Hopf Algebras in Noncommutative Geometry and Physics 9781482276718, 1482276712

Table of contents :
Content: Cover
Half Title
Series Page
Title Page
Copyright Page
Contents
Preface
Conference Participants
Morita Contexts for Corings and Equivalences
Hopf Order Module Algebra Orders
An alternative Notion of Hopf Algebroid
Topological Hopf Algebras, Quantum Groups and Deformation Quantization
On Coseparable and Biseparable Corings
More Properties of Yetter-Drinfeld Modules over Quasi-Hopf Algebras
Rationality Properties for Morita Contexts associated to Corings
Morita Duality for Corings over Quasi-Frobenius Rings
Quantized Coinvariants at Transcendental q Classification of Differentials on Quantum Doubles and Finite Noncommutative GeometryNoncommutative Differentials and Yang-Mills on Permutation Groups S[sub(n)]
The Affineness Criterion for Doi-Koppinen Modules
Algebra Properties invariant under Twisting
Quantum SL(3, C)'s: the missing case
Cuntz Algebras and Dynamical Quantum Group SU(2)
On Symbolic Computations in Braided Monoidal Categories
Quotients of Finite Quasi-Hopf Algebras
Adjointable Monoidal Functors and Quantum Groupoids
On Galois Corings

Citation preview

Hopf Algebras in Noncommutative Geometry and Physics

PURE AND APPLIED MATHEMATICS A Program of Monographs, Textbooks, and Lecture Notes EXECUTIVE EDITORS Earl J. Taft Rutgers University New Brunswick, New Jersey

Zuhair Nashed University of Central Florida Orlando, Florida

EDITORIAL BOARD M. S. Baouendi University of California, San Diego Jane Cronin Rutgers University Jack K. Hale Georgia Institute of Technology S. Kobayashi University of California, Berkeley Marvin Marcus University of California, Santa Barbara W. S. Masse Yale University

Anil Nerode Cornell University Donald Passman University of Wisconsin, Madison Fred S. Roberts Rutgers University David L. Russell Virginia Polytechnic Institute and State University Walter Schempp Universitãt Siegen Mark Teply University of Wisconsin, Milwaukee

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K. Hrbacek and T. Jech, Introduction to Set Theory (1978) W. S. Massey, Homology and Cohomology Theory (1978) M. Marcus, Introduction to Modem Algebra (1978) E. C. Young, Vector and Tensor Analysis (1978) S. B. Nadler, Jr., Hyperspaces of Sets (1978) S. K. Segal, Topics in Group Rings (1978) A. C. M. van Rooij, Non-Archimedean Functional Analysis (1978) L Corwin and R. Szczarba, Calculus in Vector Spaces (1979) C. Sadosky, Interpolation of Operators and Singular Integrals (1979) J. Cronin, Differential Equations (1980) C. W. Groetsch, Elements of Applicable Functional Analysis (1980) /. Vaisman, Foundations of Three-Dimensional Euclidean Geometry (1980) H. I. Freedan, Deterministic Mathematical Models in Population Ecology (1980) S. B. Chae, Lebesgue Integration (1980) C. S. Rees et al., Theory and Applications of Fourier Analysis (1981) L Nachbin, Introduction to Functional Analysis (R. M. Aron, trans.) (1981) G. Orzech and M. Orzech, Plane Algebraic Curves (1981 ) R. Johnsonbaugh and W. E. Pfaffenberger, Foundations of Mathematical Analysis (1981) W. L. Voxman and R. H. Goetschel, Advanced Calculus (1981) L J. Corwin and R. H. Szczarba, Multivariate Calculus (1982) V. I. Istratescu, Introduction to Linear Operator Theory (1981) R. D. Jàrvinen, Finite and Infinite Dimensional Linear Spaces (1981) J. K. Beem and P. E. Ehrlich, Global Lorentzian Geometry (1981) D. L Armacost, The Structure of Locally Compact Abelian Groups (1981) J. W. Brewer and M. K. Smith, eds., Emmy Noether: A Tribute (1981) K. H. Kim, Boolean Matrix Theory and Applications (1982) T.W. Wieting, The Mathematical Theory of Chromatic Plane Ornaments (1982) D. B.Gauld, Differential Topology (1982) R. L Faber, Foundations of Euclidean and Non-Euclidean Geometry (1983) M. Carmeli, Statistical Theory and Random Matrices (1983) J. H. Carruth et al., The Theory of Topological Semigroups (1983) R. L. Faber, Differential Geometry and Relativity Theory (1983) S. Barnett, Polynomials and Linear Control Systems (1983) G. Karpilovsky, Commutative Group Algebras (1983) F Van Oystaeyen and A. Verschoren, Relative Invariants of Rings (1983) /. Vaisman, A First Course in Differential Geometry (1984) G. W. Swan, Applications of Optimal Control Theory in Biomedicine (1984) T. Pétrie and J. D. Randall, Transformation Groups on Manifolds (1984) K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings (1984) T. Albu and C. Nastasescu, Relative Finiteness in Module Theory (1984) K. Hrbacek and T. Jech, Introduction to Set Theory: Second Edition (1984) F Van Oystaeyen and A. Verschoren, Relative Invariants of Rings (1984) B. R. McDonald, Linear Algebra Over Commutative Rings (1984) M. Namba, Geometry of Projective Algebraic Curves (1984) G. F Webb, Theory of Nonlinear Age-Dependent Population Dynamics (1985) M. R. Bremner et al., Tables of Dominant Weight Multiplicities for Representations of Simple Lie Algebras (1985) A. E. Fekete, Real Linear Algebra (1985) S. B. Chae, Holomorphy and Calculus in Normed Spaces (1985) A. J. Jerri, Introduction to Integral Equations with Applications (1985)

