Shuffle Approach Towards Quantum Affine and Toroidal Algebras 9789819931491, 9789819931507

Table of contents :
Preface
Acknowledgments
Contents
1 Quantum Loop mathfraksln, Its Two Integral Forms, and Generalizations
1.1 Algebras Uv(Lmathfraksln),mathfrakUv(Lmathfraksln),Uv(Lmathfraksln) and Their Bases
1.1.1 Quantum Loop Algebra Uv(Lmathfraksln) and Its PBWD Bases
1.1.2 Integral Form mathfrakUv(Lmathfraksln) and Its PBWD Bases
1.1.3 Integral Form Uv(Lmathfraksln) and Its PBWD Bases
1.2 Shuffle Realizations and Applications to the PBWD Bases
1.2.1 Shuffle Algebra S(n)
1.2.2 Proofs of Theorems 1.1 and 1.8
1.2.3 Integral Form mathfrakS(n) and a Proof of Theorem 1.3
1.2.4 Integral Form S(n) and a Proof of Theorem 1.6
1.3 Generalizations to Two-Parameter Quantum Loop Algebras
1.3.1 Quantum Loop Algebra U>v1,v2(Lmathfraksln) and Its PBWD Bases
1.3.2 Integral Form mathfrakU>v1,v2(Lmathfraksln) and Its PBWD Bases
1.3.3 Shuffle Algebra widetildeS(n)
1.3.4 Proofs of Theorems 1.16 and 1.18
1.4 Generalizations to Quantum Loop Superalgebras
1.4.1 Quantum Loop Superalgebra U>v(Lmathfraksl(m|n))
1.4.2 PBWD Bases of U>v(Lmathfraksl(m|n))
1.4.3 Integral Form mathfrakU>v(Lmathfraksl(m|n)) and Its PBWD Bases
1.4.4 Shuffle Algebra S(m|n)
1.4.5 Proofs of Theorems 1.19 and 1.21
References
2 Quantum Toroidal mathfrakgl1, Its Representations, and Geometric Realization
2.1 Quantum Toroidal Algebras of mathfrakgl1
2.1.1 Quantum Toroidal mathfrakgl1
2.1.2 Elliptic Hall Algebra
2.1.3 ``90 Degree Rotation'' Automorphism
2.1.4 Shuffle Algebra Realization
2.1.5 Neguţ's Proof of Theorem 2.3
2.1.6 Commutative Subalgebra mathcalA
2.2 Representations of Quantum Toroidal mathfrakgl1
2.2.1 Categories mathcalOpm
2.2.2 Vector, Fock, Macmahon Modules, and Their Tensor Products
2.2.3 Vertex Representations and Their Relation to Fock Modules
2.2.4 Shuffle Bimodules and Their Relation to Fock Modules
2.3 Geometric Realizations
2.3.1 Correspondences and Fixed Points for (mathbbA2)[n]
2.3.2 Geometric Action I
2.3.3 Heisenberg Algebra Action on the Equivariant K-Theory
2.3.4 Correspondences and Fixed Points for M(r,n)
2.3.5 Geometric Action II
2.3.6 Whittaker Vector
References
3 Quantum Toroidal mathfraksln, Its Representations, and Bethe Subalgebras
3.1 Quantum Toroidal Algebras of mathfraksln
3.1.1 Quantum Toroidal mathfraksln (n3)
3.1.2 Hopf Pairing, Drinfeld Double, and a Universal R-Matrix
3.1.3 Horizontal and Vertical Copies of Quantum Affine mathfraksln
3.1.4 Miki's Isomorphism
3.1.5 Horizontal and Vertical Extra Heisenberg Subalgebras
3.2 Shuffle Algebra and Its Commutative Subalgebras
3.2.1 Shuffle Algebra Realization
3.2.2 Commutative Subalgebras mathcalA(s0,…,sn-1)
3.2.3 Proof of Theorem 3.7
3.2.4 Special Limit mathcalAh
3.2.5 Identification of Two Extra Horizontal Heisenbergs
3.3 Representations of Quantum Toroidal mathfraksln
3.3.1 Vector, Fock, and Macmahon Modules
3.3.2 Vertex Representations
3.3.3 Shuffle Bimodules
3.4 Identification of Representations
3.4.1 Shuffle Realization of Fock Modules
3.4.2 Generalization for Tensor Products of Fock and Macmahon
3.4.3 Twisting Vertex Representations by Miki's Isomorphism
3.5 Bethe Algebra Realization of mathcalA(s0,…,sn-1)
3.5.1 Trace Functionals
3.5.2 Functionals via Pairing
3.5.3 Transfer Matrices and Bethe Subalgebras
3.5.4 Bethe Algebra Incarnation of mathcalA(s0,…,sn-1)
3.5.5 Shuffle Formulas for Miki's Twist of Cartan Generators
References

