) = q1 · χ (β S(q2 ) · v) = χ (q1 β S(q2 ) · v) = χ (v), where in the third equality we used the fact that χ is H-linear and in the fourth equality we used the definition (3.2.19) of qR and (3.2.2). Now let p1 ⊗ p2 be the element pR defined in (3.2.19), ψ ∈ Homk (V,W )H and v ∈ V . We have (ν −1 ◦ ν )(ψ )(v)

=

ν (ψ )(β · v)

=

q1 · ψ (S(q2 )β · v)

476

The Quantum Dimension and Involutory Quasi-Hopf Algebras (3.2.19),(3.1.11)

=

(12.6.1)

ε (p1 )q1 · ψ (S(q2 )p2 · v) (3.2.23)

q1 p11 · ψ (S(q2 p12 )p2 · v) = ψ (v),

=

and this finishes the proof. Lemma 12.26 Let H be an involutory quasi-Hopf algebra, V a finite-dimensional left H-module and {vi }i=1,n a basis in V with dual basis {vi }i=1,n . Then the map n

Tr : Endk (V ) ζ → ∑ vi , β −1 · ζ (vi ) ∈ k i=1

is H-linear. Moreover, the relation between Tr and the classical trace function Tr is

 Tr(ζ ) = Tr v → ζ (β −1 · v) , ∀ ζ ∈ Endk (V ). Proof

Indeed, for any h ∈ H and ζ ∈ Endk (V ) we have Tr(h · ζ )

n

=

∑ vi , β −1 · (h · ζ )(vi )

i=1 n

=

∑ vi , β −1 h1 · ζ (S(h2 ) · vi )

i=1 n

=

∑ v j , ζ (S(h2 ) · vi )vi , β −1 h1 · v j 

i, j=1 n

=

∑ v j , ζ (S(h2 )β −1 h1 · v j )

j=1 (12.5.4) =

n

 ε (h) ∑ v j , ζ (β −1 · v j ) = ε (h)Tr v → ζ (β −1 · v) . j=1

On the other hand, by using dual bases we have that 

Tr v → ζ (β −1 · v) =

n

∑ v j , ζ (β −1 · v j )

j=1 n

=

∑ vi , β −1 · v j v j , ζ (vi )

i, j=1 n

= ∑ vi , β −1 · ζ (vi ) = Tr(ζ ), i=1

which is the second assertion in the statement. The first one follows now from the two computations above. By using Lemma 12.25 one can associate to any linear map an H-linear one. This can be done by using the integrals in H. Let V and W be two left H-modules and ζ ∈ Homk (V,W ). Since Homk (V,W ) is a left H-module, by the definition of a left integral t in H it follows that t · ζ belongs to Homk (V,W )H . Keeping the same notation as

12.6 Representations of Involutory Quasi-Hopf Algebras

477

in Lemma 12.25, we get that ζ˜ := ν (t · ζ ) is an H-linear map. Explicitly, to any ζ ∈ Homk (V,W ) we associated the map ζ˜ ∈ HomH (V,W ) defined by

ζ˜ (v) = q1t1 · ζ (S(q2t2 ) · v),

(12.6.2)

for all v ∈ V , where t is a left integral in H. We are now able to prove one of the main results of this section. Theorem 12.27 Let H be a semisimple involutory quasi-Hopf algebra over a field k of characteristic p ≥ 0. Then p does not divide the dimension of any finite-dimensional absolutely simple H-module. Proof The assertion follows from the fact that the map Tr defined in Lemma 12.26 is H-linear. Actually, since H is semisimple we know from Theorem 7.28 that there is a left integral t in H such that ε (t) = 1. Let now V be a finite-dimensional absolutely simple H-module, that is, V is finite dimensional and any element of EndH (V ) := HomH (V,V ) is of the form cIdV , for some scalar c ∈ k. Therefore, for any ζ ∈ Endk (V ) there exists a scalar cζ ∈ k such that ζ˜ = cζ IdV , where ζ˜ ∈ EndH (V ) is the associated H-linear map of ζ as in (12.6.2). In other words, for any ζ ∈ Endk (V ) there is a scalar cζ such that q1t1 · ζ (S(q2t2 ) · v) = cζ v, for all v ∈ V . By (7.2.3) we have t1 ⊗ S(t2 ) = q1t1 ⊗ S(q2t2 )β , so the above relation is equivalent to t1 · ζ (S(t2 )β −1 · v) = cζ v, for all v ∈ V , and this implies:

 cζ dimk (V ) = Tr(v → cζ v) = Tr v → t1 · ζ (S(t2 )β −1 · v) 

= Tr v → (t · ζ )(β −1 · v) = Tr(t · ζ ) = ε (t)Tr(ζ ) = Tr(ζ ).

 Clearly, we can choose a map ζ ∈ Endk (V ) such that Tr(ζ ) = Tr v → ζ (β −1 · v) = 1, so we conclude that dimk (V ) = 0 in k. Corollary 12.28 Let H be a semisimple involutory quasi-Hopf algebra over an algebraically closed field of characteristic p ≥ 0. Then p does not divide the dimension of any finite-dimensional simple H-module. Proof

This follows from Schur’s lemma and Theorem 12.27.

We will focus now on the case when H is not semisimple. In order to simplify the proof of the next theorem we first show the following result: Proposition 12.29 Let H be a finite-dimensional quasi-Hopf algebra and P, Q finite-dimensional projective left H-modules. Then: (i) P∗ = Homk (P, k) is a projective left H-module; (ii) P ⊗ Q is a projective left H-module, where the H-module structure of P ⊗ Q is defined by the comultiplication Δ of H. Consequently, we obtain that Endk (P) is a projective left H-module. Proof (i) Recall that, if V is a left H-module then V ∗ , the linear dual of V, is a left H-module with H-action (h · v∗ )(v) = v∗ (S(h) · v), for all v∗ ∈ V ∗ , h ∈ H, v ∈ V .

478

The Quantum Dimension and Involutory Quasi-Hopf Algebras

Since P is finite dimensional it follows that P is a finitely generated projective left H-module. Therefore, there exist a natural number n and a left H-module P such that P ⊕ P ∼ = H n as left H-modules. Thus (H n )∗ = Homk (H n , k) ∼ = Homk (P ⊕ P , k) ∼ Homk (P, k) ⊕ Homk (P , k) = P∗ ⊕ P∗ , = as left H-modules. Now, H is finite dimensional, so from the proof of Theorem 7.48 we know that the application H h → (h → λ (h S(h))) ∈ H ∗ is bijective, where λ is a non-zero left cointegral on H. Replacing H by H op,cop we get that H ∼ = H ∗ as left H-modules, where the left H-module structures on H and H ∗ are given by the regular multiplication on H and by the corresponding left H-module structure induced on its dual, respectively, namely (h · h∗ )(h ) = h∗ (S(h)h ), for h, h ∈ H and h∗ ∈ H ∗ . From the above we obtain that H n ∼ = (H ∗ )n ∼ = (H n )∗ ∼ = P∗ ⊕P∗ , as left H-modules, so P∗ is a projective left H-module, too. (ii) We follow the same line as above. There exist two natural numbers n and m and two left H-modules P and Q such that P ⊕ P ∼ = H n and Q ⊗ Q ∼ = H m , as left H-modules. We then have (H ⊗ H)nm ∼ = Hn ⊗ Hm ∼ = (P ⊗ Q) ⊕ (P ⊗ Q ) ⊕ (P ⊗ Q) ⊕ (P ⊗ Q ), as left H-modules. We now prove that H ⊗ H with the diagonal H-module structure is a free left H-module. Thus, as a consequence, we will obtain that P ⊗ Q is a projective left H-module. To this end, we claim that the map

μ : · H ⊗ · H → · H ⊗ H,

μ (h ⊗ h ) = q˜2 h2 ⊗ S−1 (q˜1 h1 )h,

for all h, h ∈ H, is a left H-linear isomorphism. Here we denote by · H ⊗ · H and · H ⊗ H the k-vector space H ⊗ H endowed with the left H-module structure given by Δ and by the left regular multiplication on H, respectively. In addition, qL = q˜1 ⊗ q˜2 is the element defined in (3.2.20). Indeed, for all h, h , h ∈ H we have

μ (h · (h ⊗ h )) = =

μ (h1 h ⊗ h2 h ) q˜2 h(2,2) h2 ⊗ S−1 (q˜1 h(2,1) h1 )h1 h

(3.2.22)  2  = h q˜ h2 ⊗ S−1 (q˜1 h1 )h  

=

h μ (h ⊗ h ),

and this means that μ is left H-linear. Next, one can easily check that the map μ −1 defined for all h, h ∈ H by

μ −1 : · H ⊗ H → · H ⊗ · H,

μ −1 (h ⊗ h ) = h1 p˜1 h ⊗ h2 p˜2

is the inverse of μ . More precisely, (3.2.22) and (3.2.24) imply that μ −1 ◦ μ = Id, while (3.2.22) and (3.2.24) imply that μ ◦ μ −1 = Id, we leave the verification of the details to the reader.

12.7 Notes

479

Finally, if V is a finite-dimensional k-vector space then Endk (V ) ∼ = V ⊗ V ∗ as Hmodules. Indeed, one can prove that the maps n

Endk (V ) χ → ∑ χ (vi ) ⊗ vi ∈ V ⊗V ∗ , i=1 

 V ⊗V v ⊗ v → v → v∗ (v )v ∈ Endk (V ) ∗

∗



are H-linear and inverse to each other (the details are left to the reader). Thus, if P is a finite-dimensional projective left H-module then by part (i) P∗ is a projective left H-module, and then we deduce that Endk (P) ∼ = P ⊗ P∗ is a projective left H-module as well; see (ii). We are now able to prove the second important result of this section. Theorem 12.30 Let H be a finite-dimensional non-semisimple involutory quasiHopf algebra over a field of characteristic p ≥ 0. Then p divides the dimension of any finite-dimensional projective left H-module. Proof Let P be a finite-dimensional projective left H-module and suppose that p does not divide dimk (P). Then the map k c → (dimk (P))−1 c (v → β · v) ∈ Endk (P) is H-linear; see (3.2.1). Obviously, the map above is a section for the H-linear map Tr defined in Lemma 12.26, specialized for V = P. So k is isomorphic to a direct summand of Endk (P), which is a projective left H-module; see Proposition 12.29. Thus k is a projective left H-module. By Theorem 7.28 we obtain that H is semisimple, a contradiction.

12.7 Notes Section 12.1 is taken from [49], where an answer to a conjecture raised by Hausser and Nill in [109] was given. The form of the cointegrals on D(H) is also taken from [49], as well as the explicit form of the modular element of D(H). The explicit form of the (co)integrals in and on D(H) leads to a characterization of the semisimplicity of D(H), a property closely related to its representation-theoretic rank. The computation of the representation-theoretic rank for H and D(H) was performed in [62]. The goal was to find a plausible definition for the involutory notion in the quasi-Hopf case, a notion that in the Hopf case is closely related to the fifth conjecture of Kaplansky [126]: a Hopf algebra H is semisimple as an algebra if and only if it is involutory. It is well known that, over a field of characteristic zero, H is semisimple if and only if it is cosemisimple, if and only if it is involutory, that is, S2 = IdH . These remarkable results were proved by Larson and Radford in [133, 134], answering in the positive, in characteristic zero, the fifth conjecture of Kaplansky. They have also proved that in characteristic p sufficiently large

480

The Quantum Dimension and Involutory Quasi-Hopf Algebras

a semisimple cosemisimple Hopf algebra is involutory. Afterwards, using this result and a lifting theorem, Etingof and Gelaki proved in [90] that the antipode of a semisimple cosemisimple Hopf algebra over any field is an involution. Trying to generalize the above results for quasi-Hopf algebras, the first problem which occurs is: what could be an involutory quasi-Hopf algebra? According to Majid [143], Tr(S2 ) arises in a very natural way as the representation-theoretic rank of the Schr¨odinger representation of H, dim(H), or as the representation-theoretic rank of the canonical representation of the quantum double, dim(D(H)), and this stimulated performing similar computations in the quasi-Hopf case. This also led to the involutory notion in Section 12.5, a definition that agrees with the point of view in [96, Prop. 8.24 and 8.23] or [153]. We end by pointing out that the fifth conjecture of Kaplansky is still an open problem in quasi-Hopf algebra theory. Apart from [49, 62], in the presentation of this chapter we also used [52].

13 Ribbon Quasi-Hopf Algebras

We define and characterize ribbon quasi-Hopf algebras by using properties of a ribbon category. Consequently, we have a one-to-one correspondence between ribbon elements for a quasi-Hopf algebra and its grouplike elements. We also construct ribbon categories from left or right rigid monoidal categories and use this construction to introduce a special class of ribbon quasi-Hopf algebras.

13.1 Ribbon Categories The concept of ribbon category is defined as follows. Definition 13.1 Let (C , c) be a braided category. (1) (C , c) is called balanced if there exists a natural isomorphism η = (ηV : V → V )V ∈Ob(C ) such that, for all V,W ∈ C ,

ηV ⊗W = (ηV ⊗ ηW ) ◦ cW,V ◦ cV,W .

(13.1.1)

(2) A balanced category (C , c, η ) is called ribbon if, in addition, C is left rigid and

ηV ∗ = (ηV )∗

(13.1.2)

for all V ∈ C . If this is the case then η is called a twist on C . Examples 13.2 (1) Any symmetric category C is ribbon with η defined as the identity natural transformation of C . In particular, the category of representations of a triangular quasi-Hopf algebra is ribbon. (2) If (C , c, η ) is balanced then so are C = (C , c, η ), C in = (C , c, η −1 ) and C opp = (C opp , copp , η −1 ). (3) Let k be a field, G a multiplicative finite abelian group and R : G × G → k∗ a bilinear form. Then, endowed with the strict monoidal structure and with the braiding c defined by (1.5.9), the category of G-graded vector spaces VectG is braided; see Proposition 1.44. Furthermore, if we restrict to the finite-dimensional case, then vectG is ribbon. The braided structure on vectG is induced by that of VectG described above, and a twist on it is given by ηV : V v → R(|v|, |v|)v ∈ V , extended by linearity, where we assumed that v is a homogenous element in V of degree |v|.

482

Ribbon Quasi-Hopf Algebras

Indeed, one can see that the bilinearity of R implies (13.1.1). Also, by Example 1.65 the category vectG is rigid, and (13.1.2) is satisfied by η since, on the one hand, for all v∗ ∈ V ∗ homogenous of degree g we have (ηV )∗ (v∗ )(v) = v∗ (ηV (v)) = R(g−1 , g−1 )v∗ (vg−1 ) and ηV ∗ (v∗ )(v) = R(g, g)v∗ (v) = R(g, g)v∗ (vg−1 ), for all v = ∑x∈G vx ∈ V . On the other hand, R(g, g) = R(g−1 , g)−1 = R(g−1 , g−1 ), for all g ∈ G. More examples of balanced categories can be obtained as follows. Proposition 13.3 Let (C , c) be a braided category. Denote by B(C , c) the category whose • objects are pairs (V, ηV ) consisting of an object V ∈ C and an automorphism ηV : V → V of V in C ; • morphisms between (V, ηV ) and (W, ηW ) are morphisms f : V → W in C fulfilling ηW f = f ηV . Then B(C , c) is a balanced category via the following structure: (i) The tensor product on B(C , c) is given by (V, ηV )⊗(W, ηW ) = (V ⊗W, ηV ⊗W ), with ηV ⊗W = (ηV ⊗ ηW )cW,V cV,W ; on morphisms it acts as the tensor product of C . The unit object in B(C , c) is (1, η1 := Id1 ). Together with the associativity and the left and right unit constraints of C these give the monoidal structure on B(C , c); (ii) The braiding on B(C , c) is determined by the braiding c of C ; (iii) The balancing on B(C , c) is produced by η := (ηV )(V,ηV )∈B(C ,c) . Proof We check that the associativity constraint of C is a morphism in B(C , c); the remaining details are trivial. Assuming C is strict monoidal, this reduces to the equality η(V ⊗W )⊗T = ηV ⊗(W ⊗T ) , for all V,W, T ∈ C . In diagrammatic notation, the latter comes out as V W T

V W T

= h

ηT

hh

ηV ηW

V W T

. h

ηV

hh

ηW ηT

V W T

To prove the above equality we compute the right-hand side of it as follows: apply (1.5.10) twice, and then apply the naturality of the braiding c to the morphism cW,V cV.W : V ⊗W → V ⊗W . In this way we get the left-hand side of the equality. Remark 13.4 By Proposition 13.3 we can associate to any monoidal category C two balanced ones: namely, Bl (C ) := B(Zl (C ), c) and Br (C ) := B(Zr (C ), c), where Zl/r (C ) is the left/right center of C as in Section 8.1. As a concrete example, we can take C = H M , H a quasi-Hopf algebra with bijective antipode, in which case we have Bl (C ) = B(H H YD, c) with c as in (8.2.15), and Br (C ) = B(H YD H , c) with c given by (8.2.18).

13.1 Ribbon Categories

483

Note that Proposition 13.3 can also be applied to the category H M , where (H, R) is a QT quasi-Hopf algebra. We will exploit this fact in the forthcoming sections. The following result says that the inverse of the square of the twist is completely determined by the rigid braided structure of the category. Proposition 13.5

Let (C , c, η ) be a left rigid balanced category. Then V

V

 ηVh ∗

ηV−1 =

 ηVh ∗ , ∀V ∈ C.

=





V

V

(13.1.3)

Consequently, a left rigid balanced category (C , c, η ) is ribbon if and only if −1 ηV−2 = (evV ⊗ IdV )(IdV ∗ ⊗ cV,V )(cV,V ∗ coevV ⊗ IdV ), ∀ V ∈ C .