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P. L. Sachdev, Nonlinear Ordinary Differential Equations and Their Applications (1991) T. Husain, Orthogonal Schauder Bases (1991) J. Foran, Fundamentals of Real Analysis (1991) W. C. Brown, Matrices and Vector Spaces (1991) M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces (1991) J. S. Golan and T. Head, Modules and the Structures of Rings (1991) C. Small, Arithmetic of Finite Fields (1991 ) K. Yang, Complex Algebraic Geometry (1991) D. G. 'Hoffman et al., Coding Theory (1991 ) M. O. González, Classical Complex Analysis (1992) M. O. González, Complex Analysis (1992) L W. Baggett, Functional Analysis (1992) M. Sniedovich, Dynamic Programming (1992) R. P. Agarwal, Difference Equations and Inequalities (1992) C. Brezinski, Biorthogonality and Its Applications to Numerical Analysis (1992) C. Swartz, An Introduction to Functional Analysis (1992) S. B. Nadler, Jr., Continuum Theory (1992) M. A. Al-Gwaiz, Theory of Distributions (1992) E. Perry, Geometry: Axiomatic Developments with Problem Solving (1992) E. Castillo and M. R. Ruiz-Cobo, Functional Equations and Modelling in Science and Engineering (1992) A. J. Jerri, Integral and Discrete Transforms with Applications and Error Analysis (1992) A. Charlier et al., Tensors and the Clifford Algebra (1992) P. Bilerand T. Nadzieja, Problems and Examples in Differential Equations (1992) E. Hansen, Global Optimization Using Interval Analysis (1992) S. Guerre-Delabrière, Classical Sequences in Banach Spaces (1992) Y C. Wong, Introductory Theory of Topological Vector Spaces (1992) S. H. Kulkarni and B. V. Limaye, Real Function Algebras (1992) W. C. Brown, Matrices Over Commutative Rings (1993) J. Loustau and M. Dillon, Linear Geometry with Computer Graphics (1993) W. V. Petryshyn, Approximation-Solvability of Nonlinear Functional and Differential Equations (1993) E. C. Young, Vector and Tensor Analysis: Second Edition (1993) T. A. Bick, Elementary Boundary Value Problems (1993) M. Pavel, Fundamentals of Pattern Recognition: Second Edition (1993) S. A. Albeverio et al., Noncommutative Distributions (1993) W. Fulks, Complex Variables (1993) M. M. Rao, Conditional Measures and Applications (1993) A. Janicki and A. Weron, Simulation and Chaotic Behavior of a-Stable Stochastic Processes (1994) P. Neittaanmaki and D. Tiba, Optimal Control of Nonlinear Parabolic Systems (1994) J. Cronin, Differential Equations: Introduction and Qualitative Theory, Second Edition (1994) S. Heikkilà and V. Lakshmikantham, Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations (1994) X. Mao, Exponential Stability of Stochastic Differential Equations (1994) B. S. Thomson, Symmetric Properties of Real Functions (1994) J. E. Rubio, Optimization and Nonstandard Analysis (1994) J. L Bueso et al., Compatibility, Stability, and Sheaves (1995) A. N. Michel and K. Wang, Qualitative Theory of Dynamical Systems (1995) M. R. Darnel, Theory of Lattice-Ordered Groups (1995) Z Naniewicz and P. D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications (1995)

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Additional Volumes in Preparation

Hopf Algebras in Noncommutative Geometry and Physics

edited by

Stefaan Caenepeel

Free University of Brussels VUB, Belgium

Freddy Van Oystaeyen

University of Antwerp UIA, Belgium

Boca Raton London New York

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First published 2005 by Marcel Dekker Published 2018 by CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2005 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business

No claim to original U.S. Government works ISBN 13: 978-0-8247-5759-5 (pbk) ISBN 13: 978-1-138-45430-9 (hbk) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www. copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MAO 1923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress.