Citation preview

SpringerBriefs in Mathematical Physics 49 Alexander Tsymbaliuk

Shuffle Approach Towards Quantum Affine and Toroidal Algebras

SpringerBriefs in Mathematical Physics Volume 49

Series Editors Nathanaël Berestycki, Fakultät für Mathematik, University of Vienna, Vienna, Austria Mihalis Dafermos, Mathematics Department, Princeton University, Princeton, NJ, USA Atsuo Kuniba, Institute of Physics, The University of Tokyo, Tokyo, Japan Matilde Marcolli, Department of Mathematics, University of Toronto, Toronto, Canada Bruno Nachtergaele, Department of Mathematics, Davis, CA, USA Hal Tasaki, Department of Physics, Gakushuin University, Tokyo, Japan

SpringerBriefs are characterized in general by their size (50–125 pages) and fast production time (2–3 months compared to 6 months for a monograph). Briefs are available in print but are intended as a primarily electronic publication to be included in Springer’s e-book package. Typical works might include: • An extended survey of a field • A link between new research papers published in journal articles • A presentation of core concepts that doctoral students must understand in order to make independent contributions • Lecture notes making a specialist topic accessible for non-specialist readers. SpringerBriefs in Mathematical Physics showcase, in a compact format, topics of current relevance in the field of mathematical physics. Published titles will encompass all areas of theoretical and mathematical physics. This series is intended for mathematicians, physicists, and other scientists, as well as doctoral students in related areas. Editorial Board • • • • • • • • • •

Nathanaël Berestycki (University of Cambridge, UK) Mihalis Dafermos (University of Cambridge, UK/Princeton University, US) Atsuo Kuniba (University of Tokyo, Japan) Matilde Marcolli (CALTECH, US) Bruno Nachtergaele (UC Davis, US) Hal Tasaki (Gakushuin University, Japan) 50 – 125 published pages, including all tables, figures, and references Softcover binding Copyright to remain in author’s name Versions in print, eBook, and MyCopy

Alexander Tsymbaliuk

Shuffle Approach Towards Quantum Affine and Toroidal Algebras

Alexander Tsymbaliuk Department of Mathematics Purdue University West Lafayette, IN, USA

ISSN 2197-1757 ISSN 2197-1765 (electronic) SpringerBriefs in Mathematical Physics ISBN 978-981-99-3149-1 ISBN 978-981-99-3150-7 (eBook) https://doi.org/10.1007/978-981-99-3150-7 © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

These are detailed lecture notes of the crash course on shuffle algebras delivered by the author at Tokyo University of Marine Science and Technology during the second week of March 2019. Historically, the shuffle approach to Drinfeld-Jimbo quantum groups Uq (g) (embedding the “positive” subalgebras Uq> (g) into q-deformed shuffle algebras) was first developed independently in the 1990s by J. Green, M. Rosso, and P. Schauenburg. Motivated by similar ideas, B. Feigin and A. Odesskii proposed a shuffle approach to elliptic quantum groups around the same time. The shuffle algebras featuring in the present notes may be viewed as trigonometric degenerations of the aforementioned Feigin-Odesskii elliptic algebras. They provide combinatorial models for the “positive” subalgebras of various quantum affinized algebras (such as affine or toroidal). Conceptually, they make it possible to work conveniently with the elements of quantum affinized algebras that are given by rather complicated non-commutative polynomials in the original loop generators. Starting from our joint work with B. Feigin (where an action of the Heisenberg algebra on the equivariant K-theory of the Hilbert schemes of points on a plane was constructed, generalizing the classical result of H. Nakajima in the cohomological setup), and being immensely generalized in a profound series of work by A. Negu¸t (in his study of affine Laumon spaces, Nakajima quiver varieties associated with a cyclic quiver, and W-algebras associated with surfaces), the shuffle approach has found major applications in the geometric representation theory within the last decade. Explicitly, while the action of the loop generators is classically given by the simplest Hecke correspondences equipped with tautological line bundles (following Nakajima’s geometric construction of representations for quantum affine algebras), the geometry of the corresponding moduli spaces (such as Nakajima quiver varieties or Laumon spaces) allows for a much wider family of natural geometric correspondences that are best handled using the shuffle realization. Yet another interesting algebro-geometric application of the shuffle algebras has recently been presented in joint work with M. Finkelberg in the study of quantized K-theoretic Coulomb branches. In the simplest rank one case, the latter also provides a link to type A quantum Q-systems, as first noticed by P. Di Francesco and R. Kedem.