(13.1.4)

Proof Assume that C is strict monoidal. By taking V = W = 1 in (13.1.1), by Proposition 1.49 we get that η1⊗1 = η1 ⊗ η1 . By the naturality of η and l we see that

η1 l1 = l1 η1⊗1 = l1 (Id1 ⊗ η1 )(η1 ⊗ Id1 ) = η1 r1 (η1 ⊗ Id1 ) = η12 l1 , where in the last two equalities we used the naturality of r and the fact that l1 = r1 ; see Proposition 1.5. We conclude that η1 = Id1 , and so by the the naturality of η we get that ηV ⊗V ∗ coevV = coevV , for any object V of C . Therefore, V



V

 IdV = V

(13.1.1)

ηV ⊗V ∗

=



V



V

=



 ηVh ∗

⇔ ηV−1 =

hh

ηV ηV ∗

(∗)

=



V

V

 ηVh ∗

;



V

V

V

in (*) we applied the naturality of c to evV . The second equality in (13.1.3) follows from the naturality of c applied to cV,V ∗ (IdV ⊗ ηV ∗ )coevV : 1 → V ∗ ⊗V . If ηV ∗

1

1

V V∗

V V∗

  = (ηV )∗ then = ηh , and therefore by (13.1.3) we have that ηVh ∗ V V

ηV−1 =

 ηVh

V

V

V

 ⇔ ηV−2 =



=

,





V

V

(13.1.5)

484

Ribbon Quasi-Hopf Algebras

as desired. For the converse, if (13.1.4) holds then again by (13.1.3) we have that V

ηV−1 =

1

V

 ηVh

 ηVh ∗ =





V

V∗

V

V V∗

1

 ηVh ⇔



V 1

1

  ηVh ∗ ηVh ⇔ =

 ηVh ∗

1

= 







V V∗

1

  ηVh ηVh ∗ ⇔ = ⇔ V

V∗

V

1

1

V V∗

V V∗

  = ηh , ηVh V∗

V∗

V V∗

which is equivalent to ηV ∗ = (ηV )∗ , as needed. So our proof is complete. Any ribbon category is rigid; see Proposition 1.74. Furthermore, the next results say that the right rigid structure can be constructed from the left rigid structure, braiding and twisting in such a way that the left and right dual functors coincide. Thus the choice of the left duals in the definition of a ribbon category is irrelevant. Proposition 13.6 Let (C , c, η ) be a left rigid balanced category. If V is an object of C and V ∗ is the left dual of V in C with evaluation and coevaluation morphisms evV : V ∗ ⊗V → 1 and coevV : 1 → V ⊗V ∗ , then (V ∗ , evV := evV cV,V ∗ (ηV ⊗ IdV ∗ ), coevV := (ηV ∗ ⊗ IdV )cV,V ∗ coevV )

(13.1.6)

is a right dual for V in C . Furthermore, with respect to it the left and right dual functors coincide as strong monoidal functors. Consequently, any left rigid balanced category is sovereign. Proof By using the equalities in (13.1.3) one can easily verify that V ∗ with ev and coev as in (13.1.6) is a right dual for V in C . Also, with respect to this right duality we have ∗ f = f ∗ , for any morphism f : V → W in C , since W∗

W∗

 ∗

h fh f= h ηW ηV ∗

W∗

 =



hh

ηV ∗ ηV

=

= h

h

f

V∗

V∗

V∗

W∗



h

f

f





W∗

 fh = = f ∗,



V∗

because of (13.1.1) and the naturality of c and η , respectively.

V∗

13.1 Ribbon Categories

485

The only thing left to show is that λ in (1.7.1) equals λ  , its right-handed version. With notation as in Section 1.7 we have that ∗ (V

∗ (V

•  •

 λV,W =

⊗W )

=

V ⊗W



•



•  •

hh

ηV ηW

hh

ηV ηW

λV,W

λV,W



∗W ∗V

⊗W )

⊗W )

=

∗W ∗V

∗ (V

∗ (V

⊗W )

•  •

∗ (V

•  •

⊗W )

λV,W



∗W ∗V ∗ (V

•  •

⊗W )

λV,W

•  •

λV,W

=

= hh

= hh

ηV ηW



∗W ∗V ∗ (V



∗W ∗V

⊗W )

∗ (V

 •

λV,W

 •

⊗W )

 •



λV,W

∗ (V

 • ηVh

h=

h

h



ηW





⊗W )

λV,W

=

ηW

∗W ∗V

ηW

∗W ∗V

ηV

=

ηVh h

ηV ηW

 • h ηW

= λV,W ,  • ηVh





∗W

∗V

∗W ∗V

for all V,W ∈ C , since X X

 • ηXh=

 X h h=

ηX ∗ ηX





X X

 X =



 = IdX ,

X

X

for all X ∈ C . This ends the proof. Corollary 13.7 Remark 13.8

Any ribbon category is sovereign. If (C , c, η ) is ribbon, from ηV ∗ = (ηV )∗ it follows that coevV in

486

Ribbon Quasi-Hopf Algebras

(13.1.6) can be restated as coevV = (IdV ∗ ⊗ ηV )cV,V ∗ coevV .

(13.1.7)

Corollary 13.9 A braided category (C , c) is ribbon if and only if it is right rigid and there exists a natural isomorphism θ = (θV : V → V )V ∈Ob(C ) such that

θV ⊗W = (θV ⊗ θW ) ◦ cW,V ◦ cV,W , ∀ V, W ∈ C , θ∗V = ∗ (θV ), ∀ V ∈ C . Proof The direct implication follows from Proposition 13.6. Conversely, we have that C is ribbon and by applying again Proposition 13.6 (this time to C , the reverse braided category associated to (C , c)) the converse follows. The result in Proposition 13.6 has a converse. In particular, we can characterize ribbon categories in terms of sovereign categories. Theorem 13.10 Let (C , c) be a left rigid braided category. Then (i) (C , c) is balanced if and only if C is sovereign; (ii) (C , c) is ribbon if and only if it is sovereign and, with respect to the rigid structure given by the fact that C is sovereign, we have V∗

 •

V∗



= , ∀V ∈ C.

• V∗

(13.1.8)

V∗

Proof (i) By Proposition 13.6, a left rigid balanced category is sovereign. Let (C , c) be a braided sovereign category. We claim that, for all V ∈ C , V

V

 •

ηV :=

and



ηV−1

V



:=

(13.1.9)

• V

are inverses of each other in C and, moreover, (C , c) with η := (ηV )V ∈Ob(C ) becomes a balanced category. As before, the graphical notation is as in Sections 1.6 and 1.8. Indeed, we use the naturality of c−1 to compute that V

ηV ηV−1 =

 •

V





=

•

V

  V •

• V

=

  V •

•  = = IdV .

•



• V

V

13.1 Ribbon Categories

487

Likewise, by the naturality of c and its inverse c−1 we have V



V

 •

ηV−1 ηV =



 =

 •

•

V

=

  • =

• V

V

V

  •

= IdV ,

• 

• V

V

and so ηV is an isomorphism in C with inverse ηV−1 , as claimed. For f : V → W a morphism in C , by the naturality of c and f ∗ = ∗ f , we get that V

V

V

V

W

W

W

W

 •

•    • • •  fh fh ηW f = = = = = f ηV , fh fh

•







and therefore η = (ηV )V ∈Ob(C ) is a natural isomorphism. Since ∗ (−) and (−)∗ are equal as strong monoidal functors, we calculate V W

•  •

V W

 •

−1 λV,W

ηV ⊗W =

h

ηV

=

V W

 • =

λV,W





V W

= h

ηV



V W

V W

, hh

ηV ηW

V W

V W

for all V,W ∈ C , as needed. (ii) By part (i), we have to show that, for all V ∈ C , ηV ∗ = (ηV )∗ if and only if (13.1.8) holds. To this end, note that (13.1.6) and the naturality of c imply V∗

 ηVh

V∗

  • =

• V∗

V∗

 • =

•

V∗

= IdV ∗ , 

•

V∗

488

Ribbon Quasi-Hopf Algebras

and therefore V∗

V∗





=

•

ηV ⊗V ∗

= ηhη h V V∗

•

V∗

V∗



V∗

 ηVh =

=

V∗ V∗ ∗

(ηV ) =

  V ∗ •



= ηV ∗ ,

h •

ηV ∗

ηVh ∗

•

•

V∗

V∗

 ηVh

V∗

V∗

  •

=

V∗

 •

=

=



V∗

 • ,





V∗

V∗

V∗

V∗

and this finishes the proof. Corollary 13.11 If H is a sovereign quasi-Hopf algebra then the category of finitefd dimensional left-right Yetter–Drinfeld modules H YD H is balanced. fd

Proof The category H YD H is sovereign; see Theorem 8.25. As it is braided, the result follows from part (i) of Theorem 13.10.

13.2 Ribbon Categories Obtained from Rigid Monoidal Categories To any left rigid braided category (C , c) (which is consequently rigid) we assign a ribbon category R(C , c). Thus, to any left (resp. right) rigid monoidal category C we can associate a ribbon monoidal one, that will be denoted by Rl (C ) (resp. Rr (C )). The latter are possible due to the left and right center constructions presented in Section 8.1. Inspired by the formula in (13.1.4) we introduce the following category, which will turn out to be a ribbon category. Definition 13.12 If (C , c) is a left rigid braided (strict) monoidal category then R(C , c) is the category whose • objects are pairs (V, ηV ) consisting of an object V of C and an automorphism ηV of V in C satisfying V

V



ηV−2 =

 =

V

;

V

(13.2.1)

13.2 Ribbon Categories Obtained from Rigid Monoidal Categories

489

• morphisms f : (V, ηV ) → (W, ηW ) are morphisms f : V → W in C such that ηW f = f ηV . The composition in R(C ) is given by the composition in C , and the identity morphism of an object (V, ηV ) is IdV . In other words, R(C , c) is the full subcategory of B(C , c) considered in Proposition 13.3 determined by those objects (V, ηV ) of B(C , c) for which ηV obeys (13.2.1). As we pointed out in the second part of Proposition 13.5, this is the necessary and sufficient condition that turns B(C , c) into a ribbon category. Actually, the ribbon structure of R(C , c) is encoded in the following result. Theorem 13.13 Let (C , c) be a left rigid braided (strict) monoidal category. Then R(C , c) is a ribbon monoidal category as follows: • if (V, ηV ), (W, ηW ) ∈ R(C , c) then (V, ηV ) ⊗ (W, ηW ) = (V ⊗W, ηV ⊗W ), where

ηV ⊗W = (ηV ⊗ ηW )cW,V cV,W ;

(13.2.2)

• the unit object is (1, η1 = Id1 ), and the associativity and the left and right unit constraints are the same as those of C ; • the braiding equals c, regarded as an isomorphism in R(C , c); • for (V, ηV ) an object in R(C , c), a left dual object for it is (V ∗ , ηV ∗ ), where

ηV ∗ = (ηV )∗ ,

(13.2.3)

with evaluation and coevaluation morphisms equal to evV and coevV , viewed now as morphisms in R(C , c); • the twist is given by

ηV : (V, ηV ) → (V, ηV ),

(13.2.4)

an automorphism in R(C , c). Proof We start by proving that (V ⊗ W, ηV ⊗W ) is an object of R(C , c), that is, ηV ⊗W in (13.2.2) obeys (13.2.1). To this end, we need the equalities V W X Y

V W X Y f

(a)

=

V

,

X g

(b)

V X

=

g

,

(13.2.5)

f

Z V W

Z

Y

V W

Z V T

Y

Z V T

for any morphisms f : X ⊗Y → Z and g : X → Y ⊗ Z ⊗ T in C , and X Y

Z

X Y

Z

= Z Y X

, Z Y X

(13.2.6)

490

Ribbon Quasi-Hopf Algebras

valid for all X,Y, Z ∈ C , respectively, which follow from the fact that c−,− is a natural isomorphism. Note that (13.2.6) is nothing but an equivalent form of the categorical version of the Yang–Baxter equation (1.5.10). Now, if λV,W : (V ⊗ W )∗ → W ∗ ⊗ V ∗ is the isomorphism in C defined in (1.7.1) −1 is its inverse as in (1.7.2), then we compute and λV,W V W

V W



V W

 

V ⊗W

 

−1 λV,W

(13.2.5.a)

=

=

λV,W



V ⊗W

 V W



V W V W



(13.2.5.b)

V W V W



=

=

(13.2.6)







V W





V W

V W

V W V W





V W



V W

hh

−1 ηV−1 ηW

h

ηV−2

h

ηV−2

(∗)

=

(13.2.6)

=

(13.2.1)

h

ηV−2

(13.2.6)

(13.2.1)

=

=







V W

W



(13.2.5.b)

(13.2.6)

=

V



V W

V W

V W

−2 h h= ηV ⊗W ,

−1 ηV−1 ηW

where (*) is V W

V W X

X

V W 1

h

=

for h =

h

 , i.e.



V W



=

.

W∗ W Y

Z V W

Y

Z V W

W∗ V W W

W∗ V W W

13.2 Ribbon Categories Obtained from Rigid Monoidal Categories

491

This ends the proof of the fact that the tensor product of R(C , c) is well defined at the level of objects. It is easy to see that for any two morphisms f , f  in R(C , c) their tensor product f ⊗ f  in C is actually a morphism in R(C , c). Therefore, we have a tensor product functor on R(C , c) that together with the associativity and the left and right unit constraints of C defines on R(C , c) a monoidal structure; the unchecked details are left to the reader. We next show that c provides a braiding on R(C , c). The only thing that must be verified is that cV,W is a morphism in R(C , c), for any objects (V, ηV ), (W, ηW ) of R(C , c). This follows directly from the definitions and from the naturality of c. Let (V ∗ , ηV ∗ ) be as in (13.2.3). By using the naturality of c one sees that V

h 

ηV

V

V

V   ηVh ηVh ηVh ηVh ηVh= ηVh = = = IdV ,

ηVh







V

V

V

V V



 h η −2 V

ηV−2 =

h

ηV−2

=

V

V

 

V

V

(13.2.7)

(13.2.1)

=

  =

, (13.2.8)





V

V



V

−1 where in the last equality of the second computation we applied the naturality of cV,− to the morphism evV cV,V ∗ : V ⊗V ∗ → 1. These formulas allow us to prove that

V∗

 V V∗  V



h

ηV2 V∗



V∗

V∗

V  V

 V V∗  V =

V



V∗

V V



V∗

h

=

h

ηV2

ηV2 V



V∗ V∗

V∗

 V =

=

V

V∗

V



V∗

 h = (η −2 )∗ = (ηV ∗ )−2 .

ηV−2



V∗

V

492

Ribbon Quasi-Hopf Algebras

−1 In the first equality we used an equivalent form of the naturality of cV,− applied to ∗ evV , in the second equality (1.6.6), in the third equality (13.2.7) and in the fourth equality (1.6.6) and (13.2.8). In other words, the relation in (13.2.1) is satisfied by ηV ∗ , and thus (V ∗ , ηV ∗ ) is an object of R(C , c). That it is a left dual of (V, ηV ) in R(C , c) reduces to the fact that evV and coevV are morphisms in R(C , c). Towards this end, we need the equivalence

W∗ V

W∗ V





W∗

 W∗ V

h

ηV2

=

fh

=

h ηV2



h

f

(13.2.1)



h

ηV2

⇐⇒

h



f



1

(13.2.9)

f

1

h

= f ∗,

V∗

1

true for any morphism f : V → W in C with (V, ηV ) ∈ R(C , c), which follows from −1 applied to evV (IdV ∗ ⊗ f ηV2 )cV,W ∗ : V ⊗W ∗ → 1 and the definithe naturality of cV,− ∗ tion of f . Now, that evV is a morphism in R(C , c) is a consequence of the computation

V∗ V

V∗

V∗

V

ηV ∗ ⊗V

= ηh h= V ∗ ηV



1

1

V∗ V

V



h



=

h

ηV2

ηV2





1

=

V∗ V

 h

ηV2

(13.2.9)

= evV .



1

1

Likewise, we compute that 1

1

   (13.2.1) ηV ⊗V ∗ = = = coevV , ηVh ∗ ηVhηVh ηVh 1

V V∗



V V∗

V

V∗

and so coevV is a morphism in R(C , c), as stated. Finally, from the left rigid monoidal structure of R(C , c) we get that η := (ηV )V is a twist on R(C , c), and so the proof is finished. More generally, Theorem 13.13 allows us to construct ribbon categories from left or right rigid monoidal categories.

13.2 Ribbon Categories Obtained from Rigid Monoidal Categories Proposition 13.14 a category whose

493

Let C be a left rigid (strict) monoidal category. Then Rl (C ) is

• objects are triples (V, cV,− , ηV ) consisting of an object V of C , a natural isomorphism cV,− = (cV,X : V ⊗ X → X ⊗V )X∈Ob(C ) and an automorphism ηV of V in C such that (8.1.4) and (13.1.4) hold, and (IdX ⊗ ηV )cV,X = cV,X (ηV ⊗ IdX ), ∀ X ∈ Ob(C );

(13.2.10)

• morphisms f : (V, cV,− , ηV ) → (V  , cV  ,− , ηV  ) are morphisms f : V → V  in C obeying (IdX ⊗ f )cV,X = cV  ,X ( f ⊗ IdX ), for any object X of C , and f ηV = ηV  f . Proof One can easily check that Rl (C ) = R(Zl (C ), c), where Zl (C ) is the left center of C as in Section 8.1, a braided category. We need only note that (13.1.4) is nothing but the second equality in (13.2.1). We now uncover the ribbon structure of Rl (C ). For the choice of the natural isomorphism cV ∗ ,− below see (1.8.1) or Theorem 8.17. Theorem 13.15 Let C be a left rigid monoidal category. Then Rl (C ) is a ribbon category with the following structure: • if (V, cV,− , ηV ), (W, cW,− , ηW ) ∈ Rl (C ) then (V, cV,− , ηV ) ⊗ (W, cW,− , ηW ) = (V ⊗W, cV ⊗W,− , ηV ⊗W ),

(13.2.11)

where cV ⊗W,− is as in (8.1.5) and

ηV ⊗W = (ηV ⊗ ηW )cW,V cV,W ;

(13.2.12)

• the unit object is (1, c1,− = (rX−1 lX )X∈C ≡ Id, η1 = Id1 ), and the associativity and the left and right unit constraints are the same as those of C ; • the braiding c is determined by cV,W : (V, cV,− , ηV ) ⊗ (W, cW,− , ηW ) → (W, cW,− , ηW ) ⊗ (V, cV,− , ηV ), (13.2.13) an isomorphism in Rl (C ); • for (V, cV,− , ηV ) ∈ Rl (C ), a left dual object for it is (V ∗ , cV ∗ ,− , ηV ∗ ), where −1 cV ∗ ,X = (evV ⊗ IdX⊗V ∗ )(IdV ∗ ⊗ cV,X ⊗ IdV ∗ )(IdV ∗ ⊗X ⊗ coevV ),

(13.2.14)

for all X ∈ Ob(C ), and

ηV ∗ = (ηV )∗ ,

(13.2.15)

and the evaluation and coevaluation morphisms are precisely evV and coevV , viewed now as morphisms in Rl (C ); • the twist is given by

ηV : (V, cV,− , ηV ) → (V, cV,− , ηV ), an automorphism in Rl (C ).