Preface This volume contains the proceedings of meeting "Hopf algebras and quantum groups", which was held from May 28 until June 1, 2002, at the Koninklijke Vlaamse Académie van België voor Wetenschappen en Kunsten, the Royal Academy located in the historic centre of Brussels. The conference focussed at new results on classical Hopf algebras, with emphasis on the classification theory of finite dimensional Hopf algebras, categorical aspects of Hopf algebras, connections with Mathematical Physics, and the recent developments in the theory of corings and of quasi-Hopf algebras. The meeting was supported by: the project "Noncommutative Geometry" (NOG) of the European Science Foundation (ESF); the bilateral project "New computational, geometric and algebraic methods applied to quantum groups and differential operators" of the Flemish and Chinese governments; the bilateral project "Hopf algebras in Algebra, Topology, Geometry and Physics" of the Flemish and Romanian governments; the Flemish Science Foundation FWO-Vlaanderen; the Royal Flemish Academy of Belgium (KVAB); the Free University of Brussels (VUB); the TMR project "Algebraic Lie Representations". We wish to thank the ESF, the Flemish government, the FWO-Vlaanderen, the KVAB, VUB and TMR for enabling us to organize this meeting. Our thanks also go to the participants, the speakers and the authors. J.-P. Tignol kindly gave us the permission to use his "Fesart style file", used previously in volume 208 of the Lecture Notes in Pure and Applied Mathematics Series. All papers have been individually refereed, and we thank the anonymous referees. Stefaan Caenepeel Freddy Van Oystaeyen December 2003

Contents Preface Conference Participants Morita Contexts for Corings and Equivalences J. Abuhlail

1

Hopf Order Module Algebra Orders F. Aly and F. Van Oystaeyen

21

An alternative Notion of Hopf Algebroid G. Bõhm

31

Topological Hopf Algebras, Quantum Groups and Deformation Quantization 55 Ph. Bonneau and D. Sternheimer On Coseparable and Biseparable Corings T. Brzezinski, L. Kadison and R. Wisbauer

71

More Properties of Yetter-Drinfeld Modules over Quasi-Hopf Algebras D. Bulacu, S.Caenepeel and F. Panaite

89

Rationality Properties for Morita Contexts associated to Corings S. Caenepeel, J. Vercruysse and S.H. Wang

113

Morita Duality for Corings over Quasi-Frobenius Rings L. El Kaoutit and J. Gómez-Torrecillas

137

Quantized Coinvariants at Transcendental q K. R. Goodearl and T. H. Lenagan

155

Classification of Differentials on Quantum Doubles and Finite Noncommutative Geometry S. Majid