v

vi

Preface

These notes consist of three chapters, providing a separate treatment for (1) the quantum loop algebras of sln and their generalizations; (2) the quantum toroidal algebras of gl1 ; and (3) the quantum toroidal algebras of sln . The shuffle realization of the corresponding “positive” subalgebras is established as well as of the commutative subalgebras and some combinatorial representations for the toroidal algebras. One of the key techniques involved is that of “specialization maps”. Each chapter emphasizes a different aspect of the theory: in the first chapter, shuffle algebras are used to construct a family of new PBWD bases for type A quantum loop algebras and their integral forms; in the second chapter, a geometric interpretation of the Fock modules is provided and the shuffle description of a commutative subalgebra is used to construct an action of the Heisenberg algebra on the equivariant K-theory of the Hilbert schemes of points on a plane; in the last chapter, vertex and combinatorial representations of the quantum toroidal algebras of sln are related using Miki’s isomorphism, and shuffle realization is used to explicitly compute commutative Bethe subalgebras and their limits. The latter construction is inspired by B. Enriquez’s work relating shuffle algebras to the correlation functions of quantum affinized algebras. Given the size restrictions of the present volume, the focus is mostly on the algebraic aspects of the above theory (with a single exception made in Chap. 2 to provide a flavor of the geometry involved). To the reader interested in the beautiful geometric applications, we highly recommend a series of papers by A. Negu¸t. While most of the presented results have previously appeared in the literature, the aim here is to unify the exposition by simplifying some arguments as well as adding new details. These notes are intended as an introduction to the subject of shuffle algebras with an indication of the key algebraic techniques involved and are appropriate for any interested researcher, including both graduate and advanced undergraduate students. West Lafayette, USA

Alexander Tsymbaliuk

Acknowledgments

I am indebted to Pavel Etingof, Boris Feigin, Michael Finkelberg, and Andrei Negu¸t for numerous stimulating discussions and collaboration over the years; to Luan Bezerra and Evgeny Mukhin for their correspondence on quantum affine superalgebras; to Naihuan Jing for correspondence on two-parameter quantum algebras; to Joshua Wen for a discussion of a beautiful application of the results from the last chapter to the study of wreath Macdonald polynomials and operators; and to the anonymous referees for useful suggestions that improved the overall exposition. Special thanks are due to Hitoshi Konno for inviting me to deliver this crash course on shuffle algebras at Tokyo University of Marine Science and Technology (Japan) in March 2019; and to Hitoshi Konno, Atsuo Kuniba, Tomoki Nakanishi, and Masato Okado for inviting me to speak at the workshop “Infinite Analysis 2019: Quantum symmetries in integrable systems”, held at the University of Tokyo, a week prior to the course. I am also grateful to the audience for their interest and questions. I gratefully acknowledge NSF grants DMS-1821185 and DMS-2037602. West Lafayette, Indiana, USA May 2022

Alexander Tsymbaliuk

vii

Contents

1 Quantum Loop sln , Its Two Integral Forms, and Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Algebras Uv (Lsln ), Uv (Lsln ), Uv (Lsln ) and Their Bases . . . . . . . . . 1.1.1 Quantum Loop Algebra Uv (Lsln ) and Its PBWD Bases . . . . 1.1.2 Integral Form Uv (Lsln ) and Its PBWD Bases . . . . . . . . . . . . . 1.1.3 Integral Form Uv (Lsln ) and Its PBWD Bases . . . . . . . . . . . . 1.2 Shuffle Realizations and Applications to the PBWD Bases . . . . . . . 1.2.1 Shuffle Algebra S (n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Proofs of Theorems 1.1 and 1.8 . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Integral Form S(n) and a Proof of Theorem 1.3 . . . . . . . . . . . 1.2.4 Integral Form S(n) and a Proof of Theorem 1.6 . . . . . . . . . . . 1.3 Generalizations to Two-Parameter Quantum Loop Algebras . . . . . . 1.3.1 Quantum Loop Algebra Uv>1 , v2 (Lsln ) and Its PBWD Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Integral Form U> v1 , v2 (Lsln ) and Its PBWD Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Shuffle Algebra  S (n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.4 Proofs of Theorems 1.16 and 1.18 . . . . . . . . . . . . . . . . . . . . . . 1.4 Generalizations to Quantum Loop Superalgebras . . . . . . . . . . . . . . . . 1.4.1 Quantum Loop Superalgebra Uv> (Lsl(m|n)) . . . . . . . . . . . . . 1.4.2 PBWD Bases of Uv> (Lsl(m|n)) . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 Integral Form U> v (Lsl(m|n)) and Its PBWD Bases . . . . . . . . 1.4.4 Shuffle Algebra S (m|n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.5 Proofs of Theorems 1.19 and 1.21 . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 5 7 9 9 13 19 24 26 27 28 29 29 30 30 31 32 33 34 36