(13.2.16)

494

Ribbon Quasi-Hopf Algebras

Proof Since Rl (C ) = R(Zl (C ), c), the only thing we must check is the fact that cV ∗ ,X in (13.2.14) is an isomorphism in C , for all X ∈ Ob(C ). In particular, Theorem 8.17 applies. In the computations below, we use graphical notations similar to those used in Section 8.1. Namely, we denote ⎞ ⎞ ⎛ ⎛ cV,− := ⎝

V X

⎠

•

X V

−1 and cV,− := ⎝

X V •

⎠

V X

X∈Ob(C )

.

X∈Ob(C )

Let cV ∗ ,− be defined by (13.2.14). We claim that, for all X ∈ Ob(C ), cV ∗ ,X is an isomorphism in C with inverse given by X V∗

X V∗

 X V∗

•

 • cV−1∗ ,X

=

•

 •

h

ηV

=

•

•

=

h

h

ηV2

(13.2.10)

•

•

V∗ X

,

•

ηV





V∗ X

V∗ X

where, as the graphical notation suggests, the evaluation and coevaluation morphisms with a black dot are evV and coevV defined as in (13.1.6). Since ηV ∗ = (ηV )∗ , it follows that coevV can be restated as in (13.1.7). In fact, (13.2.9) still holds if we replace

with

. Specializing it for

•

f = IdV , we get that X V∗

 •

h

ηV2

cV ∗ ,X ◦ cV−1∗ ,X =

V∗

X

 2 ηVh

• •

(13.2.10)



=

(1.6.6)

•



•



(13.2.9)

= IdX⊗V ∗ ,

•

(13.2.10)

•

V∗

 2 ηVh

=

•

(∗)

X

•



•





X V∗

X V∗

X V∗

where (*) refers to the naturality of cV,− applied to cV ∗ ,X . Likewise, again using the naturality of cV,− applied to cV ∗ ,X , one can show that cV−1∗ ,X ◦ cV ∗ ,X = IdV ∗ ⊗X . Definition 13.16 If (C , c, η ) and (D, d, θ ) are ribbon categories then a functor F : C → D is called a ribbon functor if it is a braided monoidal functor such that F(ηV ) = θF(V ) , for all V ∈ Ob(C ), and is compatible with the duality.

13.2 Ribbon Categories Obtained from Rigid Monoidal Categories

495

The left-handed version of the Universal Property in Proposition 8.3 leads to a similar property for Rl (C ). Proposition 13.17 Let (D, d, θ ) be a ribbon category, C a left rigid monoidal category and F : D → C a strong monoidal functor which is bijective on objects and surjective on morphisms. Then there exists a unique ribbon functor Rl (F) : D → Rl (C ) such that Πl ◦ Rl (F) = F, where Πl : Rl (C ) → C is the forgetful functor. Proof If Π : Rl (C ) → Zl (C ) is the functor that forgets the twist then Π is a braided monoidal functor and Π ◦ Π = Πl , where Π : Zl (C ) → C is the strict monoidal functor that forgets the natural isomorphism. According to the left-handed version of Proposition 8.3, there exists a unique braided monoidal functor Zl (F) : D → Zl (C ) such that Π◦Zl (F) = F. Then Rl (F) : D → Rl (C ) given by Rl (F)(Y ) = (Zl (F)(Y ), F(θY )), for all Y ∈ D, and Rl (F)( f ) = F( f ), for any morphism f in D, is a well-defined functor. It follows that Rl (F) is a ribbon functor satisfying Πl ◦ Rl (F) = F; it is, moreover, unique with these properties because of the uniqueness of Zl (F) and of its ribbon property. Corollary 13.18 If (C , c, η ) is a ribbon category then there exists a unique ribbon functor Rl : C → Rl (C ) such that Πl ◦ Rl = IdC . Proof

In Proposition 13.17, take D = C and F = IdC . Then Rl = Rl (IdC ).

In what follows we also need the right-handed version of Rl (C ). In fact, if we start with a right rigid monoidal category then C is a left rigid monoidal category, and so we can consider Rl (C ). Thus Rr (C ) := Rl (C ) is a ribbon category, too. More precisely, we have the following. Proposition 13.19 Let C be a right rigid monoidal category. Then the objects of Rr (C ) are triples (V, c−,V , θV ) consisting of an object V of C , a natural isomorphism c−,V = (cX,V : X ⊗V → V ⊗ X)X∈Ob(C ) and a morphism θV : V → V in C , subject to the following conditions: • c1,V ≡ IdV and cX⊗Y,V = (cX,V ⊗ IdY )(IdX ⊗ cY,V ), for all X,Y ∈ C ; • θV is an automorphism of V in C obeying cX,V (IdX ⊗ θV ) = (θV ⊗ IdX )cX,V , for all X ∈ C , and −1 θV−2 := (IdV ⊗ evV )(cV,V ⊗ Id∗V )(IdV ⊗ c∗V,V )(IdV ⊗ coevV ).

(13.2.17)

A morphism f : (V, c−,V , θV ) → (W, c−,W , θW ) in Zr (C ) is a morphism f : V → W in C such that cX,W ( f ⊗ IdX ) = (IdX ⊗ f )cX,V , for all X ∈ C . The category Rr (C ) is ribbon via the following structure: • the tensor product of two objects (V, c−,V , θV ) and (W, c−,W , θW ) in Rr (C ) is (V ⊗ W, c−,V ⊗W , θV ⊗W ), where c−,V ⊗W is defined by (8.1.1) and θV ⊗W = (θV ⊗ θW )cW,V cV,W , and the tensor product of two morphisms in Rr (C ) is their tensor product in C , while the associativity and the left and right unit constraints are the same as those of C ;

496

Ribbon Quasi-Hopf Algebras

• the braiding between two objects (V, c−,V , θV ) and (W, c−,W , θW ) in Rr (C ) is given by cV,W ; • the right dual object of (V, c−,V , θV ) in Rr (C ) is (∗V, c−,∗V , θ∗V ) determined by −1 cX,∗V = (Id∗V ⊗X ⊗ evV )(IdV ∗ ⊗ cV,X ⊗ Id∗V )(coevV ⊗ IdX⊗∗V ),

for all X ∈ C , θ∗V = ∗ (θV ), and the evaluation and coevaluation morphisms equal evV and coevV , respectively; • the twist is given by θV : (V, c−,V , θV ) → (V, c−,V , θV ), for all V ∈ C . Proof As Rr (C ) := Rl (C ), everything follows by applying Corollary 13.9, after we specialize Proposition 13.14 and Theorem 13.15 to C .

13.3 Ribbon Quasi-Hopf Algebras This section can be viewed as an extension of Section 10.1, as we continue to investigate when the category of representations over a QT quasi-bialgebra is a balanced or a ribbon category. Let H be a quasi-bialgebra, so H M is monoidal. Then we have seen that H M fd is left rigid braided monoidal (and so right rigid as well; see Theorem 1.77), provided that H is a QT quasi-Hopf algebra. Furthermore, the converse is also true if we assume H is finite dimensional; see the results in Section 3.5. Hence, in the finitedimensional case, to see when H M fd is a balanced/ribbon category such that the forgetful functor F : H M fd → k M fd is a left rigid quasi-monoidal functor reduces to the following problem: for a finite-dimensional QT quasi-Hopf algebra (H, R) describe all the balanced structures/twists on H M fd . The answer to the above question is given by the result below. Proposition 13.20 Let (H, R) be a finite-dimensional QT quasi-Hopf algebra, so fd fd H M is a rigid braided category. Then H M is, moreover, (i) a balanced category if and only if there exists an invertible central element η ∈ H such that Δ(η ) = (η ⊗ η )R21 R,

(13.3.1)

where, if R = R1 ⊗ R2 then R21 = R2 ⊗ R1 ; (ii) a ribbon category if and only if there exists an invertible central element η ∈ H obeying (13.3.1) and such that S(η ) = η .

(13.3.2)

Proof Since H is finite dimensional we can regard H ∈ H M fd via its multiplication. Suppose that (ηV : V → V )V , indexed by V ∈ H M fd , defines a balanced structure on H M fd . Then ηH is completely determined by η := η (1H ). Actually, η describes ηV completely, for any V ∈ H M fd . To see this, take v ∈ V ∈ H M fd and define ϕv : H → V by ϕv (h) = h · v, for all h ∈ H. Since ϕv is left H-linear, by the naturality of

13.3 Ribbon Quasi-Hopf Algebras

497

the family (ηV : V → V )V we have ηV ◦ ϕv = ϕv ◦ ηH . By evaluating both sides of this relation on 1H we deduce that

ηV (v) = η · v, ∀ v ∈ V.

(13.3.3)

Since ηV is an isomorphism for any V ∈ H M fd it follows that η is invertible (this follows from the fact that ηH is a left H-linear isomorphism). Also, it can be easily seen that ηV is left H-linear, for all V ∈ H M fd , if and only if η is a central element of H. Furthermore, for all v ∈ V ∈ H M fd and w ∈ W ∈ H M fd we have (ηV ⊗ ηW ) ◦ cW,V ◦ cV,W (v ⊗ w) = (ηV ⊗ ηW )(r2 R1 · v ⊗ r1 R2 · w) = η r2 R1 · v ⊗ η r1 R2 · w and ηV ⊗W (v ⊗ w) = η · (v ⊗ w) = η1 · v ⊗ η2 · w. Thus (13.1.1) is satisfied for all V,W ∈ H M fd if and only if (13.3.1) holds (for the direct implication take v = w = 1H ∈ V = W = H). For the ribbon case, we have ηV ∗ (v∗ )(v) = (η · v∗ )(v) = v∗ (S(η ) · v), for all v∗ ∈ ∗ V , v ∈ V . By Proposition 3.34 the left transpose of ηV in H M fd coincides with the usual transpose map of ηV in k M fd , and so (ηV )∗ (v∗ )(v) = (v∗ ◦ ηV )(v) = v∗ (η · v). We conclude that (13.1.2) is equivalent to (13.3.2), and this ends the direct assertion in both (i) and (ii). Conversely, from the above computation it follows that (ηV )V , with ηV defined by an invertible central element η of H as in (13.3.3) that also obeys (13.3.1), provides a balanced structure on H M fd . If (13.3.2) is satisfied as well then (ηV )V is a twist on H M fd . The next definitions are imposed by the characterization that we have just proved in Proposition 13.20. Definition 13.21 (i) We call a QT quasi-bialgebra (resp. quasi-Hopf algebra) (H, R) a balanced quasi-bialgebra (resp. quasi-Hopf algebra) if there exists an invertible central element η ∈ H satisfying (13.3.1). (ii) A QT quasi-Hopf algebra (H, R) is called a ribbon quasi-Hopf algebra if there exists an invertible central element η ∈ H satisfying (13.3.1) and (13.3.2). By Theorem 3.38 and Proposition 13.20 we have the following consequence. Corollary 13.22 For a finite-dimensional k-algebra H there exists a bijective correspondence between • balanced/ribbon structures on H M fd such that the forgetful functor F : H M fd → fd k M is a left rigid quasi-monoidal functor; • balanced/ribbon quasi-Hopf algebra structures on H. According to Theorem 13.10, a ribbon category C is sovereign. When C = H M fd , with H a ribbon quasi-Hopf algebra, this implies the existence of an invertible element g ∈ H fulfilling (3.6.3) and (3.6.4); see Proposition 3.41. In other words, we can construct sovereign/pivotal elements for H out of ribbon elements. In what follows,

498

Ribbon Quasi-Hopf Algebras

we will give a concrete description for this correspondence. In particular, we will measure how far an element η as in Proposition 13.20 is from being u, the element considered in (10.3.4). To see all these, we need the following technical result. Lemma 13.23 Let (H, R) be a QT quasi-Hopf algebra and u the element defined in (10.3.4). Then Δ(u) = f −1 (S ⊗ S)( f21 )(u ⊗ u)(R21 R)−1 ,

(13.3.4)

where, as always, f = f 1 ⊗ f 2 is the element defined in (3.2.15) and f −1 is its inverse as in (3.2.16). 1

2

Proof We set R = R1 ⊗ R2 = r1 ⊗ r2 = R 1 ⊗ R 2 , R−1 = R ⊗ R = r1 ⊗ r2 , γ = γ 1 ⊗ γ 2 and δ = δ 1 ⊗ δ 2 as in (3.2.5) and (3.2.6), and f −1 = g1 ⊗ g2 . Then the formula in (13.3.4) is a consequence of the following technical computation: Δ(u) (10.3.4),(3.2.14)

=

(3.2.13)

=

(3.2.5),(3.1.9)

=

(3.2.1),(3.2.1) (10.1.2)

=

S(R2 p2 )1 g1 γ 1 R11 p11 ⊗ S(R2 p2 )2 g2 γ 2 R12 p12 g1 S(R22 p22 )γ 1 R11 p11 ⊗ g2 S(R21 p21 )γ 2 R12 p12 g1 S(x1 X 2 R22 p22 )α x2 (X 3 R1 )1 p11 ⊗ g2 S(X 1 R21 p21 )α x3 (X 3 R1 )2 p12 g1 S(x1 R2 X 3 y3 p22 )α x2 R11 X12 r11 y11 p11 ⊗ g2 S(X 1 r2 y2 p21 )α x3 R12 X22 r21 y12 p12

(10.1.1)

=

g1 S(R 2 x2 R2Y 3 X 3 y3 p22 )α R 1 x1Y 1 X12 r11 y11 p11 ⊗ g2 S(X 1 r2 y2 p21 )α x3 R1Y 2 X22 r21 y12 p12

(10.3.8)

=

g1 S(α x2 R2Y 3 X 3 y3 p22 )ux1Y 1 X12 r11 y11 p11 ⊗ g2 S(X 1 r2 y2 p21 )α x3 R1Y 2 X22 r21 y12 p12

(10.3.10)

=

g1 S(S(x1 )α x2 R2Y 3 X 3 y3 p22 )uY 1 X12 r11 y11 p11 ⊗ g2 S(X 1 r2 y2 p21 )α x3 R1Y 2 X22 r21 y12 p12

(3.2.14),(3.2.13)

=

f −1 (S ⊗ S)( f21 )[S(S(x1 )α x2 R2Y 3 X 3 y3 z22 δ 2 S(z31 ))u Y 1 X12 r11 y11 z11 ⊗ S(X 1 r2 y2 z21 δ 1 S(z32 ))α x3 R1Y 2 X22 r21 y12 z12 ]

(3.2.6),(3.1.9)

=

(3.2.1),(3.1.10) (3.1.9)

=

f −1 (S ⊗ S)( f21 )[S(S(x1 )α x2 R2Y 3 X 3 y3 z22t 2 Z 1 β S(z31t13 Z 2 ))u Y 1 X12 r11 y11 z11 ⊗ S(X 1 r2 y2 z21t 1 β S(z32t23 Z 3 ))α x3 R1Y 2 X22 r21 y12 z12 ] f −1 (S ⊗ S)( f21 )[S(S(x1 )α x2 R2Y 3 X 3 y2 z31 Z 1 β S(y31 z3(2,1) Z 2 ))uY 1 X12 (r1 y11 )1 z11 ⊗ S(X 1 r2 y12 z2 β S(y32 z3(2,2) Z 3 ))α x3 R1Y 2 X22 (r1 y11 )2 z12 ]

(3.1.7),(3.2.1)

=

(10.1.3) (3.1.9)

=

f −1 (S ⊗ S)( f21 )[S(S(x1 )α x2 R2Y 3 X 3 y2 Z 1 β S(y31 Z 2 ))uY 1 (X 2 y12 )1 r11 z11 ⊗ S(X 1 y11 r2 z2 β S(y32 Z 3 z3 ))α x3 R1Y 2 (X 2 y12 )2 r21 z12 ] f −1 (S ⊗ S)( f21 )[S(S(x1 )α x2 R2Y 3 y22t 2 X13 Z 1 β S((y3t 3 X23 )1 Z 2 ))uY 1 (y21t 1 X 2 r1 z1 )1 ⊗ S(y1 X 1 r2 z2 β S((y3t 3 X23 )2 Z 3 z3 ))α x3 R1Y 2 (y21t 1 X 2 r1 z1 )2 ]

13.3 Ribbon Quasi-Hopf Algebras (3.1.7),(3.2.1)

=

499

f −1 (S ⊗ S)( f21 )[S(S(x1 )α x2 R2 y2(2,2)Y 3t 2 Z 1 β S(y31t13 Z 2 ))uy21Y 1 t11 X12 r11 z11 ⊗ S(y1 X 1 r2 z2 β S(y32t23 Z 3 X 3 z3 ))α x3 R1 y2(2,1)Y 2t21 X22 r21 z12 ]

(10.1.3),(10.3.10)

=

f −1 (S ⊗ S)( f21 )[S(S(x1 y21 )α x2 y2(2,1) R2Y 3t 2 Z 1 β S(y31t13 Z 2 ))uY 1t11 X12 r11 z11 ⊗ S(y1 X 1 r2 z2 β S(y32t23 Z 3 X 3 z3 ))α x3 y2(2,2) R1Y 2t21 X22 r21 z12 ]

(3.1.7),(3.2.1)