167

Noncommutative Differentials and Yang-Mills on Permutation Groups Sn 5. Majid

189

The Affineness Criterion for Doi-Koppinen Modules C Menini and G. Militaru

215

Algebra Properties invariant under Twisting S. Montgomery

229

Quantum SL(3, C)'s: the missing case C.Ohn

245

Cuntz Algebras and Dynamical Quantum Group SU(2) A. Paolucci

257

On Symbolic Computations in Braided Monoidal Categories B. Pareigis

269

Quotients of Finite Quasi-Hopf Algebras P. Schauenburg

281

Adjointable Monoidal Functors and Quantum Groupoids K. Szlachányi

291

On Galois Corings R. Wisbauer

309

Conference Participants Abuhlail, Jawad (Birzeit); abuhlail@ kfupm.edu.sa Adriaenssens, Jan (Antwerp); [email protected] Ardizzoni, Alessandra (Ferrara); ardiz@ dm.unife.it Backelin, Erik (Antwerp); backelin@ uia.ua.ac.be Balan, Adriana (Bucharest); steleanu@ lycos.com Beattie, Margaret (Mount Allison, Sackville); [email protected] Bõhm, Gabriella (Budapest); bgabr@ rmki.kfki.hu Bonneau, Philippe (Bourgogne); bonneau@ u-Bourgogne.fr Brown, Ken (Glasgow); [email protected] Brzezinski, Thomas (Wales at Swansea); T.Brzezinski@ swansea.ac.uk Bulacu, Daniel (Bucharest); dbulacu@ al.math.unibuc.ro Caenepeel, Stef (Brussels); scaenepe@ vub.ac.be Chin, William (De Paul, Chicago); wchin@ condor.depaul.edu Cibils, Clause (Montpellier); cibils@ math.univ-montp2.fr Cohen, Mia (Beer Sheva); [email protected] Cuadra, Juan (Antwerp and Almería; [email protected], [email protected] De Groot, Erwin (Brussels); [email protected] Didt, Daniel (Miinchen); didt@ mathematik.uni-muenchen.de Farinati, Marco (Buenos Aires); mfarinat@ dm.uba.ar Fr0nsdahl, Christian (UCLA); fronsdal@ physics.ucla.edu Garcia, Socorro (Tenerife); sgarcia@ ull.es Gomez Torrecillas, José (Granada); [email protected] Gomez, Xavier (Queen Mary, London); X.Gomez@ qmul.ac.uk Gonzalez, Ramon (Vigo); rgrodri@ correo.uvigo.es Guédénon, Thomas (Brussels); thomas.guedenon@ vub.ac.be Hajac, Pjotr (Miinchen); Piotr.Hajac@ fuw.edu.pl Heckenberger, Istvan (Leipzig): heckenbe@ mathematik.uni-leipzig.de Iovanov, Miodrag (Bucharest); myo30@ lycos.com Iyer, Uma (Bonn); uiyer@ mpim-bonn.mpg.de Janviere, Ndirahista (Antwerp); Ndirahisha.Janviere@ ua.ac.be Kadison, Lars (Gõteborg) ; [email protected] Kolb, Stefan (Leipzig) ; [email protected] Lebruyn, Lieven (Antwerp); lieven.lebruyn@ ua.ac.be Legagneux, Jean-Louis ; Legagneux.Coudier@ Wanadoo.fr Lenagan, Tom (Edinburgh); tom@ maths.ed.ac.uk Majid, Shawn (Queen Mary, London); s.majid@ qmul.ac.uk Mendoza, Judit (Tenerife); [email protected] Menini, Claudia (Ferrara); men@ dns.unife.it Militaru, Gigel (Bucharest); gmilit@ al.math.unibuc.ro Montgomery, Susan (Southern California, Los Angeles); [email protected] Nãstãsescu, Constantin (Bucharest); cnastase@ al.math.unibuc.ro Natale, Sonja (Córdoba); natale@ mate.uncor.edu

Ohn, Christian (Reims); christian.ohn@ univ-reims.fr Panaite, Florin (Bucharest); [email protected] Paolucci, Anna (Leeds) ; [email protected] Pareigis, Bodo (München); [email protected] Rúan, Zhong-Jin (Illinois at Urbana); ruan@ math.uiuc.edu Schauenburgh, Peter (München); schauen@ rz.mathematik.uni-muenchen.de Schneider, Hans-Jürgen (München); Hans-Juergen.Schneider@ mathematik.unimuenchen.de Slingerland, Joost Amsterdam; slinger@ science.uva.nl Solotar, Andrea (Buenos Aires); asolotar@ dm.uba.ar Sommerhauser, Yorck (München); sommerh@ rz.mathematik.uni-muenchen.de Sternheimer, Daniel (Bourgogne); Daniel.Sternheimer@ u-Bourgogne.fr Szlachanyi, Kornel (Budapest); szlach@ rmki.kfki.hu Ufer, Stefan (München); ufer@ rz.mathematik.uni-muenchen.de Van Oystaeyen, Fred (Antwerp); fred.vanoystaeyen@ ua.ac.be Vercruysse, Joost (Brussels); [email protected] Vidunas, Raimundas (Antwerp); vidunas@ math.kyushu-u.ac.jp Wang, Dingguo (Brussels and Henan); [email protected] Wisbauer, Robert (Dusseldorf); wisbauer@ math.uni-duesseldorf.de Wisniewski, Piotr (Torun); pikonrad@ mat.uni.torun.pl Zhang, Yinhuo (Antwerp and Fiji); Yinhuo.Zhang@ vuw.ac.nz

Program Tuesday, May 28 10.00-10.50 10.50-11.20 11.20-11.55 12.00-12.35 12.35-14.20 14.20-15.10 15.15-15.50 15.50-16.20 16.20-16.55 17.00-17.35 17.45-19.00

B. Pareigis (München) On Symbolic Computations with Elements in Braided Monoidal Categories Coffee L. Kadison (Gõteborg) Antipodes at Depth Two A. Solotar (Buenos Aires) Hochschild homology oftrivial extensions Lunch T. Brzeziñski (Wales at Swansea) Why corings? Y. Sommerhãuser (München) Self-dual modules ofsemisimple Hopf algebras Coffee M. Farinati (Buenos Aires) Hochschild homology of Generalized Weyl Algebras S. Natale (Córdoba) On the semi-solvability ofsemisimple Hopf algebras of small dimension Reception in the Marble Room