ix

x

Contents

2 Quantum Toroidal gl1 , Its Representations, and Geometric Realization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Quantum Toroidal Algebras of gl1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Quantum Toroidal gl1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Elliptic Hall Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 “90 Degree Rotation” Automorphism . . . . . . . . . . . . . . . . . . . 2.1.4 Shuffle Algebra Realization . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.5 Negu¸t’s Proof of Theorem 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.6 Commutative Subalgebra A . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Representations of Quantum Toroidal gl1 . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Categories O± . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Vector, Fock, Macmahon Modules, and Their Tensor Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Vertex Representations and Their Relation to Fock Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Shuffle Bimodules and Their Relation to Fock Modules . . . 2.3 Geometric Realizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Correspondences and Fixed Points for (A2 )[n] . . . . . . . . . . . . 2.3.2 Geometric Action I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Heisenberg Algebra Action on the Equivariant K -Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Correspondences and Fixed Points for M(r, n) . . . . . . . . . . . 2.3.5 Geometric Action II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.6 Whittaker Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Quantum Toroidal sln , Its Representations, and Bethe Subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Quantum Toroidal Algebras of sln . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Quantum Toroidal sln (n ≥ 3) . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Hopf Pairing, Drinfeld Double, and a Universal R-Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Horizontal and Vertical Copies of Quantum Affine sln . . . . . 3.1.4 Miki’s Isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.5 Horizontal and Vertical Extra Heisenberg Subalgebras . . . . . 3.2 Shuffle Algebra and Its Commutative Subalgebras . . . . . . . . . . . . . . . 3.2.1 Shuffle Algebra Realization . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Commutative Subalgebras A(s0 , . . . , sn−1 ) . . . . . . . . . . . . . . 3.2.3 Proof of Theorem 3.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Special Limit Ah . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.5 Identification of Two Extra Horizontal Heisenbergs . . . . . . .

37 37 37 41 42 43 45 48 49 50 51 58 61 63 63 64 65 67 68 70 71 73 73 73 76 79 82 85 86 86 88 90 95 96

Contents

3.3 Representations of Quantum Toroidal sln . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Vector, Fock, and Macmahon Modules . . . . . . . . . . . . . . . . . . 3.3.2 Vertex Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Shuffle Bimodules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Identification of Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Shuffle Realization of Fock Modules . . . . . . . . . . . . . . . . . . . . 3.4.2 Generalization for Tensor Products of Fock and Macmahon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Twisting Vertex Representations by Miki’s Isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Bethe Algebra Realization of A(s0 , . . . , sn−1 ) . . . . . . . . . . . . . . . . . . 3.5.1 Trace Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Functionals via Pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.3 Transfer Matrices and Bethe Subalgebras . . . . . . . . . . . . . . . . 3.5.4 Bethe Algebra Incarnation of A(s0 , . . . , sn−1 ) . . . . . . . . . . . . 3.5.5 Shuffle Formulas for Miki’s Twist of Cartan Generators . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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97 98 101 103 106 106 110 111 117 117 123 125 127 128 129

Chapter 1

Quantum Loop sln , Its Two Integral Forms, and Generalizations

Abstract In this chapter, we establish the shuffle algebra realization in the simplest case of type A quantum loop algebra Uv> (Lsln ). As an important application we construct a family of PBWD (Poincaré-Birkhoff-Witt-Drinfeld) bases for the quantum loop algebra Uv (Lsln ) in the new Drinfeld realization. The shuffle approach also allows to strengthen this by constructing a family of PBWD bases for the RTT form (arising naturally from a different, historically the first, realization [9] of Uv (Lsln ) shown in [4] to be equivalent to the one presented below) and Lusztig form (introduced in [15]) of Uv (Lsln ), both defined over the ring Z[v, v−1 ]. We also discuss the generalizations of these results to two-parameter and super-cases. Let us emphasize that the previously known PBW result (see [1]) for the quantum loop algebras g)) was established in the DrinfeldUv (Lg) (more generally, for quantum affine Uv ( Jimbo realization of the latter, and thus it is compatible with the Drinfeld-Jimbo triangular decomposition, while ours is compatible with the new Drinfeld triangular decomposition. The exposition of this chapter closely follows the author’s paper [19].