=

(3.1.9)

f −1 (S ⊗ S)( f21 )[S(S(x1 )α x2 R2t22 w2Y13 Z 1 β S((y3t 3 w3Y23 )1 Z 2 ))u t 1Y 1 X12 r11 z11 ⊗ S(y1 X 1 r2 z2 β S((y3t 3 w3Y23 )2 Z 3 X 3 z3 ))α y2 x3 R1 t12 w1Y 2 X22 r21 z12 ]

(3.1.7),(3.2.1)

=

f −1 (S ⊗ S)( f21 )[S(S(x1 )α x2 R2t22 w2 Z 1 β S(y31t13 w31 Z 2 ))ut 1Y 1 X12 r11 z11 ⊗ S(y1 X 1 r2 z2 β S(y32t23 w32 Z 3Y 3 X 3 z3 ))α y2 x3 R1t12 w1Y 2 X22 r21 z12 ]

(3.1.9),(3.2.1)

=

(3.1.10),(10.3.10)

f −1 (S ⊗ S)( f21 )[S(S(x1t 1 )α x2 R2 (t 2 Z 1 )2 w2 β S(y31t13 Z 2 w3 ))uY 1 X12 r1 z11 ⊗ S(y1 X 1 r2 z2 β S(y32t23 Z 3Y 3 X 3 z3 ))α y2 x3 R1 (t 2 Z 1 )1 w1Y 2 X22 r21 z12 ]

(3.1.9),(10.1.3)

=

1 f −1 (S ⊗ S)( f21 )[S(S(x1 Z11t 1 v1 )α x2 Z(2,1) t12 R2 v2(1,2) w2 β

S(y31 Z 2t 3 v22 w3 ))uY 1 X12 r11 z11 ⊗ S(y1 X 1 r2 z2 β S(y32 Z 3 v3Y 3 X 3 z3 ))α 1 y2 x3 Z(2,2) t22 R1 v2(1,1) w1Y 2 X22 r21 z12 ] (3.1.7),(3.2.1)

=

f −1 (S ⊗ S)( f21 )[S(S(x1t 1 v1 )α x2t12 R2 w2 β S(y31 Z 2t 3 w3 ))uY 1 X12 r11 z11 ⊗ S(y1 X 1 r2 z2 β S(y32 Z 3 v3Y 3 X 3 z3 ))α y2 Z 1 x3t22 R1 w1 v2Y 2 X22 r21 z12 ]

(10.3.12)

=

f −1 (S ⊗ S)( f21 )[S(S(v1 )S(R )S(y31 Z 2 ))uY 1 X12 r11 z11 ⊗ S(y1 X 1 r2 2

1

z2 β S(y32 Z 3 v3Y 3 X 3 z3 ))α y2 Z 1 R v2Y 2 X22 r21 z12 ] (10.3.10),(10.1.3)

=

f −1 (S ⊗ S)( f21 )[uy31 Z 2 X22 R r11 z11 2

1

⊗ S(y1 X 1 r2 z2 β S(y32 Z 3 X 3 z3 ))α y2 Z 1 X12 R r21 z12 ] (3.1.9),(3.2.1)

=

(3.1.11),(10.1.3) (10.1.1)

=

f −1 (S ⊗ S)( f21 )[uy3 r21 z12 R

2 1

⊗ S(y1 r2 z2 β S(z3 ))α y2 r11 z11 R ] f −1 (S ⊗ S)( f21 )[ux3 r1Y 2 z12 R

2 1

⊗ S(R2 x2 r2Y 3 z2 β S(z3 ))α R1 x1Y 1 z11 R ] (3.1.9),(3.2.1)

=

(3.1.11),(10.3.8) (10.3.10),(10.1.3)

=

f −1 (S ⊗ S)( f21 )[ux3 r1 y21 z1 R

2 1

⊗ S(α x2 r2 y22 z2 β S(y3 z3 ))ux1 y1 R f −1 (S ⊗ S)( f21 )(u ⊗ u)[x3 y22 r1 z1 R

2

⊗ S−1 (S(x1 y1 )α x2 y21 r2 z2 β S(y3 z3 ))R ] 1

(10.3.12)

=

f −1 (S ⊗ S)( f21 )(u ⊗ u)r1 R ⊗ r2 R

=

f −1 (S ⊗ S)( f21 )(u ⊗ u)(R21 R)−1 ,

as stated.

2

1

500

Ribbon Quasi-Hopf Algebras

From the formula of Δ(u) we can derive other useful formulas. Corollary 13.24

In the hypotheses of Lemma 13.23 we have:

Δ(u) = (R21 R)−1 f −1 (S ⊗ S)( f21 )(u ⊗ u), Δ(S(u)) = (R21 R)

−1

(13.3.5)

−1 (S(u) ⊗ S(u))(S ⊗ S)( f21 )f,

(13.3.6)

−1

Δ(S(u) u) = f −1(S ⊗ S)( f21 )(S2 ⊗ S2 )( f −1 )(S3 ⊗S3 )( f21 )(S(u)−1 u⊗S(u)−1 u), (13.3.7) S4 (h) = (S(u)−1 u)h(S(u)−1 u)−1 , ∀ h ∈ H. Proof

(13.3.8)

Let us start by noting that (10.3.6) implies (S ⊗ S)(RR21 ) = f R21 R f −1 .

(13.3.9)

−1 . By applying S ⊗ S to By (13.3.9) we have (S ⊗ S)((R21 R)−1 ) = f21 (RR21 )−1 f21 both sides of this relation and taking into account (13.3.9) we deduce that −1 ) f (R21 R)−1 f −1 (S ⊗ S)( f21 ). (S2 ⊗ S2 )((R21 R)−1 ) = (S ⊗ S)( f21

This together with (10.3.10) and (13.3.4) guarantees the fact that Δ(u) = f −1 (S ⊗ S)( f21 )(S2 ⊗ S2 )((R21 R)−1 )(u ⊗ u) = (R21 R)−1 f −1 (S ⊗ S)( f21 )(u ⊗ u), proving (13.3.5). By using (13.3.9), (3.2.13) and (13.3.4), we compute: Δ(S(u))

=

f −1 (S ⊗ S)(Δcop (u)) f

=

−1 )f f −1 (S ⊗ S)((RR21 )−1 )(S(u) ⊗ S(u))(S2 ⊗ S2 )( f )(S ⊗ S)( f21

= = (10.3.10)

=

−1 )f (R21 R)−1 f −1 (S(u) ⊗ S(u))(S2 ⊗ S2 )( f )(S ⊗ S)( f21

 −1 1 −1 1 2 −1 2 −1 S(S( f )uS (g )) ⊗ S(S( f )uS (g )) (S ⊗ S)( f21 )f (R21 R) −1 (R21 R)−1 (S(u) ⊗ S(u))(S ⊗ S)( f21 )f,

and so (13.3.6) is proved. Now (13.3.7) follows easily from (13.3.6), (13.3.5) and (10.3.10); the details are left to the reader. Finally, for all h ∈ H, we have: (S(u)−1 u)h(S(u)−1 u)−1 = S(u)−1 S2 (h)S(u) = S(uS(h)u−1 ) = S4 (h). This finishes the proof of the corollary. We can prove the following characterization for ribbon quasi-Hopf algebras. Theorem 13.25 A QT quasi-Hopf algebra (H, R) is ribbon if and only if there exists a central element ν ∈ H such that

ν 2 = uS(u),

(13.3.10) −1

Δ(ν ) = (ν ⊗ ν )(R21 R) ,

(13.3.11)

S(ν ) = ν ,

(13.3.12)

where, as before, u is the element defined in (10.3.4).

13.3 Ribbon Quasi-Hopf Algebras

501

Proof Suppose that (H, R) is ribbon and let η ∈ H be an invertible central element such that (13.3.1) and (13.3.2) are satisfied. It follows that ν = η −1 is a central element in H, and it satisfies (13.3.11) and (13.3.12). To see that (13.3.10) is satisfied as well, observe first that by applying ε ⊗ ε to both sides of (13.3.1) we get ε (η ) = 1; see (10.1.5). Thus, by using (13.3.1) again and the fact that η is a central element in H, we deduce that

α

=

ε (η )α

=

S(η1 )αη2

=

S(η R2 r1 )αη R1 r2

(13.3.2)

η 2 S(R2 r1 )α R1 r2

(10.3.8)

η 2 S(α r1 )ur2

(10.3.10)

η 2 S(S(r2 )α r1 )u

(10.3.8)

η 2 S(S(α )u)u

(10.3.10)

η 2 S(u)uα .

= = = = =

This fact allows to compute, for all A, B ∈ H: Aα B

=

Aη 2 S(u)uα B

=

η 2 S(uS−1 (A))uα B

(10.3.10)

η 2 S(u)S2 (A)uα B

(10.3.10)

η 2 S(u)uAα B.

= =

In particular, by taking A ⊗ B = S(x1 ) ⊗ x2 β S(x3 ), by (3.2.2) we conclude that 1H = η 2 S(u)u, and therefore ν 2 = S(u)u, as needed. Note that S(u)u = uS(u) by Corollary 10.18. Conversely, let ν be a central element of H such that (13.3.10), (13.3.11) and (13.3.12) are fulfilled. Then ν is invertible because u is, and therefore η := ν −1 is an invertible central element of H. It follows easily that (13.3.2) and (13.3.1) are satisfied, so (H, R) is indeed a ribbon quasi-Hopf algebra. Definition 13.26 An element ν of a QT quasi-Hopf algebra (H, R) is called a quasi-ribbon element if ν satisfies the equations (13.3.10), (13.3.11) and (13.3.12). If, moreover, ν is central then it is called a ribbon element of H. Remark 13.27 As in the proof of Theorem 13.25 we get that any (quasi-)ribbon element ν of (H, R) also has the property that ε (ν ) = 1. In order to achieve our main goal we need one more result. For a quasi-Hopf algebra H denote −1 ) f }. G(H) = {l ∈ H | l is invertible with l −1 = S(l) and Δ(l) = (l ⊗ l)(S ⊗ S)( f21

For l ∈ G(H) one obtains ε (l) = 1, by applying ε ⊗ ε to both sides of the relation −1 )f. Δ(l) = (l ⊗ l)(S ⊗ S)( f21

502

Ribbon Quasi-Hopf Algebras

For a QT quasi-Hopf algebra (H, R) we prove that there exists a one-to-one correspondence between quasi-ribbon (or ribbon) elements of H and certain elements of G(H). Lemma 13.28 Let (H, R) be a QT quasi-Hopf algebra and u the element defined by (10.3.4). Suppose that ν is a quasi-ribbon element of H. If we set h¯ = u−1 S(u) and l = u−1 ν , then l 2 = h¯ and l ∈ G(H). Proof By (13.3.12) it follows that S2 (ν ) = ν . Hence, by applying (10.3.10), we obtain that u and ν commute. By (13.3.10) we have ν 2 = uS(u) = u2 h¯ . Thus l 2 = u−2 ν 2 = h¯ since u and ν commute. Because uS(u) = S(u)u is central we have S(l)l = S(u−1 ν )u−1 ν = S(ν )S(u−1 )u−1 ν = S(ν )(uS(u))−1 ν = (uS(u))−1 S(ν )ν . Now, by (13.3.10) and (13.3.12) it follows that S(l)l = 1H . By again using (13.3.10) and (13.3.12) we obtain that lS(l) = u−1 ν S(ν )S(u−1 ) = u−1 ν 2 S(u)−1 = u−1 uS(u)S(u)−1 = 1H . Hence, l is invertible and l −1 = S(l). By (13.3.11) and because u and ν commute, and by using also (13.3.4), it follows −1 ) f , and therefore l ∈ G(H). that Δ(l) = (l ⊗ l)(S ⊗ S)( f21 The next result says that any (quasi-)ribbon element of (H, R) comes from a deformation of u by a suitable element of G(H). It also says that H is sovereign via l = u−1 ν , if ν is a ribbon element for it. Theorem 13.29 Let (H, R) be a QT quasi-Hopf algebra and let u and h¯ be as in Lemma 13.28. Then the following hold. (a) l → ul defines a one-to-one correspondence {l ∈ G(H) | l 2 = h¯ } ←→ {quasi-ribbon elements of H}. (b) Suppose that l ∈ G(H) satisfies l 2 = h¯ . Then ν = ul is a ribbon element of H if and only if S2 (h) = l −1 hl, for all h ∈ H. Proof Let l ∈ G(H) with l 2 = h¯ . By Lemma 13.28, to prove part (a) we need only to show that ν = ul is a quasi-ribbon element of H. By S(l) = l −1 , we get S2 (l) = l, hence, by (10.3.10), it follows that u and l commute. Thus ν 2 = u2 l 2 = u2 h¯ = u2 u−1 S(u) = uS(u), so (13.3.10) holds for ν . To show that (13.3.12) holds for ν , we first note that h¯ is invertible with h¯ −1 = S(u)−1 u. Now l −1 = h¯ −1 l, which follows from the equation l 2 = h¯ . Therefore S(ν ) = S(lu) = S(u)S(l) = S(u)l −1 = S(u)¯h−1 l = S(u)S(u)−1 ul = ul = ν . Since l ∈ G(H) and u and ν commute (because u and l commute) it follows that Δ(uν −1 ) = Δ(l −1 ) = f −1 (S ⊗ S)( f21 )(l −1 ⊗ l −1 )

13.3 Ribbon Quasi-Hopf Algebras

503

= f −1 (S ⊗ S)( f21 )(uν −1 ⊗ uν −1 ), and thus (13.3.11) holds since Δ(ν )

=

Δ(ν u−1 )Δ(u)

=

−1 ) f Δ(u) (ν u−1 ⊗ ν u−1 )(S ⊗ S)( f21

(13.3.4)

=

(ν ⊗ ν )(R21 R)−1 ,

as needed. Since ν = ul is central if and only if l −1 hl = uhu−1 for all h ∈ H, part (b) follows by part (a) and (10.3.10). We end by showing that in the ribbon case the condition l −1 = S(l) in the definition of an element l ∈ G(H) is redundant. Corollary 13.30 Let (H, R) be a QT quasi-Hopf algebra and u the element defined in (10.3.4). If h¯ = u−1 S(u) then l → ul defines a one-to-one correspondence between −1 ) f and lS2 (h) = hl, ∀ h ∈ H} R(H) := {l ∈ H | l 2 = h¯ , Δ(l) = (l ⊗ l)(S ⊗ S)( f21

and ribbon elements of H. Proof

According to Theorem 13.29 we only have to prove that R(H) = {l ∈ G(H) | l 2 = h¯ and lS2 (h) = hl, ∀ h ∈ H}.

We will show this by double inclusion. By the definitions of R(H) and G(H) it follows that {l ∈ G(H) | l 2 = h¯ and lS2 (h) = hl, ∀ h ∈ H} ⊆ R(H). To show the converse inclusion it suffices to show that any element l ∈ R(H) is invertible and l −1 = S(l). Indeed, if l ∈ R(H) then by l 2 = h¯ = u−1 S(u) and from the fact that u is invertible it follows that l is invertible, too. We claim now that lS(l) = 1H , and this will end the proof. To prove the claim, observe that by applying ε ⊗ ε to both sides of Δ(l) = (l ⊗ −1 ) f we get ε (l) = ε (l)2 in k, and since l is invertible we deduce that l)(S ⊗ S)( f21 ε (l) = 1. By the same equality we have now that

β = ε (l)β = l1 β S(l2 ) = lS(g2 ) f 1 β S(lS(g1 ) f 2 ) (8.7.7)

(8.7.7)

= lS(g2 )S(α )S(lS(g1 )) = lS2 (β )S(l) = β lS(l),

where for the last equality we used the identity lS2 (h) = hl, applied to h = β . By using again the equality lS2 (h) = hl, valid for all h ∈ H, we compute that Aβ B = Aβ lS(l)B = Aβ lS(S−1 (B)l) = Aβ lS(lS(B)) = Aβ lS2 (B)S(l) = Aβ BlS(l), for all A ⊗ B ∈ H ⊗ H. By taking A ⊗ B = X 1 ⊗ S(X 2 )α X 3 in the above equality, by (3.2.2) we conclude that 1H = lS(l). So our proof is finished.

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Ribbon Quasi-Hopf Algebras

Corollary 13.31 If (H, R) is a unimodular QT quasi-Hopf algebra and u is as in (10.3.4) then l → ul defines a one-to-one correspondence between −1 ) f and lS2 (h) = hl, ∀ h ∈ H} {l ∈ H | l 2 = g , Δ(l) = (l ⊗ l)(S ⊗ S)( f21

and ribbon elements of H. Here g is the modular element of H; see (7.6.2). Proof When H is unimodular, that is, μ = ε , the formula in (11.6.5) gives the equality u−1 S(u)S(g) = 1H or, equivalently, u−1 S(u) = S(g−1 ). As uS(u) = S(u)u and S2 (u) = u this implies S2 (g−1 ) = S(u−1 S(u)) = S2 (u)S(u−1 ) = uS(u)−1 = S(u)−1 u = (u−1 S(u))−1 = S(g). By using the bijectivity of S, we obtain that S(g−1 ) = g, and so h¯ = u−1 S(u) = g. Now everything follows from Corollary 13.30. With the help of Corollary 13.30 we can compute all the (quasi-)ribbon elements of H(2)± , the QT quasi-Hopf algebras constructed in Example 10.7. 1 Example 13.32 H(2)± has exactly four ribbon elements: namely, v± 1 = 2 (1 ∓ ig), ± ± ± 1 ∓i ±i v2 = − 2 (1 ∓ ig), v3 = 2 (1 ± ig) and v4 = 2 (1 ± ig).