Wednesday, May 29 09.00-09.50 09.55-10.30 10.30-11.00 11.00-11.35 11.40-12.15 12.15-14.00 14.00-14.35 14.40-15.15 15.20-15.55 16.00-16.30

H.-J. Schneider (München) On the classification of pointed Hopf algebras R. Wisbauer (Dusseldorf) Galois corings Coffee J. Gomez Torrecillas (Granada) Semisimple corings S. Caenepeel (Brussels) Cleft entwining structures andMorita contexts associated to a coring Lunch P. Schauenburgh (München) Some properties of finite quasi-Hopf algebras T. Lenagan (Edinburgh) Coinvariants and co-orbits for quantum matrices P. Hajac (München) Fredholm Index and locally trivial noncommutative Hopf fibration Coffee

Thursday, May 30 09.30-10.20 10.20-10.50 10.50-11.25 11.30-12.05 12.05-14.00 14.00-14.50 14.55-15.30 15.30-16.00 16.00-16.35 16.40-17.15 17.20-17.55 20.00

S. Montgomery (Univ. of Southern California) Properties of algebras invariant under twisting Coffee C. Menini (Ferrara) Quantum groups of dimension 16 with the Chevalley property D. Bulacu (Bucharest) Hopf modules for quasi-Hopf algebras Lunch break K. Szlachányi (Budapest) Adjointable lax monoidal functors and bialgebroids G. Bõhm (Budapest) Hopf algebrólas with bijective antipode Coffee J. Cuadra (Almería and Antwerp) The Brauer group of the Dihedral group respect to a quasi-triangular structure Y. Zhang (Antwerp and Fiji) The equi variant Brauer group of an infinite group D. Wang (Brussels and Qufu Univ.) Twistings, crossed coproducts and Hopf Galois coextensions Congress Dinner

Friday, May 31 09.30-10.20

S. Majid (Queen Mary College, London) Dirac operators and Riemannian geometry on Hopf algebras 10.20-10.50 Coffee 10.50-11.25 W. Chin (De Paul Univ., Chicago) Prime Spectra of Quantized Hyperalgebras 11.30-12.05 C. Cibils (Montpellier) Hochschild cohomology ofHopfalgebras 12.05-14.00 Lunch break 14.00-14.35 M. Beattie (Mount Allison University) Hopf algebras of dimension 14 14.40-15.15 G. Militaru (Bucharest) 15.15-15.45 Coffee

15.45-16.20 16.25-17.00 17.05-17.40

F. Panaite (Bucharest) Hopf bimodules as modules over a diagonal crossed product algebra X. Gomez (Queen Mary College, London) Relating Woronowicz's quantum Lie algebras and Majid's braided Lie algebras P. Wisniewski (Torun) The Lasker-Noether theorem for commutative and noetherian module algebras over a pointed Hopf algebra

Saturday, June 1 09.30-10.20 10.20-10.50 10.50-11.25 11.30-12.05 12.10-12.40

D. Sternheimer and Ph. Bonneau (Univ. de Bourgogne) Topological Hopf algebras, quantum groups and deformation quantization Coffee C. Fr0nsdahl (UCLA) A. Paolucci (Leeds) Cuntz Algebras and Dynamical Quantum Group SL(2) C.Ohn (Reims) Quantum SL(3) 's: the missing case

Morita Contexts for Corings and Equivalences JAWAD ABUHLAIL* Mathematics Department, Birzeit University PO Box 14, Birzeit, Palestine e-mail: [email protected], [email protected] ABSTRACT. In this note we study Morita contexts and Galois extensions for corings. For a coring C over a (not necessarily commutative) ground ring A we give equivalent conditions for Mc to satisfy the weak. resp. the strong structure theorem. We also characterize the so called cleft C-Galois extensions over commutative rings. Our approach is similar to that of Y. Doi and A. Masuoka in their work on (cleft) fi-Galois extensions (see for example [10, 11]).

1.

INTRODUCTION

Let C be a coring over a not necessarily commutative ring A and assume A to be a right C-comodule through QA ' A —> A xa for some grouplike element x G C (see [4, Lemma 5.1]). In Section 2, we study from the viewpoint of Morita theory the relationship between A and its subring of coinvariants B := AcoC = | fe e A | g ^ = 6 x j W e c o n s ider the A-ring *C := Hom,4_(C,,4) and its left ideal Q := {q G *C | X^citf(c2) =