1.1 Algebras Uv (Lsln ), Uv (Lsln ), Uv (Lsln ) and Their Bases In this section, we recall the definition of the quantum loop algebra Uv (Lsln ), defined over C(v), as well as of its two integral forms, defined over Z[v, v−1 ]. Evoking the triangular decompositions of these algebras, we provide a family of PBWD-type bases for their “positive” subalgebras (the proofs are postponed till Sect. 1.2).

1.1.1 Quantum Loop Algebra Uv (Lsln ) and Its PBWD Bases For an integer n ≥ 2, let I = {1, . . . , n − 1}, (ci j )i, j∈I be the Cartan matrix of sln (explicitly: cii = 2, ci,i±1 = −1, and ci j = 0 if |i − j| > 1), and v be a formal variable. Following [3], we define the quantum loop algebra of sln (a.k.a. the quantum © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 A. Tsymbaliuk, Shuffle Approach Towards Quantum Affine and Toroidal Algebras, SpringerBriefs in Mathematical Physics 49, https://doi.org/10.1007/978-981-99-3150-7_1

1

1 Quantum Loop sln , and its Integral Forms

2

affine algebra with a trivial central charge), denoted by Uv (Lsln ), to be the associa∈Z,s∈N ± }ri∈I with the following defining tive C(v)-algebra generated by {ei,r , f i,r , ψi,±s relations: 

± ∓ · ψi,0 = 1, [ψi (z), ψ j (w)] = 0 , ψi,0

(1.1)

(z − vci j w)ei (z)e j (w) = (vci j z − w)e j (w)ei (z) ,

(1.2)

(vci j z − w) f i (z) f j (w) = (z − vci j w) f j (w) f i (z) ,

(1.3)

(z − vci j w)ψi (z)e j (w) = (vci j z − w)e j (w)ψi (z) ,

(1.4)

(vci j z − w)ψi (z) f j (w) = (z − vci j w) f j (w)ψi (z) ,

(1.5)

[ei (z), f j (w)] =

 z   δi j + − ψ δ (z) − ψ (z) , i i v − v−1 w

(1.6)

ei (z)e j (w) = e j (w)ei (z) if ci j = 0 ,

(1.7)

[ei (z 1 ), [ei (z 2 ), e j (w)]v−1 ]v + [ei (z 2 ), [ei (z 1 ), e j (w)]v−1 ]v = 0 if ci j = −1 , f i (z) f j (w) = f j (w) f i (z) if ci j = 0 ,

(1.8)

[ f i (z 1 ), [ f i (z 2 ), f j (w)]v−1 ]v + [ f i (z 2 ), [ f i (z 1 ), f j (w)]v−1 ]v = 0 if ci j = −1 , where [a, b]x := ab − x · ba and the generating series are defined as follows: ei (z) :=



ei,r z −r , f i (z) :=

r ∈Z

 r ∈Z

f i,r z −r , ψi± (z) :=



± ψi,±s z ∓s , δ(z) :=

s≥0



zr .

r ∈Z

The last two relations (1.7, 1.8) are usually referred to as quantum Serre relations. It ± }s>0 , defined via: is convenient to use the generators {h i,±r }r >0 instead of {ψi,±s  −1

exp ±(v − v )



 h i,±r z

∓r

± −1 ± ) ψi (z) = ψ¯ i± (z) := (ψi,0

(1.9)

r >0 ∓ ± ± with h i,±r ∈ C[ψi,0 , ψi,±1 , ψi,±2 , . . .]. Then, the relations (1.4, 1.5) are equivalent to:

[kci j ]v e j,l+k (k = 0) , k [kci j ]v f j,l+k (k = 0) , f j,l ψi,0 , [h i,k , f j,l ] = − k

ψi,0 e j,l = vci j e j,l ψi,0 , [h i,k , e j,l ] =

(1.10)

ψi,0 f j,l = v−ci j

(1.11)