Proof We have seen in Example 3.44 that in H(2) we have (S ⊗ S)( f21 ) f = 1. Also, as S is the identity morphism of H(2), we have h¯ = u−1 ± S(u± ) = 1 in H(2). Therefore R(H(2)± ) = {l ∈ H(2) | l 2 = 1 and Δ(l) = l ⊗ l} = {−1, 1, g, −g}. By (10.3.8) and the commutativity of H(2) it follows that u± = S(R2± )R1± = R2± R1± = 1 − (1 ± i)p− =

1 (1 ∓ ig) . 2

Corollary 13.30 now says that lu± with l ∈ {−1, 1, g, −g} are all the (quasi-)ribbon elements of H(2)± , so we are done. We prove that a twisting of a QT quasi-Hopf algebra preserves ribbon elements. Proposition 13.33 Let (H, R) be a QT quasi-Hopf algebra, F ∈ H ⊗ H a gauge transformation for H and ν ∈ H a ribbon element for (H, R). Then ν is a ribbon element for (HF , RF ) as well. Proof Since u = uF by Proposition 10.19 and SF = S, the only thing we need to prove is that ΔF (ν ) = (ν ⊗ ν )((RF )21 RF )−1 (see Proposition 10.10 for the definition of RF ). One immediately sees that (RF )21 RF = F(R21 R)F −1 , so we can compute: (ν ⊗ ν )((RF )21 RF )−1

=

(ν ⊗ ν )F(R21 R)−1 F −1

=

F(ν ⊗ ν )(R21 R)−1 F −1

(13.3.11)

=

FΔ(ν )F −1 = ΔF (ν ),

where for the second equality we used the fact that ν is central.

13.4 A Class of Ribbon Quasi-Hopf Algebras

505

13.4 A Class of Ribbon Quasi-Hopf Algebras We have seen that to any finite-dimensional quasi-Hopf algebra we can associate a QT one, its quantum double. In this section we will go further by showing that to any QT quasi-bialgebra (resp. quasi-Hopf algebra) we can associate a balanced (resp. ribbon) one. We start with the quasi-bialgebra case. In what follows, for H a quasi-bialgebra, we denote by H[θ , θ −1 ] the free k-algebra generated by H and θ , with relations hθ = θ h, for all h ∈ H, and θ θ −1 = θ −1 θ = 1; by analogy with the commutative case, we call H[θ , θ −1 ] the Laurent polynomial algebra over H. Proposition 13.34 Let (H, R) be a QT quasi-bialgebra. Then H[θ , θ −1 ] is a balanced quasi-bialgebra with structure given by Δ |H = ΔH , Δ(θ ± ) = (R21 R)∓ (θ ± ⊗ θ ± ) ,

ε |H = εH , ε (θ ± ) = 1,

where, for simplicity, we denote θ = θ + and θ − = θ −1 , and similarly for (R21 R)± . Proof The only thing we have to prove is the quasi-coassociativity of Δ on θ . It will then follow that the natural inclusion of H into H[θ , θ −1 ] is a quasi-bialgebra morphism, and that H[θ , θ −1 ] is balanced via θ . Now, since Δ(h)R21 R = R21 RΔ(h), for all h ∈ H, the quasi-coassociativity of Δ on θ reduces to Φ(R21 R ⊗ 1H )(ΔH ⊗ IdH )(R21 R)Φ−1 = (1H ⊗ R21 R)(IdH ⊗ ΔH )(R21 R). With notation as in Remark 10.4(3), this can be restated as Φ(ΔH ⊗ IdH )(R21 R)R21 R12 Φ−1 = R32 R23 (IdH ⊗ ΔH )(R21 R). By using (10.1.1) and (10.1.2), the left-hand side of the above equality equals −1 Φ(Δ(R2 ) ⊗ R1 )(Δ(r1 ) ⊗ r2 )R21 R12 Φ−1 = R32 Φ132 R31 R13 Φ−1 132 R23 ΦR21 R12 Φ ,

while its right-hand side is equal to −1 R32 R23 (R2 ⊗ ΔH (R1 ))(r1 ⊗ ΔH (r2 )) = R32 R23 ΦR21 Φ−1 213 R31 R13 Φ213 R12 Φ .

Thus we must show that −1 Φ132 R31 R13 Φ−1 132 R23 ΦR21 = R23 ΦR21 Φ213 R31 R13 Φ213 .

If τ is the usual switch for the category of k-vector spaces, this follows from R23 ΦR21 Φ−1 213 R31 R13 Φ213 = (10.1.2)

(τ ⊗ IdH )(R13 Φ213 R12 Φ−1 )(τ ⊗ IdH )(1H ⊗ R21 R)Φ213

=

(τ ⊗ IdH )(Φ231 (IdH ⊗ ΔH )(R)(1H ⊗ R21 R))Φ231

=

Φ132 (τ ⊗ IdH )(1H ⊗ R21 R)(τ ⊗ IdH )(IdH ⊗ Δ(R))Φ213

(10.1.2)

=

−1 Φ132 R31 R13 (τ ⊗ IdH )(Φ−1 231 R13 Φ213 R12 Φ )Φ213

=

Φ132 R31 R13 Φ−1 132 R23 ΦR21 ,

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Ribbon Quasi-Hopf Algebras

as desired. Remark 13.35 Let (H, R) be a QT quasi-bialgebra and C = H M . Then B(C , c) and H[θ ,θ −1 ] M are isomorphic as balanced categories, where c is as in (10.1.4) and B(C , c) is as in the last part of Remark 13.4. Indeed, the desired isomorphism is produced by the following correspondence. To (V, ηV ) in B(C , c) we associate V regarded as a left H[θ , θ −1 ]-module via the Hmodule structure of it and θ ± · v = ηV∓ (v), for all v ∈ V . In this way a morphism in B(C , c) becomes a morphism in H[θ ,θ −1 ] M . In general, H[θ , θ −1 ] is not a quasi-Hopf algebra, and so not a ribbon quasi-Hopf algebra. To “make” it ribbon, we have to consider a quotient of it. Actually, we have θ] instead of H[θ , θ −1 ], where H[θ ] is the to consider the k-algebra H(θ ) := θ 2 H[ −uS(u) free k-algebra generated by H and θ with relations hθ = θ h, for all h ∈ H, and u is as in (10.3.4). In what follows, we still denote by θ the class of θ in H(θ ). Proposition 13.36 If (H, R) is a QT quasi-Hopf algebra then H(θ ) is a quasi-Hopf algebra with structure determined by Δ |H = ΔH , ε |H = εH , S |H = S, Δ(θ ) = (θ ⊗ θ )(R21 R)−1 ,

ε (θ ) = 1, S(θ ) = θ ,

and the reassociator and distinguished elements that define the antipode equal to those of H. Furthermore, (H(θ ), R) is QT and θ is a ribbon element for it. Proof As we have already pointed out, Δ(h)R21 R = R21 RΔ(h), for all h ∈ H. So by (13.3.5) and (13.3.6) one can see that −1 Δ(uS(u)) = f −1 (S ⊗ S)( f21 )(uS(u) ⊗ uS(u))(S ⊗ S)( f21 ) f (R21 R)−2 ,

and because uS(u) is central in H we get Δ(uS(u)) = (uS(u) ⊗ uS(u))(R21 R)−2 . This shows that Δ is well defined on θ . Also, it follows from Proposition 13.34 that H(θ ) is a quasi-bialgebra. Furthermore, since S(θ 2 − uS(u)) = S(θ )2 − S2 (u)S(u) = θ 2 − uS(u) we deduce that S is well defined on θ , too. So it remains to show the equalities S(θ1 )αθ2 = α

and θ1 β S(θ2 ) = β . 2

1

The equality in (10.3.8) can be rewritten as S(α R )uR = α or, equivalently, as 1 2 S(R )α R = S−1 (α u−1 ) = S(u−1 α ); see (10.3.10). Hence S(θ1 )αθ2

1

2

=

S(R r2 )α R r1 θ 2

=

S(u−1 α r2 )r1 θ 2

(10.3.10)

=

S(u−1 S(r1 )α r2 )θ 2

=

S2 (u−1 α )S(u−1 )θ 2

(10.3.10)

=

α u−1 S(u−1 )θ 2 = α ,

as required; in the last equality we used that uS(u) = S(u)u.

13.4 A Class of Ribbon Quasi-Hopf Algebras 1

507

2

Notice that the formula in (11.6.3) is equivalent to R S(R β u) = β , and therefore 2 1 to R β S(R ) = u−1 S(β ), because of (10.3.10). This gives us

θ1 β S(θ2 )

1

2

=

R r2 β S(R r1 )θ 2

=

R u−1 S(R β )θ 2 1

(10.3.10) −1

2

2

1

=

u S(R β S(R ))θ 2

=

u−1 S(u−1 S(β ))θ 2

(10.3.10) −1

=

u S(u−1 )θ 2 β = β ,

since uS(u) = S(u)u. Finally, R is an R-matrix for H(θ ) because it is for H and RΔ(θ ) = R(R21 R)−1 (θ ⊗ θ ) = R−1 21 (θ ⊗ θ ) = (RR21 )−1 R(θ ⊗ θ ) = Δcop (θ )R. Clearly θ is a ribbon element for (H(θ ), R), and this completes the proof. Our next goal is to show that the category of finite-dimensional left modules over H(θ ) can be identified with the category R(H M fd , c), where c is defined by (10.1.4). Theorem 13.37 Let (H, R) be a QT quasi-Hopf algebra, so H M fd is a rigid braided category. Then R(H M fd , c) identifies as a ribbon category with H(θ ) M fd . Proof Specializing Definition 13.12 for C = H M fd , we deduce that an object of the category R(H M fd , c) is a pair (V, ηV ) consisting of a finite-dimensional left Hmodule V and an H-automorphism ηV of V such that

η −2 (v)

=

X 1 r2 R1 β S(X 2 r1 R2 )α X 3 · v

=

q1 r2 R1 β S(q2 r1 R2 ) · v

(11.6.3)

S(q2 r1 β uS−1 (q1 r2 )) · v

(10.3.10)

S(q2 r1 β S(q1 r2 )u) · v

(11.6.3)

S(q2 S(q1 β u)u) · v

(10.3.10)

S(u)S2 (q1 β S(q2 )u) · v

= = = =

(3.2.2)

=

S(u)u · v = uS(u) · v,

for all v ∈ V . Thus objects of R(H M fd , c) are pairs (V, ηV ) consisting of a finitedimensional left H-module and an H-automorphism ηV of V satisfying η 2 (v) = (uS(u))−1 · v, for all v ∈ V . Clearly, a morphism f : (V, ηV ) → (W, ηW ) is a left H-linear morphism f : V → W such that ηW f = f ηV . Let F : R(H M fd , c) → H(θ ) M fd be the functor defined as follows: F((V, ηV )) = V regarded as an H(θ )-module via the H-action on V and θ · v = ηV−1 (v); F acts as identity on morphisms. We can easily see that θ 2 = uS(u) together with η −2 (v) = (uS(u))−1 · v, for all v ∈

508

Ribbon Quasi-Hopf Algebras

V , implies that F is a well-defined functor. It provides an isomorphism of categories, its inverse being the functor G : H(θ ) M fd → R(H M fd , c) given by G(V ) = (V, ηV : V v → θ −1 · v ∈ V ), for all V ∈ H(θ ) M fd . The functor F is monoidal since

θ · (v ⊗ w) = ηV−1 ⊗W (v ⊗ w)

= (R21 R)−1 (ηV−1 (v) ⊗ ηW−1 (w)) = (R21 R)−1 (θ · v ⊗ θ · w) = θ1 · v ⊗ θ2 · w,

for all V,W ∈ R(H M fd , c), v ∈ V and w ∈ W . Furthermore, F is braided because for both categories the braiding is defined by c.

of H(θ ) M fd is induced by the element θ −1 , in the sense The ribbon structure η −1

V (v) = θ · v = ηV (v), for all v ∈ V ∈ H(θ ) M fd . In other words, the two that η categories have the same ribbon structure, and therefore F is a ribbon isomorphism functor. Observe also that F is compatible with the left rigid monoidal structures on R(H M fd , c) and H(θ ) M fd . More precisely, we have F((V ∗ , ηV ∗ )) = V ∗ with left ∗ H(θ )-module structure induced by that of the left H-module and θ · v∗ = ηV−1 ∗ (v ) = (ηV−1 )∗ (v∗ ) = v∗ ◦ ηV−1 = v∗ (θ ·) = v∗ (S(θ )·), for all v∗ ∈ V ∗ , as required. Remark 13.38 As in the proof of Theorem 11.30, if (H, R) is QT then R˜ = R−1 21 = 2 1 R ⊗R is another R-matrix for H. In addition, if u˜ is the element (10.3.4) correspond˜ then we have seen that u˜ = S(u−1 ), so uS( ˜ u) ˜ = (uS(u))−1 . Therefore, ing to (H, R) ˜ is a ribbon category that if c˜ is the braiding on H M fd defined by R˜ then R(H M fd , c) ˜ is isomorphic to H(θ ) M fd , too. To see this, observe that an object of R(H M fd , c) is a pair (V, θV ) consisting of V ∈ H M fd and an automorphism θV of the left Hmodule V such that θV2 (v) = uS(u) · v, for all v ∈ V . It is clear at this point that (V, ηV ) → (V, θV := ηV−1 ) defines the desired isomorphism of categories. Corollary 13.39 Let H be a finite-dimensional quasi-Hopf algebra and D(H) its quantum double. If D(H, θ ) := D(H)(θ ) then Rr (H M fd ) ∼ = R(H YD Hfd , c) ∼ = D(H,θ ) M fd as ribbon categories, where c is the braiding defined in (8.2.18). Proof The first isomorphism can be deduced from the definition Rr (H M fd ) = R(Zr (H M fd ), c) and the braided isomorphism between Zr (H M fd ) and H YD Hfd induced by Theorem 8.8 and Proposition 8.12. The second one follows from the braided isomorphism (H YD Hfd , c) ∼ = (D(H) M fd , RD ) established in Theorem 10.20, and Theorem 13.37.

13.5 Some Ribbon Elements for Dω (H) and Dω (G) Let H be a finite-dimensional cocommutative Hopf algebra, ω a normalized 3-cocycle on H and Dω (H) the quasi-Hopf algebra constructed in Section 8.6. Recall from

13.5 Some Ribbon Elements for Dω (H) and Dω (G)

509

Proposition 10.22 that Dω (H) is a QT quasi-Hopf algebra isomorphic to the quantum double D(Hω∗ ). In what follows, in order to avoid any confusion, we denote by μH ∈ H ∗ and gH ∈ H the modular elements of H as a Hopf algebra. Similar notation, μHω∗ ∈ H and gH ∗ ∈ H ∗ , is used for the modular elements of the quasi-Hopf algebra Hω∗ . As H is ω cocommutative it follows that gH = μHω∗ = 1H , that is, H ∗ and Hω∗ are unimodular as Hopf and quasi-Hopf algebras, respectively. Also, it is clear that a left (and at the same time right) integral in the quasi-Hopf algebra Hω∗ is nothing but a left (and at the same time right) integral on the Hopf algebra H, that is, an element λ ∈ H ∗ obeying λ (h)h = λ (h)1H , for all h ∈ H. In this section we show that the element ν = u(ζ #1H )βDω (H) = u(ζ β #1H ) is a ribbon element for Dω (H), provided that ζ : H → k is an algebra map such that ζ 2 = μH ; here, as before, u is the element in (10.3.4) corresponding to Dω (H) and the other notation is as in Section 8.6. Note that when dimk H = 0 in k or H is unimodular we can take ζ = ε , and therefore Dω (H) is ribbon with ribbon element ν = u(β #1H ). This applies for instance to a finite-dimensional Hopf group algebra H = k[G], and so Dω (G) is always a ribbon quasi-Hopf algebra. To this end, we start by describing the space of left cointegrals for Hω∗ . Lemma 13.40 Let t ∈ H be a non-zero left integral in H, that is, ht = ε (h)t, for all h ∈ H. Then t := β (S(t))t ∈ H is a non-zero left cointegral for Hω∗ . Proof Let λ ∈ H ∗ be a non-zero (left and right) integral for Hω∗ . As α , β are invertible in Hω∗ , by the comments made before Example 7.53 we have that a non-zero left cointegral for Hω∗ is a non-zero element t ∈ H satisfying λ (ht) = λ (t)β (h), for all h ∈ H. Since the antipode of H is bijective, by using the Hopf version of the isomorphism (7.2.11) we find a unique element h∗ ∈ H ∗ such that h∗ (t)t = t; in particular, h∗ (t) = 0. We have, for all h ∈ H, that h∗ (t)λ (ht) = h∗ (t)λ (t)β (h), which is equivalent to h∗ (S(h)ht)λ (ht) = λ (t)h∗ (1H )β (h), which in turn is equivalent to λ (t)h∗ (S(h)) = λ (t)h∗ (1H )β (h). As λ (t) = 0, it follows that h∗ = h∗ (1H )β ◦ S, which implies β (S(t)) = 0. Therefore, by rescaling, we can assume without loss of generality that t = β (S(t))t, as stated. Conversely, t = β (S(t))t is a left cointegral for Hω∗ since

λ (ht) = β (S(t))λ (ht) = β (S(ht)h)λ (ht) = β (h)λ (ht) = β (h)λ (t), and this is equal to λ (t)β (h) because

λ (t)β (h) = β (S(t))λ (t)β (h) = λ (t)β (S(1H ))β (h) = λ (t)β (h). So our proof ends. Corollary 13.41

We have that gH ∗ = β 2 μH . Consequently, the modular element ω

 β 2 μH . gD(H ∗ ) of the quantum double D(Hω∗ ) equals 1H  ω

510

Ribbon Quasi-Hopf Algebras

Proof Recall that μH is defined by th = μH (h)t, for all h ∈ H and t ∈ H a non-zero left integral. Also, since H ∗ is unimodular we have λ ◦ S = λ . Thus, by specializing (7.6.2) to Hω∗ we compute, for all h ∈ H, that gH ∗ = q2 (S(t))p2 (S(t))λ (S(th))q1 (S(h))p1 (S(h)) ω

= ω (S(h), S(t), t)β (S(t))ω −1 (S(h), t, S(t))λ (S(th)) = ω (S(h), hS(th), thS(h))β (hS(th))ω −1 (S(h), thS(h), hS(th))λ (th) = ω (S(h), h, S(h))β (h)ω −1 (S(h), h, S(h))λ (th) = β (h)β (S(t))λ (th) = β (h)β (hS(th))λ (th) = β (h)2 μH (h)λ (t).

But the pair (λ , t) obeys λ (t) = 1 or, equivalently, β (S(t))λ (t) = 1. The latter is equivalent to λ (t) = 1, and so gH ∗ = β 2 μH , as desired. ω Finally, we have μHω∗ = 1H , hence the formula in Proposition 12.12 yields  g−1 ◦ S−3 = 1H   g−1 ◦ S = 1H   gH ∗ , gD(H ∗ ) = 1H  H∗ H∗ ω

ω

ω

ω

since g−1 = β −2 μH−1 with β −1 = β ◦ S and μH−1 = μH ◦ S, and combined with S2 = H∗ ω

◦ S = gH ∗ . IdH this gives g−1 H∗ ω

ω

Another preliminary result that we need is the following. Lemma 13.42 The distinguished element βDω (H) = β #1H ∈ Dω (H) satisfies Δ(βDω (H) ) = (βDω (H) ⊗ βDω (H) )(s ⊗ s)(f−1 21 )f,

(13.5.1)

where f ∈ Dω (H) ⊗ Dω (H) is the Drinfeld element. Consequently, the same relation is satisfied by any element of the form ζ β #1H ∈ Dω (H), provided that ζ : H → k is an algebra map. Proof

By (3.2.14), the relation in (13.5.1) is equivalent to

δDω (H) (s ⊗ s)(f21 ) = βDω (H) ⊗ βDω (H) ,

(13.5.2)

where δDω (H) is the element defined in (3.2.6), specialized for the quasi-Hopf algebra Dω (H). To prove this relation we proceed as follows. Since for Dω (H) we have αDω (H) = ε #1H , the unit of Dω (H), by the relation (3.2.14) we get γDω (H) = f. It follows that f and δ := δDω (H) can be written as: f = ( f1 #1H ) ⊗ ( f2 #1H ),

δ = (δ1 #1H ) ⊗ (δ2 #1H ),

(13.5.3)

with f1 (x) f2 (y) = ω (S(x), x, y)ω −1 (S(y), S(x), xy),

δ1 (x)δ2 (y) = β (x)β (y)ω (x, y, S(xy))ω

−1

(y, S(y), S(x)),

(13.5.4) (13.5.5)

13.5 Some Ribbon Elements for Dω (H) and Dω (G)

511

for all x, y ∈ H. We can now see that

δ1 S( f2 )(x)δ2 S( f1 )(y) = δ1 (x) f2 (S(x))δ2 (y) f1 (S(y)) (13.5.4)

=

(13.5.5)

=

β (x)β (y)ω (x, y, S(xy))ω −1 (y, S(y), S(x)) ω (y, S(y), S(x))ω −1 (x, y, S(xy)) β (x)β (y),

for all x, y ∈ H, and so (13.5.2) is proved. The second assertion in the statement follows easily from the fact that Δ(ζ #1H ) = ζ #1H ⊗ ζ #1H and from the fact that the comultiplication of Dω (H) is multiplicative. We now have all the necessary ingredients in order to prove the following: Theorem 13.43 Let H be a finite-dimensional cocommutative Hopf algebra H, ω a normalized 3-cocycle on H and ζ : H → k an algebra map. Then the element ν = u(ζ β #1H ) is ribbon if and only if ζ 2 = μH . Proof The quasi-Hopf algebra Dω (H) is unimodular; this follows from Theorems 8.34 and 12.6. Furthermore, by Corollary 13.41 and the definition of the isomorphism w in (8.6.7) we can see that the modular element gDω (H) of Dω (H) is gDω (H) = w(gD(H ∗ ) ) = gD(H ∗ ) #1H = β 2 μH #1H . ω

ω

So, according to Corollary 13.31, it suffices to see when the element l := ζ β #1H ∈ Dω (H) satisfies the relations l2 = β 2 μH #1H ,

(13.5.6)

Δ(l) = (l ⊗ l)(s ⊗ s)(f−1 21 )f,

(13.5.7) ω

lS (ϕ #h) = (ϕ #h)l , ∀ ϕ #h ∈ D (H). 2

(13.5.8)

It can be easily checked that (13.5.6) is equivalent to ζ 2 β 2 = β 2 μH , and the latter is equivalent to ζ 2 = μH , because H ∗ is commutative and β is convolution invertible. Also, Lemma 13.42 guarantees that (13.5.7) is always satisfied. We look at (13.5.8). By the last paragraph of the proof of Theorem 8.35 we have that s2 (ϕ #h) = (β −1 #1H )(ϕ #h)(β #1H ), for all ϕ #h ∈ Dω (H). Hence (13.5.8) is equivalent to (ζ #1H )(ϕ #h) = (ϕ #h)(ζ #1H ), for all ϕ #h ∈ Dω (H). The last equation becomes ζ ϕ #h = ϕ (h  ζ  S(h))#h, and it holds for any ϕ #h ∈ Dω (H) since ζ is an algebra map and H ∗ is commutative. We end this section with some concrete examples. By G(H ∗ ) := {ζ : H → k | ζ is an algebra map} we denote the set of grouplike elements of H ∗ , assuming, as before, that H is a finitedimensional cocommutative Hopf algebra. G(H ∗ ) is a group under convolution, and so the group Hopf algebra k[G(H ∗ )] is a Hopf subalgebra of H ∗ . By the Hopf algebraic version of the freeness theorem proved in Section 7.7 it follows that | G(H ∗ ) | divides dimk (H).

512

Ribbon Quasi-Hopf Algebras

Example 13.44 Assume that μH has odd order in G(H ∗ ). Then Dω (H) is a ribbon quasi-Hopf algebra. Proof If μH has order 2m + 1 in G(H ∗ ) then ζ := μHm+1 : H → k is an algebra map such that ζ 2 = μH . By Theorem 13.43 we obtain that u(μHm+1 β #1H ) is a ribbon element for Dω (H). Example 13.45 Suppose that either dimk (H) or |G(H ∗ )| is an odd number. Then Dω (H) is a ribbon quasi-Hopf algebra. Proof Since |G(H ∗ )| divides dimk (H), in either case we get that μH has odd order, and so Example 13.44 applies. Example 13.46 The element ν = u(β #1H ) is a ribbon element for Dω (H) if and only if H is unimodular. Proof Take ζ = ε in Theorem 13.43; we obtain that ν = u(β #1H ) is a ribbon element if and only if μH = ε , i.e. H is unimodular. Example 13.47

The quasi-Hopf algebra Dω (G) is ribbon.

Proof We have Dω (G) = Dω (k[G]) and k[G] is unimodular. Indeed, t = ∑g∈G g is a left and right integral in k[G]. Example 13.48 Suppose that dimk (H) = 0 in k (this happens for instance when k is of characteristic zero). Then ν = u(β #1H ) is a ribbon element for Dω (H). Proof By part (ii) of Theorem 12.17, specialized to the Hopf algebra H, we get that H is semisimple, and so unimodular, too. It then follows that ν = u(β #1H ) is a ribbon element for Dω (H).

13.6 Notes Ribbon (quasi-)Hopf algebras were introduced for topological reasons: they give rise to a topological invariant of knots and links in the 3-sphere; see [191]. From the algebraic point of view, ribbon quasi-Hopf algebras are Hopf-like objects defined by algebras for which their categories of representations are ribbon in the sense of Turaev [213], that is, are braided and have a twist which makes the braiding involutive (the balanced property) and at the same time relates it to duality. Note that ribbon categories were also introduced by Joyal and Street [120, 121] but under the name of tortile categories. That rigid braided balanced categories are actually braided sovereign categories was proved by Deligne [74]; see also [149, 216]. Kassel and Turaev [128] associate to any rigid monoidal category a ribbon category and specialize this construction to the category of representations of a Hopf algebra. The slightly more general construction in Section 13.2 was done in [65], and has the advantage that in the quasi-Hopf case it

13.6 Notes

513

leads to a more conceptual and less computational proof for the ribbon isomorphisms in Corollary 13.39. Note that, following the idea of Kassel and Turaev, balanced categories from monoidal ones were obtained by Drabant in [78]; his construction generalizes in the spirit of [65]; see Proposition 13.3 and Remark 13.4. When (H, R) is a ribbon quasi-Hopf algebra, the element responsible for the fact that H M fd is sovereign was found in [56, 65, 204], by using the ideas of Kauffman and Radford in [130]; this is the content of Section 13.3. Section 13.4 uses the ideas in [191, 78] to construct H[θ , θ −1 ] and H(θ ) in the quasi-Hopf setting. Proposition 13.33 is from [137]. Finally, the particular ribbon elements for Dω (H) presented in Section 13.5 are taken from [57], which had as main sources of inspiration [56] and [130].

Bibliography

[1] Abe, E. 1977. Hopf Algebras. Cambridge: Cambridge University Press. [2] Aguiar, M. and Mahajan, S. 2010. Monoidal functors, species and Hopf algebras. CRM Monograph Series 29, Providence, RI: American Mathematical Society. [3] Akrami, S. E. and Majid, S. 2004. Braided cyclic cocycles and nonassociative geometry. J. Math. Phys., 45 (10), 3883–3911. [4] Albuquerque, H., Elduque, A. and P´erez-Izquierdo, J. M. 2002. Z2 -quasialgebras. Comm. Algebra, 30 (2), 2161–2174. [5] Albuquerque, H. and Majid, S. 1999. Zn -quasialgebras. In A. P. Santana, A. L. Duarte and J. F. Queiro (eds), Matrices and Group Representations (Coimbra, 1998). Textos Mat. S´er. B 19, Coimbra: Universidade de Coimbra, pp. 57–64. [6] Albuquerque, H. and Majid, S. 1999. Quasialgebra structure of the octonions. J. Algebra, 220, 247–264. [7] Albuquerque, H. and Majid, S. 2002. Clifford algebras obtained by twisting of group algebras. J. Pure Appl. Algebra, 171, 133–148. [8] Albuquerque, H. and Panaite, F. 2009. On quasi-Hopf smash products and twisted tensor products of quasialgebras. Algebr. Represent. Theory, 12, 199–234. [9] Altschuler, D. and Coste, A. 1992. Quasi-quantum groups, knots, three manifolds and topological field theory. Comm. Math. Phys., 150, 83–107. [10] Anderson, F. W. and Fuller, K. R. 1992. Rings and Categories of Modules. Graduate Texts in Mathematics 13, 2nd edition. New York: Springer-Verlag. [11] Andruskiewitsch, N. and D˘asc˘alescu, S. 2003. Co-Frobenius Hopf algebras and the coradical filtration. Math. Z., 243 (1), 145–154. [12] Andruskiewitsch, N. and Enriquez, B. 1992. Examples of compact matrix pseudogroups arising from the twisting operation. Comm. Math. Phys., 149 (2), 195–207. [13] Angiono, I. E. 2010. Basic quasi-Hopf algebras over cyclic groups. Adv. Math., 225 (6), 3545–3575. [14] Angiono, I. E., Ardizzoni, A. and Menini, C. 2017. Cohomology and coquasibialgebras in the category of Yetter-Drinfeld modules. Ann. Sc. Norm. Super. Pisa Cl. Sci., 17 (2), 609–653. [15] Ardizzoni, A., Beattie, M. and Menini, C. 2015. Quantum lines for dual quasibialgebras. Algebr. Represent. Theory, 18 (1), 35–64. [16] Ardizzoni, A., Bulacu, D. and Menini, C. 2013. Quasi-bialgebra structures and torsionfree abelian groups. Bull. Math. Soc. Sci. Math. Roumanie (N.S.), 56(104), 247–265. [17] Ardizzoni, A., El Kaoutit, L. and Saracco, P. 2016. Functorial constructions for nonassociative algebras with applications to quasi-bialgebras. J. Algebra, 449, 460–496.

516

Bibliography

[18] Ardizzoni, A., Menini, C. and S¸tefan, D. 2007. A monoidal approach to splitting morphisms of bialgebras. Trans. Amer. Math. Soc., 359 (3), 991–1044. [19] Ardizzoni, A. and Pavarin, A. 2012. Preantipodes for dual quasi-bialgebras. Israel J. Math., 192 (1), 281–295. [20] Ardizzoni, A. and Pavarin, A. 2013. Bosonization for dual quasi-bialgebras and preantipode. J. Algebra, 390, 126–159. [21] Babelon, O., Bernard, D. and Billey, E. 1996. A quasi-Hopf algebra interpretation of quantum 3-j and 6-j symbols and difference equations. Phys. Lett. B, 375 (1-4), 89–97. [22] Bagheri, S. 2014. Adjunctions of Hom and tensor as endofunctors of (bi-)module categories over quasi-Hopf algebras. Comm. Algebra, 42 (2), 488–510. [23] Bagheri, S. and Wisbauer, R. 2012. Hom-tensor relations for two-sided Hopf modules over quasi-Hopf algebras. Comm. Algebra, 40 (9), 3257–3287. [24] Balan, A. 2009. A Morita context and Galois extensions for quasi-Hopf algebras. Comm. Algebra, 37 (4), 1129–1150. [25] Balan, A. 2010. Galois extensions for coquasi-Hopf algebras. Comm. Algebra, 38 (4), 1491–1525. [26] Balodi, M, Huang, H.-L. and Kumar, S. D. 2017. On the classification of finite quasiquantum groups. Rev. Math. Phys., 29 (10), 1730003, 20pp. [27] Beattie, M., Bulacu, D. and Torrecillas, B. 2007. Radford’s S4 formula for co-Frobenius Hopf algebras. J. Algebra, 307 (1), 330–342. [28] Beattie, M., D˘asc˘alescu, S. and Gr¨unenfelder, L. 1999. On the number of types of finite-dimensional Hopf algebras. Invent. Math., 136 (1), 1–7. [29] Beattie, M., Iovanov, M. and Raianu, S¸. 2009. The antipode of a dual quasi-Hopf algebra with nonzero integrals is bijective. Algebr. Represent. Theory, 12 (2-5), 251–255. [30] B´enabou, J. 1963. Cat´egories avec multiplication. C. R. Acad. Sci. Paris S´er. I Math., 256, 1887–1890. [31] Bespalov, Y. and Drabant, B. 1998. Hopf (bi-)modules and crossed modules in braided monoidal categories. J. Pure Appl. Algebra, 123, 105–129. [32] Bespalov, Y. and Drabant, B. 1999. Cross product bialgebras I. J. Algebra, 219, 466–505. [33] Bespalov, Y. and Drabant, B. 1999. Cross product bialgebras II. J. Algebra, 240, 445–504. [34] Boboc, C., D˘asc˘alescu, S. and Van Wyk, L. 2012. Isomorphisms between Morita context rings. Linear & Multilinear Algebra, 60 (5), 545–563. [35] B¨ohm, G. and Lack, S. 2016. Hopf comonads on naturally Frobenius map-monoidales. J. Pure Appl. Algebra, 220 (6), 2177–2213. [36] B¨ohm, G. and S¸tefan, D. 2012. A categorical approach to cyclic duality. J. Noncommut. Geom., 6 (3), 481–538. [37] Brauer, R. and Nesbitt, C. 1937. On the regular representations of algebras. Proc. Nat. Acad. Sci. USA, 23, 236–240. [38] Brzezi´nski, T. 2005. Galois comodules. J. Algebra, 290 (2), 503–537. [39] Brzezi´nski, T. and Majid, S. 2000. Quantum geometry of algebra factorisations and coalgebra bundles. Comm. Math. Phys., 213 (3), 491–521. [40] Brzezi´nski, T., Marquez, A. V. and Vercruysse, J. 2011. The Eilenberg-Moore category and a Beck-type theorem for a Morita context. Appl. Categ. Structures, 19 (5), 821–858. [41] Bulacu, D. 1999. On the antipode of semi-Hopf algebras and braided semi-Hopf algebras. Rev. Roum. Math. Pures Appl., 44, 329–340. [42] Bulacu, D. 2009. The weak braided Hopf algebra structure of some Cayley-Dickson algebras. J. Algebra, 322, 2404–2427.

Bibliography

517

[43] Bulacu, D. 2011. A Clifford algebra is a weak Hopf algebra in a suitable symmetric monoidal category. J. Algebra, 332, 244–284. [44] Bulacu, D. 2017. A structure theorem for quasi-Hopf bimodule coalgebras. Theory Appl. Categ. (TAC), 32 (1), 1–30. [45] Bulacu, D. 2018. Quasi-quantum groups obtained from tensor braided Hopf algebras. Preprint, submitted. [46] Bulacu, D. and Caenepeel, S. 2003. Two-sided two-cosided Hopf modules and DoiHopf modules for quasi-Hopf algebras. J. Algebra, 270, 55–95. [47] Bulacu, D. and Caenepeel, S. 2003. Integrals for (dual) quasi-Hopf algebras: Applications. J. Algebra, 266, 552–583. [48] Bulacu, D. and Caenepeel, S. 2003. The quantum double for quasitriangular quasi-Hopf algebras. Comm. Algebra, 31 (3), 1403–1425. [49] Bulacu, D. and Caenepeel, S. 2012. On integrals and cointegrals for quasi-Hopf algebras. J. Algebra, 351, 390–425. [50] Bulacu, D., Caenepeel, S. and Panaite, F. 2005. More properties of Yetter-Drinfeld modules over quasi-Hopf algebras, in S. Caenepeel and F. Van Oystaeyen (eds.), Hopf Algebras in Noncommutative Geometry and Physics. Lecture Notes in Pure and Applied Mathematics 239. New York: Marcel Dekker, pp. 89–112. [51] Bulacu, D., Caenepeel, S. and Panaite, F. 2006. Yetter-Drinfeld categories for quasiHopf algebras. Comm. Algebra, 34, 1–35. [52] Bulacu, D., Caenepeel, S. and Torrecillas, B. 2009. Involutory quasi-Hopf algebras. Algebr. Represent. Theory, 12, 257–285. [53] Bulacu, D., Caenepeel, S. and Torrecillas, B. 2011. The braided monoidal structures on the category of vector spaces graded by the Klein group. Proc. Edinburgh Math. Soc., 54, 613–641. [54] Bulacu, D. and Nauwelaerts, E. 2002. Radford’s biproduct for quasi-Hopf algebras and bosonization. J. Pure Appl. Algebra, 174, 1–42. [55] Bulacu D. and Nauwelaerts, E. 2003. Quasitriangular and ribbon quasi-Hopf algebras. Comm. Algebra, 31 (2), 1–16. [56] Bulacu, D. and Panaite, F. 1998. A generalization of the quasi-Hopf algebra Dω (G). Comm. Algebra, 26, 4125–4141. [57] Bulacu, D. and Panaite, F. 2018. Some ribbon elements for the quasi-Hopf algebra Dω (H). Preprint, submitted. [58] Bulacu, D., Panaite, F. and Van Oystaeyen, F. 1999. Quantum traces and quantum dimensions for quasi-Hopf algebras. Comm. Algebra, 27, 6103–6122. [59] Bulacu, D., Panaite, F. and Van Oystaeyen, F. 2000. Quasi-Hopf algebra actions and smash products. Comm. Algebra, 28, 631–651. [60] Bulacu, D., Panaite, F. and Van Oystaeyen, F. 2006. Generalized diagonal crossed products and smash products for quasi-Hopf algebras. Applications. Comm. Math. Phys., 266, 355–399. [61] Bulacu, D. and Torrecillas, B. 2004. Factorizable quasi-Hopf algebras: Applications. J. Pure Appl. Algebra, 194 (1-2), 39–84. [62] Bulacu, D. and Torrecillas, B. 2008. The representation-theoretic rank of the doubles of quasi-quantum groups. J. Pure Appl. Algebra, 212, 919–940. [63] Bulacu, D. and Torrecillas, B. 2006. Two-sided two-cosided Hopf modules and YetterDrinfeld modules for quasi-Hopf algebras. Appl. Categor. Structures, 28, 503–530. [64] Bulacu, D. and Torrecillas, B. 2018. Quasi-quantum groups obtained from the TannakaKrein reconstruction theorem. Preprint, submitted. [65] Bulacu, D. and Torrecillas, B. 2018. On sovereign, balanced and ribbon quasi-Hopf algebras. Preprint, submitted.