−s

−v where [s]v = vv−v −1 for s ∈ Z. < Let Uv (Lsln ), Uv> (Lsln ), Uv0 (Lsln ) be the C(v)-subalgebras of Uv (Lsln ) gener± ∈Z ∈Z s∈N , {ei,r }ri∈I , {ψi,±s }i∈I . The following result is standard: ated respectively by { f i,r }ri∈I s

1.1 Algebras Uv (Lsln ), Uv (Lsln ), Uv (Lsln ) and Their Bases

3

Proposition 1.1 (a) (Triangular decomposition of Uv (Lsln )) The multiplication map m : Uv< (Lsln ) ⊗C(v) Uv0 (Lsln ) ⊗C(v) Uv> (Lsln ) −→ Uv (Lsln ) is an isomorphism of C(v)-vector spaces. (b) The algebra Uv> (Lsln ) (resp. Uv< (Lsln ) and Uv0 (Lsln )) is isomorphic to the ± ∈Z ∈Z s∈N (resp. { f i,r }ri∈I and {ψi,±s }i∈I ) with associative C(v)-algebra generated by {ei,r }ri∈I the defining relations (1.2, 1.7) (resp. (1.3, 1.8) and (1.1)). For completeness of our exposition let us sketch the proof (cf. [12, Theorem 2]). ∈Z,s∈N ± v (Lsln ) be the C(v)-algebra generated by {ei,r , f i,r , ψi,±s }ri∈I with Proof Let U v0 , U v> as v< , U the defining relations (1.1), (1.4)–(1.6), and define its subalgebras U v (Lsln ) before. Let I > and J > (respectively, I < and J < ) be the 2-sided ideals of U v (Lsln ) and U v< ) generated by the following families of v> (respectively, U and U elements: ci j ci j Ar,s i, j = ei,r +1 e j,s − v ei,r e j,s+1 − v e j,s ei,r +1 + e j,s+1 ei,r if ci j  = 0 ,

Bi,r,sj = Ci,r1j,r2 ,s

(1.12)

ei,r e j,s − e j,s ei,r if ci j = 0 , (1.13)

−1 = Sym ei,r1 ei,r2 e j,s − (v + v )ei,r1 e j,s ei,r2 + e j,s ei,r1 ei,r2 if ci j = −1 , r1 ,r2

(1.14) (respectively, by their f -analogues) which encode the relations (1.2), (1.7). We have: v (Lsln )) The multiplication map Lemma 1.1 (a) (Triangular decomposition of U < 0 >     m : Uv ⊗C(v) Uv ⊗C(v) Uv → Uv (Lsln ) is an isomorphism of C(v)-vector spaces. ∈Z ∈Z v> and U v< are free associative algebras in {ei,r }ri∈I (b) U and { f i,r }ri∈I , respectively, ± s∈N 0 v is the algebra generated by {ψi,±s }i∈I subject only to (1.1). while U

v< ⊗ U v0 ⊗ J > ), I < = m(J < ⊗ U v0 ⊗ U v> ) for m from (a). (c) We have I > = m(U Proof Parts (a, b) are standard and we leave their proofs to the interested reader. The v< · U v0 · J > being a 2-sided ideal of U v (Lsln ). first equality in (c) is equivalent to U To this end, it suffices to verify that any X from (1.12)–(1.14) satisfies the following: v0 · J > for any ι ∈ I , l = 0 , [X, h ι,l ] ∈ U v0 · J > for any ι ∈ I , k ∈ Z . [X, f ι,k ] ∈ U

(1.15) (1.16) [lc ]

[lcι,i ]v r +l,s The first of those follows from (1.10), e.g. [h ι,l , Ar,s Ai, j + ι,l j v Ar,s+l i, j ] = i, j . l r1 ,r2 ,s Let us now verify (1.16) for X = Ci, j by working with the generating series (the r,s cases X = Ar,s are similar but simpler). According to (1.6) and using i, j or X = Bi, j  ψj (z) = ψj+ (z) − ψj− (z) = r ∈Z ψj,r z −r , we get:

1 Quantum Loop sln , and its Integral Forms

4



ei (z 1 )ei (z 2 )e j (w) − (v + v−1 )ei (z 1 )e j (w)ei (z 2 ) + e j (w)ei (z 1 )ei (z 2 ), f ι (y) δ j,ı δ(w/y) = × v − v−1   ei (z 1 )ei (z 2 )ψ j (w) − (v + v−1 )ei (z 1 )ψ j (w)ei (z 2 ) + ψ j (w)ei (z 1 )ei (z 2 ) δi,ı δ(z 2 /y) + × v − v−1   ei (z 1 )ψi (z 2 )e j (w) − (v + v−1 )ei (z 1 )e j (w)ψi (z 2 ) + e j (w)ei (z 1 )ψi (z 2 ) δi,ı δ(z 1 /y) + × v − v−1   ψi (z 1 )ei (z 2 )e j (w) − (v + v−1 )ψi (z 1 )e j (w)ei (z 2 ) + e j (w)ψi (z 1 )ψi (z 2 ) .