518

Bibliography

[66] Burciu, S. and Natale, S. 2013. Fusion rules of equivariantizations of fusion categories. J. Math. Phys., 54 (1), 013511, 21pp. [67] Caenepeel, S., Ion, B., Militaru, G. and Zhu, S. 2000. The factorization problem and the smash biproduct of algebras and coalgebras. Algebr. Represent. Theory, 3, 19–42. [68] Caenepeel, S., Militaru, G. and Zhu, S. 2002. Frobenius and Separable Functors for Generalized Module Categories and Nonlinear Equations. Lecture Notes in Mathematics 1787. Berlin: Springer-Verlag. [69] Cap, A., Schichl, H. and Vanˇzura, J. 1995. On twisted tensor products of algebras. Comm. Algebra, 23, 4701–4735. [70] Curtis, C. W. and Reiner, I. 1962. Representation Theory of Finite Groups and Associative Algebras. Pure and Applied Mathematics 11. London–New York: Interscience. [71] D˘asc˘alescu, S. 2008. Group gradings on diagonal algebras. Arch. Math. (Bassel), 91 (3), 212–217. [72] D˘asc˘alescu, S., Iovanov, M. C. and N˘ast˘asescu, C. 2013. Path subcoalgebras, finiteness properties and quantum groups. J. Noncommut. Geom., 7 (3), 737–766. [73] D˘asc˘alescu, S., N˘ast˘asescu, C. and Raianu, S¸. 2001. Hopf Algebras: An Introduction. Monographs Textbooks in Pure and Applied Mathematics 235, Marcel Dekker, New York: Marcel Dekker. [74] Deligne, P. 1990. Cat´egories Tannakiennes, in P. Cartier et al. (eds.), The Grothendieck Festschrift, Volume 2. Basel: Birkh¨auser, pp. 111–195. [75] Dello, J., Panaite, F., Van Oystaeyen, F. and Zhang, Y. 2016. Structure theorems for bicomodule algebras over quasi-Hopf algebras, weak Hopf algebras and braided Hopf algebras. Comm. Algebra, 44, 4609–4636. [76] Dieudonn´e, J. 1958. Remarks on quasi-Frobenius rings. Illinois J. Math., 2, 346–354. [77] Dijkgraaf, R., Pasquier, V. and Roche, P. 1990. Quasi-Hopf algebras, group cohomology and orbifold models. Nuclear Phys. B Proc. Suppl., 18 B, 60–72. [78] Drabant, B. 2001. Notes on balanced categories and Hopf algebras, in A. Pressley (ed.), Quantum Groups and Lie Theory (Durham, 1999). Cambridge: Cambridge University Press, pp. 63–88. [79] Drinfeld, V. G. 1989. Quasi-Hopf algebras and Knizhnik-Zamolodchikov equations, in A. A. Belavin and P. Kofen (eds.), Problems of Modern Quantum Field Theory. Research Report in Physics. Berlin: Springer, pp. 11–13. [80] Drinfeld, V. G. 1989. Quasi-Hopf algebras. Leningrad Math. J., 1 (6), 1419–1457. [81] Drinfeld, V. G. 1990. On almost cocommutative Hopf algebras. Leningrad Math. J., 1, 321–342. [82] Drinfeld, V. G. 1991. On quasitriangular quasi-Hopf algebras and a group closely connected with Gal(Q/Q). Leningrad Math. J., 2 (4), 829–860. [83] Drinfeld, V. G. 1992. On the structure of quasitriangular quasi-Hopf algebras. Funct. Anal. Appl., 26 (1), 63–65. [84] Eilenberg, S. and Kelly, G. M. 1966. Closed categories, in Proceedings of the Conference on Categorical Algebra (La Jolla, 1965). New York: Springer, pp. 421–562. [85] Elhamdadi, M. and Makhlouf, A. 2013. Hom-quasi-bialgebras, in N. Andruskiewitsch, J. Cuadra and B. Torrecillas (eds.), Hopf Algebras and Tensor Categories, Contemporary Mathematics 585. Providence, RI: American Mathematical Society, pp. 227–245. [86] Enriquez, B. 2003. Quasi-Hopf algebras associated with semisimple Lie algebras and complex curves. Selecta Math. (N.S.), 9 (1), 1–61. [87] Enriquez, B. and Felder, G. 1998. Elliptic quantum groups Eτ ,η (sl2 ) and quasi-Hopf algebras. Comm. Math. Phys., 195 (3), 651–689. [88] Enriquez, B. and Halbout, G. 2004. Poisson algebras associated to quasi-Hopf algebras. Adv. Math., 186 (2), 363–395.

Bibliography

519

[89] Enriquez, B. and Rubtsov, V. 1999. Quasi-Hopf algebras associated with sl2 and complex curves. Israel J. Math., 112, 61–108. [90] Etingof, P. and Gelaki, S. 1998. On finite dimensional semisimple and cosemisimple Hopf algebras in positive characteristic. Internat. Math. Res. Notices, 16, 851–864. [91] Etingof, P. and Gelaki, S. 2004. Finite dimensional quasi-Hopf algebras with radical of codimension 2. Math. Res. Lett., 11, 685–696. [92] Etingof, P. and Gelaki, S. 2005. On radically graded finite-dimensional quasi-Hopf algebras. Mosc. Math. J., 5 (2), 371–378. [93] Etingof, P. and Gelaki, S. 2006. Liftings of graded quasi-Hopf algebras with radical of prime codimension. J. Pure Appl. Algebra, 205 (2), 310–322. [94] Etingof, P. and Gelaki, S. 2009. The small quantum group as a quantum double. J. Algebra, 322 (7), 2580–2585. [95] Etingof, P., Gelaki, S., Nikshych, D. and Ostrik, V. 2015. Tensor Categories. Mathematical Surveys and Monographs 205. Providence, RI: American Mathematical Society. [96] Etingof, P., Nikshych, D. and Ostrik, V. 2005. On fusion categories. Ann. of Math., 162 (2), 581–642. [97] Etingof, P., Rowell, E. and Witherspoon, S. 2008. Braid group representations from twisted quantum doubles of finite groups. Pacific J. Math., 234 (1), 33–41. [98] Farsad, V., Gainutdinov, A.M. and Runkel, I. 2018. SL(2, Z)-action for ribbon quasiHopf algebras. Preprint, arXiv:1702.01086v3. [99] Freyd, P. and Yetter, D. 1989. Braided compact closed categories with applications to low dimensional topology. Adv. Math., 77, 156–182. [100] Freyd, P. and Yetter, D. 1992. Coherence theorems via knot theory. J. Pure Appl. Algebra, 78, 49–76. [101] Gelaki, S. 2005. Basic quasi-Hopf algebras of dimension n3 . J. Pure Appl. Algebra, 198 (1-3), 165–174. [102] Goff, C., Mason, G. and Ng, S.-H. 2007. On the gauge equivalence of twisted quantum doubles of elementary abelian and extra-special 2-groups. J. Algebra, 312 (2), 849–875. [103] Gould, M. D. and Lekatsas, T. 2005. Some twisted results. J. Phys. A, 38 (47), 10123– 10144. [104] Gould, M. D. and Lekatsas, T. 2006. Quasi-Hopf *-algebras. Unpublished, arXiv:0604520. [105] Gould, M. D. Zhang, Y.-Z. and Isaac, P. S. 2000. Casimir invariants from quasi-Hopf (super)algebras. J. Math. Phys., 41 (1), 547–568. [106] Gould, M. D., Zhang, Y.-Z. and Isaac, P. S. 2001. On quasi-Hopf superalgebras. Comm. Math. Phys., 224 (2), 341–372. [107] Hausser, F. and Nill, F. 1999. Diagonal crossed products by duals of quasi-quantum groups. Rev. Math. Phys., 11, 553–629. [108] Hausser, F. and Nill, F. 1999. Doubles of quasi-quantum groups. Comm. Math. Phys., 199, 547–589. [109] Hausser, F. and Nill, F. 1999. Integral theory for quasi-Hopf algebras. Unpublished, arXiv:9904164. [110] Hennings, M. A. 1996. Invariants of links and 3-manifolds obtained from Hopf algebras. J. Lond. Math. Soc. (2), 54 (3), 594–624. [111] Huang, H.-L. 2009. Quiver approaches to quasi-Hopf algebras. J. Math. Phys., 50 (4), 043501, 9 pp. [112] Huang, H.-L. 2012. From projective representations to quasi-quantum groups, Sci. China Math., 55 (10), 2067–2080. [113] Huang, H.-L., Liu, G. and Ye, Y. 2011. Quivers, quasi-quantum groups and finite tensor categories. Comm. Math. Phys., 303 (3), 595–612.

520

Bibliography

[114] Huang, H.-L., Liu, G., Ye, Y. 2015. Graded elementary quasi-Hopf algebras of tame representation type. Israel J. Math., 209 (1), 157–186. [115] Huang, H.-L. and Yang, Y. 2015. Quasi-quantum linear spaces. J. Noncommut. Geom., 9 (4), 1227–1259. [116] Huang, H.-L. and Yang, Y. 2015. Quasi-quantum planes and quasi-quantum groups of dimension p3 and p4 . Proc. Amer. Math. Soc., 143 (10), 4245–4260. [117] Iglesias, F. C., N˘ast˘asescu, C. and Vercruysse, J. 2010. Quasi-Frobenius functors: Applications. Comm. Algebra, 38 (8), 3057–3077. [118] Iovanov, M. C. and Vercruysse, J. 2008. Co-Frobenius corings and adjoint functors. J. Pure Appl. Algebra, 212 (9), 2027–2058. [119] Joyal, A. and Street, R. 1986. Braided Monoidal Categories. Mathematics Reports 860081. North Ryde: Macquarie University. [120] Joyal, A. and Street, R. 1991. Tortile Yang-Baxter operators in tensor categories. J. Pure Appl. Algebra, 71, 43–51. [121] Joyal, A. and Street, R. 1993. Braided tensor categories. Adv. Math., 102, 20–78. [122] Kadison, L. 2006. An approach to quasi-Hopf algebras via Frobenius coordinates. J. Algebra, 295 (1), 27–43. [123] Kadison, L. 1999. New Examples of Frobenius Extensions. University Lecture Series 14. Providence, RI: American Mathematical Society. [124] Kadison, L. and Stolin, A. A. 2001. An approach to Hopf algebras via Frobenius coordinates. Beitr¨age Algebra Geom., 42 (2), 359–384. [125] Kadison, L. and Stolin, A. A. 2002. An approach to Hopf algebras via Frobenius coordinates. II. J. Pure Appl. Algebra, 176 (2-3), 127–152. [126] Kaplansky, I. 1975. Bialgebras. Lecture Notes in Mathematics. University of Chicago. [127] Kassel, C. 1995. Quantum Groups. Graduate Texts in Mathematics 155. Berlin: Springer-Verlag. [128] Kassel, C. and Turaev, V. G. 1995. Double construction for monoidal categories. Acta Math., 175, 1–48. [129] Kauffman, L. H. 1993. Gauss codes, quantum groups and ribbon Hopf algebras. Rev. Math. Phys., 5 (4), 735–773. [130] Kauffman, L. H. and Radford, D. E. 1993. A necessary and sufficient condition for a finite-dimensional Drinfeld double to be a ribbon Hopf algebra. J. Algebra, 159, 98–114. [131] Lam, T. Y. 1999. Lectures on Modules and Rings. Graduate Texts in Mathematics 189. New York: Springer-Verlag. [132] Larson, R. G. 1971. Characters of Hopf algebras. J. Algebra, 17, 352–368. [133] Larson, R. G. and Radford, D. E. 1998. Finite-dimensional cosemisimple Hopf algebras in characteristic 0 are semisimple. J. Algebra, 117, 267–289. [134] Larson, R. G. and Radford, D. E. 1998. Semisimple cosemisimple Hopf algebras. Amer. J. Math., 110, 187–195. [135] Larson, R. G. and Sweedler, M. E. 1969. An associative orthogonal bilinear form for Hopf algebras. Amer. J. Math., 91, 75–94. [136] Laugwitz, R. 2015. Braided Drinfeld and Heisenberg doubles. J. Pure Appl. Algebra, 19 (10), 4541–4596. [137] Links, J. R., Gould, M. D. and Zhang, Y.-Z. 2000. Twisting invariance of link polynomials derived from ribbon quasi-Hopf algebras. J. Math. Phys., 41, 5020–5032. [138] Liu, G. 2014. The quasi-Hopf analogue of uq (sl2 ). Math. Res. Lett., 21 (3), 585–603. [139] Liu, G., Van Oystaeyen, F. and Zhang, Y. 2016. Representations of the small quasiquantum group Quq (sl2 ). Bull. Belg. Math. Soc. Simon Stevin, 23 (5), 779–800.

Bibliography

521

[140] Liu, G., Van Oystaeyen, F. and Zhang, Y. 2017. Quasi-Frobenius-Lusztig kernels for simple Lie algebras. Trans. Amer. Math. Soc., 369 (3), 2049–2086. [141] Mac Lane, S. 1971. Categories for the Working Mathematician. Graduate Texts in Mathematics 5. New York: Springer-Verlag. [142] Mack, G. and Schomerus, V. 1992. Quasi-Hopf quantum symmetry in quantum theory. Nuclear Phys. B, 370 (1), 185–230. [143] Majid, S. 1990. Representation-theoretic rank and double Hopf algebras. Comm. Algebra, 18 (11), 3705–3712. [144] Majid, S. 1991. Doubles of quasitriangular Hopf algebras. Comm. Algebra, 19 (11), 3061–3073. [145] Majid, S. 1991. Representations, duals and quantum doubles of monoidal categories. Rend. Circ. Math. Palermo (2) Suppl., 26, 197–206. [146] Majid, S. 1994. Algebras and Hopf algebras in braided categories. In J. Bergen and S. Montgomery (eds.), Advances in Hopf Algebras (Chicago, IL, 1992) Lecture Notes in Pure and Applied Mathematics 158. New York: Dekker, pp. 55–105. [147] Majid, S. 1998. Quantum double for quasi-Hopf algebras. Lett. Math. Phys., 45, 1–9. [148] Majid, S. 1995. Foundations of Quantum Group Theory. Cambridge: Cambridge University Press. [149] Maltsiniotis, G. 1995. Traces dans les cat´egories mono¨ıdales, dualit´e et cat´egories mono¨ıdales fibr´ees. Cahiers Topologie G´eom. Diff´erentielle Cat´eg., 36 (3), 195–288. [150] Markl, M. and Shnider, S. 1996. Cohomology of Drinfeld algebras: a homological algebra approach. Int. Math. Res. Not., 9, 431–445. [151] Markl, M. and Shnider, S. 1996. Drinfeld algebra deformations, homotopy comodules and the associahedra. Trans. Amer. Math. Soc., 348 (9), 3505–3547. [152] Mason, G. and Ng, S.-H. 2001. Group cohomology and gauge equivalence of some twisted quantum doubles. Trans. Amer. Math. Soc., 353 (9), 3465–3509. [153] Mason, G. and Ng, S.-H. 2005. Central invariants and Frobenius-Schur indicators for semisimple quasi-Hopf algebras. Adv. Math., 190 (1), 161–195. [154] Mason, G. and Ng, S.-H. 2014. Cleft extensions and quotients of twisted quantum doubles, in G. Mason, I. Penkov and J. A. Wolf (eds.), Developments and Retrospectives in Lie Theory. Developments in Mathematics 38. Cham: Springer. [155] Masuoka, A. 2003. Cohomology and coquasi-bialgebra extensions associated to a matched pair of bialgebras. Adv. Math., 173 (2), 262–315. [156] Milnor, J. and Moore, J. C. 1965. On the structure of Hopf algebras. Ann. of Math., 81, 211–264. [157] Montgomery, S. 1993. Hopf Algebras and their Actions on Rings. CBMS Regional Conference Series in Mathematics 82. Providence, RI: American Mathematical Society. [158] Naidu, D. and Nikshych, D. 2008. Lagrangian subcategories and braided tensor equivalences of twisted quantum doubles of finite groups. Comm. Math. Phys., 279, 845–872. [159] Naidu, D., Nikshych, D. and Witherspoon, S. 2009. Fusion subcategories of representation categories of twisted quantum doubles of finite groups. Int. Math. Res. Not., 22, 4183–4219. [160] Nakayama, T. 1938. Note on symmetric algebras. Ann. of Math., 39, 659–668. [161] Nakayama, T. 1939. On Frobeniusean algebras. Ann. of Math. (2), 40, 611–633. [162] Nakayama, T. 1941. On Frobeniusean algebras. II. Ann. of Math. (2), 42, 1–21. [163] Natale, S. 2003. On group theoretical Hopf algebras and exact factorizations of finite groups. J. Algebra, 270 (1), 199–211. [164] N˘ast˘asescu, C., Raianu, S¸. and Van Oystaeyen, F. 1990. Modules graded by G-sets. Math. Z., 203 (4), 605–627.