Symmetrizing with respect to z 1 ↔ z 2 and pulling the ψ∗ -factors to the leftmost part by using (1.4), we will end up with the series whose coefficients are linear r,s 0 > combinations of ψ j,m Ar,s i,i (if ı = j) or ψi,m Ai, j (if ı = i), and thus are in Uv · J . v (Lsln )/(I > + I < ) Uv (Lsln ).  This lemma implies Proposition 1.1 since U n−1 We now proceed to the construction of a family of bases for Uv (Lsln ). Let {αi }i=1 + be the standard simple positive roots of sln , and  be the set of positive roots: + = {α j + α j+1 + · · · + αi }1≤ j≤i (Lsln ) and f β (r ) ∈ Uv< (Lsln ) via: eβ (r ) := [· · · [[ei1 ,r1 , ei2 ,r2 ]λ1 , ei3 ,r3 ]λ2 , · · · , ei p ,r p ]λ p−1 , f β (r ) := [· · · [[ f i1 ,r1 , f i2 ,r2 ]λ1 , f i3 ,r3 ]λ2 , · · · , f i p ,r p ]λ p−1 .

(1.19)

In particular, we have eαi (r ) = ei,r and f αi (r ) = f i,r . Remark 1.1 (a) We note that eβ (r ), f β (r ) degenerate to the corresponding root generators eβ ⊗ t r , f β ⊗ t r of sln [t, t −1 ] = sln ⊗C C[t, t −1 ] as v → 1, hence, the terminology.

1.1 Algebras Uv (Lsln ), Uv (Lsln ), Uv (Lsln ) and Their Bases

5

(b) The following particular choice features manifestly in [10]: eα j +α j+1 +···+αi (r ) := [· · · [[e j,r , e j+1,0 ]v , e j+2,0 ]v , · · · , ei,0 ]v , f α j +α j+1 +···+αi (r ) := [· · · [[ f j,r , f j+1,0 ]v , f j+2,0 ]v , · · · , f i,0 ]v .

(1.20)

Let H be the set of all functions h : + × Z → N with finite support. The monomials eh :=

→ 

eβ (r )h(β,r ) and f h :=

(β,r )∈+ ×Z

← 

f β (r )h(β,r ) for all h ∈ H (1.21)

(β,r )∈+ ×Z

will be called the ordered PBWD monomials of Uv> (Lsln ) and Uv< (Lsln ), respectively. Here, the arrows → and ← over the product signs indicate that the products are ordered with respect to (1.18) or its opposite, respectively. Our first main result establishes the PBWD property of Uv> (Lsln ) and Uv< (Lsln ) (the proof is presented in Sect. 1.2 and is based on the shuffle approach): Theorem 1.1 The ordered PBWD monomials {eh }h∈H and { f h }h∈H form C(v)-bases of the “positive” and “negative” subalgebras Uv> (Lsln ) and Uv< (Lsln ), respectively. + ∈Z Let us relabel the Cartan generators via {ψi,r }ri∈I with ψi,r = ψi,r for r ≥ 0 and − − −1 ψi,r = ψi,r for r < 0, so that (ψi,0 ) = ψi,0 . Let H0 denote the set of all functions h 0 : I × Z → Z with finite support and such that h 0 (i, r ) ≥ 0 for r = 0. The monomials  h 0 (i,r ) ψi,r for all h 0 ∈ H0 (1.22) ψh 0 := (i,r )∈I ×Z

will be called the PBWD monomials of Uv0 (Lsln ); we note that these products are unordered due to (1.1). They form a C(v)-basis of Uv0 (Lsln ) by Proposition 1.1(b). Combining this with Theorem 1.1 and Proposition 1.1(a), we finally obtain: Theorem 1.2 For any choices (1)–(3) made prior to (1.19), the elements 

   f h − · ψh 0 · eh +  h − ∈ H , h 0 ∈ H0 , h + ∈ H

form a C(v)-basis of the quantum loop algebra Uv (Lsln ).