522

Bibliography

[165] N˘ast˘asescu, C. and Torrecillas, B. 2005. Morita duality for Grothendieck categories with applications to coalgebras. Comm. Algebra, 33 (11), 4083–4096. [166] N˘ast˘asescu, C., Van Den Bergh, M. and Van Oystaeyen, F. 1989. Separable functors applied to graded rings. J. Algebra, 123 (2), 397–413. [167] N˘ast˘asescu, C. and Van Oystaeyen, F. 1982. Graded Ring Theory. North-Holland Mathematical Library 28. Amsterdam–New York: North Holland. [168] Ng, S.-H. and Schauenburg, P. 2008. Central invariants and higher indicators for semisimple quasi-Hopf algebras. Trans. Amer. Math. Soc., 360 (4), 1839–1860. [169] Nichols, W. D. and Zoeller, M. B. 1989. A Hopf algebra freeness theorem. Amer. J. Math., 111, 381–385. [170] Oberst, U. and Schneider, H. -J. 1974, Untergruppen formeller Gruppen von endlichem Index. J. Algebra, 31, 10–44. [171] Ospel, C. 2002. Cohomological properties of the quantum shuffle product and application to the construction of quasi-Hopf algebras. J. Pure Appl. Algebra, 173 (3), 315–337. [172] Panaite, F. 1998. A Maschke-type theorem for quasi-Hopf algebras, in A. Verschoren and S. Caenepeel (eds.), Rings, Hopf Algebras and Brauer Groups (Antwerp/Brussels, 1996). Lecture Notes in Pure and Applied Mathematics 197. New York: Marcel Dekker, pp. 201–207. [173] Panaite, F. 2007. Doubles of (quasi) Hopf algebras and some examples of quantum groupoids and vertex groups related to them, in L. H. Kauffman, D. E. Radford and F. J. O. Souza (eds), Hopf Algebras and Generalizations. Contemporary Mathematics 441. Providence, RI: American Mathematical Society, pp. 91–115. [174] Panaite, F., Staic, M. D. and Van Oystaeyen, F. 2010. Pseudosymmetric braidings, twines and twisted algebras. J. Pure Appl. Algebra, 214 (6), 867–884. [175] Panaite, F. and S¸tefan, D. 1997. When is the category of comodules a braided tensor category? Rev. Roum. Math. Pures Appl., 42, 107–119. [176] Panaite, F. and Van Oystaeyen, F. 2000. Existence of integrals for finite dimensional quasi-Hopf algebras. Bull. Belg. Math. Soc. Simon Stevin, 7, 261–264. [177] Panaite, F. and Van Oystaeyen, F. 2000. Quasi-Hopf algebras and the centre of a tensor category, in S. Caenepeel and F. Van Oystaeyen (eds.), Hopf Algebras and Quantum Groups. Lecture Notes in Pure and Applied Mathematics 209. New York: Marcel Dekker, pp. 221–235. [178] Panaite, F. and Van Oystaeyen, F. 2004. Quasi-Hopf algebras and representations of octonions and other quasialgebras. J. Math. Phys., 45, 3912–3929. [179] Panaite, F. and Van Oystaeyen, F. 2007. L-R-smash product for (quasi-) Hopf algebras. J. Algebra, 309, 168–191. [180] Panaite, F. and Van Oystaeyen, F. 2007. A structure theorem for quasi-Hopf comodule algebras. Proc. Amer. Math. Soc., 135, 1669–1677. [181] Pareigis, B. 1971. When Hopf algebras are Frobenius algebras. J. Algebra, 18, 588–596. [182] Pareigis, B. 1977. Non-additive ring and module theory I. General theory of monoids. Publ. Math. Debrecen, 24, 189–204. [183] Radford, D. E. 1976. The order of the antipode of a finite-dimensional Hopf algebra is finite. Amer. J. Math., 98, 333–355. [184] Radford, D. E. 1977. Pointed Hopf algebras are free over Hopf subalgebras. J. Algebra, 45, 266–273. [185] Radford, D. E. 1977. Freeness (projectivity) criteria for Hopf algebras over Hopf subalgebras. J. Pure Appl. Algebra, 11, 15–28.

Bibliography

523

[186] Radford, D. E. 1985. The structure of Hopf algebras with a projection. J. Algebra, 92 (2), 322–347. [187] Radford, D. E. 1992. On the antipode of a quasitriangular Hopf algebra. J. Algebra, 151, 1–11. [188] Radford, D. E. 1994. On Kauffman’s knot invariants arising from finite dimensional Hopf algebras, in J. Bergen and S. Montgomery (eds.), Advances in Hopf algebras (Chicago, IL, 1992), Lecture Notes in Pure and Applied Mathematics 158, Dekker, New York: Marcel Dekker, pp. 205–266. [189] Radford, D. E. 2012. Hopf Algebras. Series on Knots and Everything 49. Hackensack, NJ: World Scientific. [190] Reshetikhin, N. Y. and Semenov-Tian-Shansky, M. A. 1988. Quantum R-matrices and factorization problems. J. Geom. Phys., 5, 533–550. [191] Reshetikhin, N. Y. and Turaev, V. G. 1990. Ribbon graphs and their invariants derived from quantum groups. Comm. Math. Phys., 127, 1–26. [192] Rotman, J. 1979. An Introduction to Homological Algebra. New York: Academic Press. ˇ 2017. On categories associated to a quasi-Hopf algebra. Comm. Algebra, [193] Sak´aloˇs, S. 45 (2), 722–748. [194] Schauenburg, P. 1994. Hopf modules and Yetter-Drinfel’d modules. J. Algebra, 169, 874–890. [195] Schauenburg, P. 2002. Hopf modules and the double of a quasi-Hopf algebra. Trans. Amer. Math. Soc., 354, 3349–3378. [196] Schauenburg, P. 2002. Hopf bimodules, coquasibialgebras, and an exact sequence of Kac. Adv. Math., 165 (2), 194–263. [197] Schauenburg, P. 2003. Actions of monoidal categories and generalized Hopf smash products. J. Algebra, 270 (2), 521–563. [198] Schauenburg, P. 2004. On the Frobenius-Schur indicators for quasi-Hopf algebras. J. Algebra, 282 (1), 129–139. [199] Schauenburg, P. 2004. Two characterizations of finite quasi-Hopf algebras. J. Algebra, 273 (2), 538–550. [200] Schauenburg, P. 2004. A quasi-Hopf algebra freeness theorem. Proc. Amer. Math. Soc., 132 (4), 965–972. [201] Schauenburg, P. 2005. Quotients of finite quasi-Hopf algebras, in S. Caenepeel and F. Van Oysaeyen (eds.), Hopf Algebras in Noncommutative Geometry and Physics. Lecture Notes in Pure and Applied Mathematics 239. New York: Marcel Dekker, pp. 281–290. [202] Schneider, H.-J. 1995. Lectures on Hopf Algebras. Trabajos de Matematica 31, University of C´ordoba. [203] Shnider, S. and Sternberg, S. 1993. Quantum Groups: From Coalgebras to Drinfeld Algebras. Graduate Texts in Mathematical Physics II. Cambridge, MA: International Press. [204] Sommerh¨auser, Y. 2010. On the notion of a ribbon quasi-Hopf algebra. Rev. Uni´on Mat. Argent., 51, 177–192. [205] Sommerh¨auser, Y. and Zhu, Y. 2013. On the central charge of a factorizable Hopf algebra. Adv. Math., 236, 158–223. [206] Stasheff, J. 1992. Drinfeld’s quasi-Hopf algebras and beyond, in J. Stasheff and M. Gerstenhaber (eds.), Deformation Theory and Quantum Groups with Applications to Mathematical Physics (Amherst, MA, 1990). Contemporary Mathematics 134. Providence, RI: American Mathematical Society, pp. 297–307. [207] Street, R. 1999. The quantum double and related constructions. J. Pure Appl. Algebra, 132, 195–206.

524

Bibliography

[208] Street, R. 2004. Frobenius algebras and monoidal categories. Annual Meeting Aust. Math. Soc. Macquarie University. [209] Sweedler, M. E. 1969. Hopf Algebras. New York: Benjamin. [210] Sweedler, M. E. 1968. Cohomology of algebras over Hopf algebras. Trans. Amer. Math. Soc., 133, 205–239. [211] S¸tefan, D. 1997. The set of types of n-dimensional semisimple and cosemisimple Hopf algebras is finite. J. Algebra, 193 (2), 571–580. [212] Takeuchi, M. 1999. Finite Hopf algebras in braided tensor categories. J. Pure Appl. Algebra, 138, 59–82. [213] Turaev, V. G. 1992. Modular categories and 3-manifold invariants. Int. J. Modern Phys. B, 6, 1807–1824. [214] Van Daele, A. 1997. The Haar measure on finite quantum groups. Proc. Amer. Math. Soc., 125, 3489–3500. [215] Van Daele, A. and Van Keer, S. 1994. The Yang–Baxter and Pentagon equation. Compositio Math., 91, 201–221. [216] Yetter, D. 1992. Framed tangles and a theorem of Deligne on braided deformations of Tannakian categories, in J. Stasheff and M. Gerstenhaber (eds.), Deformation Theory and Quantum Groups with Applications to Mathematical Physics (Amherst, MA, 1990). Contemporary Mathematics 134. Providence, RI: American Mathematical Society, pp. 325–350. [217] Yetter, D. 2001. Functorial Knot Theory: Categories of Tangles, Coherence, Categorical Deformations, and Topological Invariance. Series on Knots and Everything 26. River Edge, NJ: World Scientific. [218] Wang, Y., Zhang, L. Y. and Niu, R. F. 2013. Structure theorem for dual quasi-Hopf bicomodules and its application. Math. Notes, 94 (3-4), 470–481. [219] Weibel, C. 1994. An Introduction to Homological Algebra. Cambridge Studies in Advanced Mathematics 38. Cambridge: Cambridge University Press. [220] Woronowicz, S. 1989. Differential calculus on compact matrix pseudogroups (quantum groups). Comm. Math. Phys., 122, 125–170.

Index

absolutely simple module, 474 adjoint action, 148 adjunction, 40 algebra, 56 augmented, 276 braided commutative, 60 commutative, 60 dual to a coalgebra, 70 morphism, 56 opposite, 60 semisimple, 268 separable, 268 symmetric, 273 tensor product, 64 alternative cointegral, 288 antipode, 96, 110 associative, 253 associativity constraint, 3 augmentation morphism, 276 automorphism braided group, 414 bialgebra, 89, 106 dual, 95 op-cop dual, 95 over a field, 91 super, 91 bicomodule algebra, 168 twist equivalent, 169 bilinear map, 253 bimodule algebra, 154 bimodule coalgebra, 161 bosonisation process, 419 braided bialgebra, 89 braided group, 96 braided Hopf algebra, 96 braiding, 29 mirror-reversed, 30 symmetric, 31 category, 1 (pre-)braided equivalent, 35

(pre-)braided isomorphic, 35 balanced, 481 braided, 29 equivalent, 16 isomorphic, 2 left (right) rigid, 40 monoidal, 3 monoidally equivalent, 22 monoidally isomorphic, 20 of G-graded vector spaces, 8 of bimodules, 7 of endo-functors, 13 of sets, 7 of vector spaces, 7 opmonoidally equivalent, 22 opposite, 4 pivotal, 131 pre-braided, 29 product, 2 reverse monoidal, 3 ribbon, 481 rigid, 40 sovereign, 47 strict monoidal, 3 strong monoidally equivalent, 22 symmetric, 31 coadjoint coaction, 424 coalgebra, 65, 67 braided cocommutative, 69 cocommutative, 69 coopposite, 68 dual to an algebra, 70 morphism, 66 tensor product, 69 coassociative, 65 cochain, 151 cocycle, 9, 153, 335 abelian, 32 coboundary, 9 coboundary abelian, 32

526 cohomologous, 9 normalized, 9 coinvariants alternative, 239 of the first type, 235 of the second type, 239 subalgebra of, 179 comodule, 82 comodule algebra, 162 morphism, 163 twist equivalent, 163 comonad, 66 comultiplication, 65, 109 convolution invertible, 95 product, 95 copairing, 38 corepresentation, 83 coring, 67, 249 defined by a module coalgebra, 250 counit, 65, 109 cross product algebra, 61 coalgebra, 69 degree, 8 diagonal action, 105 diagonal crossed product, 204 distinguished grouplike element, 265 division algebra, 279 Drinfeld twist, 115 dual quasi-bialgebra, 135 co-quasitriangular, 421 dual quasi-Hopf algebra, 136 duality theorem, 223 endomorphism module algebra, 188 enveloping algebra braided group, 418 faithful module, 299 Frobenius algebra, 255 augmented algebra, 277 element, 255 morphism, 255 system, 255 function algebra braided group, 429 functor, 2 (pre-)braided monoidal, 35 left (right) dual, 47 braided equivalence, 35 corestriction of scalars, 85 equivalence, 16 essentialy surjective, 17 forgetful, 103 full image, 2 fully faithful, 17

Index identity, 2 inverse, 2 isomorphism, 2 monoidal, 18 monoidal equivalence, 22 opmonoidal, 18 opmonoidal equivalence, 22 quasi-monoidal, 103 restriction of scalars, 81 ribbon, 495 rigid quasi-monoidal, 128 strict monoidal, 19 strong monoidal, 19 strong monoidal equivalence, 22 switch, 3 gauge transformation, 110, 136 generalized diagonal crossed product, 201 generalized smash product, 186 graded algebra, 57 coalgebra, 67 quasialgebra, 57 quasicoalgebra, 67 group algebra, 9 cohomology, 9 grouplike element, 119 Haar integral, 269 Hexagon Axiom, 29 higher representability assumption for comodules, 423 for modules, 409 homogeneous element, 8 Hopf algebra, 96, 110 dual, 100 over a field, 96 super, 96 Hopf bimodule, 196 Hopf crossed product, 336 hyperplane, 254 injective module, 260 invariance under twisting for biproduct, 378 for L-R-smash product, 219 for ribbon element, 504 for smash product, 180 for two-sided smash product, 193 iterated generalized smash product, 194 Knizhnik–Zamolodchikov equation, 146 L–R-smash product, 214 left center of a category, 309 left cointegral, 280 left dual, 40 left integral, 261

Index left weak center of a category, 309 linear dual basis, 41 dual space, 40 mate of a morphism, 51 mixed double product, 142 modular element, 265, 289 module, 78 module algebra, 147, 148 module coalgebra, 154, 155 monad, 57 morphism graded, 8 left (right) transpose, 44 of braided Hopf algebras, 97 Nakayama automorphism, 257 natural transformation, 2 Godement product, 14 horizontal composition, 14 identity, 13 isomorphism, 2 monoidal, 21 monoidal isomorphism, 21 opmonoidal, 21 vertical composition, 13 non-degenerate bilinear map, 253, 254 element, 280 normalized 3-cocycle, 109 normalized cointegral, 453 normalized integral, 269 pairing, 38 exact, 38 Pentagon Axiom, 3 pivotal structure, 131 pre-braiding, 29 projective module, 270 quantum dimension, 462 quantum double, 330 quasi-bialgebra, 106 balanced, 497 biproduct, 377 isomorphism, 108 morphism, 108 projection, 371 quasitriangular, 382 triangular, 383 twist equivalent, 108 unimodular, 261 quasi-commuting pair of coactions, 176 quasi-Hopf algebra, 110 biproduct, 377 factorizable, 433 involutory, 472

isomorphism, 115 morphism, 115 projection, 371 ribbon, 497 sovereign, 134 trace formula, 469 quasi-Hopf bimodule, 225, 226 datum, 226 dual, 231 morphism, 225 quasi-Hopf ideal, 123 quasi-Hopf subalgebra, 299 quasi-ribbon element, 501 quasi-smash product, 185, 186 R-matrix, 382 reassociator, 109, 135 reconstruction theorem braided, 414 dual braided, 423 for quasi-bialgebras, 104 for quasi-Hopf algebras, 128 representability assumption for comodules, 423 for modules, 407 representation, 78 representation-theoretic rank, 462 ribbon element, 501 right center of a category, 307 right cointegral, 287 right dual, 40 right integral, 261 right weak center of a category, 305 ring, 57, 246 Schr¨odinger representation, 346 separability element, 268 sigma notation for coalgebras, 67 for comodules, 84 smash product, 177, 184 smash product coalgebra, 368 strictification, 25 structure theorem comodule algebras, 247 quasi-Hopf bimodules, 235, 238, 240 subcategory, 2 full, 2 Sweedler cohomology, 335 Sweedler notation, 67 switch map, 169 tensor category, 54 tensor product, 3 theorem Eilenberg and Nakayama, 278 Kohno, 146

527

528 Krull–Schmidt, 299 Mac Lane, 28 Maschke, 269 Nichols–Zoeller, 302 Wedderburn, 279 trace, 256, 274 Triangle Axiom, 3 trivial action, 178 twist, 110, 136, 481 twisting, 136 quasi-bialgebras, 108 twisting morphism, 61 two-sided coaction, 170 twist equivalent, 170 two-sided crossed product, 191 two-sided generalized smash product, 192 two-sided smash product, 193 two-sided two-cosided Hopf module, 353 unit constraint left, 3 right, 3

Index unit object, 3 Universal Property two-sided smash product, 214 cross product algebra, 63 diagonal crossed product, 211 of the category Rl (C ), 495 right center, 307 smash product, 182, 213 vector space graded, 8 super, 11 weakly braided cocommutative, 414 commutative, 429 Yang–Baxter equation categorical, 34 quasi-, 382 Yetter–Drinfeld datum, 325 Yetter–Drinfeld module, 310, 325, 355