1.1.2 Integral Form Uv (Lsln ) and Its PBWD Bases Following the above notations, define eβ (r ) ∈ Uv> (Lsln ) and  f β (r ) ∈ Uv< (Lsln ) via: f β (r ) := (v − v−1 ) f β (r ) for any (β, r ) ∈ + × Z .  eβ (r ) := (v − v−1 )eβ (r ) and  (1.23)

1 Quantum Loop sln , and its Integral Forms

6

We also define eh ,  f h via (1.21) but using eβ (r ),  f β (r ) instead of eβ (r ), f β (r ). Finally, < −1 (Lsl ) and U (Lsl let us define integral forms U> n n ) as the Z[v, v ]-subalgebras v v r ∈Z ∈Z > < eβ (r )}β∈+ and {  f β (r )}rβ∈ of Uv (Lsln ) and Uv (Lsln ) generated by { + of (1.23), respectively. < The above definition of U> v (Lsln ), Uv (Lsln ) seems to have an obvious flaw, due to all the choices (1)–(3) used in the construction (1.19) of eβ (r ), f β (r ) and hence < f β (r ) in (1.23). The next result shows that U>  eβ (r ),  v (Lsln ), Uv (Lsln ) are actually independent of these choices and posses PBWD bases similar to those from Theorem 1.1 (the proof is presented in Sect. 1.2 and is based on the shuffle approach): < Theorem 1.3 (a) U> v (Lsln ), Uv (Lsln ) are independent of all such choices (1)–(3).

(b) The ordered PBWD monomials { eh }h∈H and {  f h }h∈H form bases of the free −1 > < Z[v, v ]-modules Uv (Lsln ) and Uv (Lsln ), respectively. ± s∈N }i∈I . Let U0v (Lsln ) be the Z[v, v−1 ]-subalgebra of Uv0 (Lsln ) generated by {ψi,±s Due to Proposition 1.1(b), the PBWD monomials {ψh 0 }h 0 ∈H0 of (1.22) form a basis of the free Z[v, v−1 ]-module U0v (Lsln ). Finally, we define the integral form Uv (Lsln ) ∈Z,s∈N ± as the Z[v, v−1 ]-subalgebra of Uv (Lsln ) generated by { eβ (r ),  f β (r ), ψi,±s }rβ∈ + ,i∈I . < Remark 1.2 Since Uv (Lsln ) is generated by U0v (Lsln ), U> v (Lsln ), Uv (Lsln ), it is  independent of any choices (used to define  eβ (r ), f β (r )), due to Theorem 1.3(a).

The following result is proved in [10, Theorem 3.24(b)] using the RTT realization of Uv (Lsln ) (though in [10] we worked with C[v, v−1 ]-subalgebras, the argument of loc.cit. applies verbatim to Z[v, v−1 ]-subalgebras instead): Theorem 1.4 ([10, Theorem 3.24]) For the particular choice (1.20) of the PBWD basis elements eβ (r ), f β (r ) in (1.23), the elements 

    f h − · ψh 0 ·  eh +  h − ∈ H , h 0 ∈ H0 , h + ∈ H

form a basis of the free Z[v, v−1 ]-module Uv (Lsln ). Remark 1.3 Let us point out that it is often more convenient to work with the quantum loop algebra Uv (Lgln ) and its similarly defined integral form Uv (Lgln ). Following [10, Proposition 3.11], Uv (Lgln ) is identified with the RTT integral form rtt Urtt v (Lgln ) under the C(v)-algebra isomorphism Uv (Lgln ) Uv (Lgln ) ⊗Z[v,v−1 ] C(v) of [4]. Combining Theorems 1.3–1.4, we get the triangular decomposition of Uv (Lsln ): Corollary 1.1 The multiplication map 0 > m : U< v (Lsln ) ⊗Z[v,v−1 ] Uv (Lsln ) ⊗Z[v,v−1 ] Uv (Lsln ) −→ Uv (Lsln )

is an isomorphism of free Z[v, v−1 ]-modules.

1.1 Algebras Uv (Lsln ), Uv (Lsln ), Uv (Lsln ) and Their Bases

7

Combining this Corollary with Theorem 1.3 once again, we finally obtain: Theorem 1.5 For any choices (1)–(3) made prior to (1.19), the elements 

    f h − · ψh 0 ·  eh +  h − ∈ H , h 0 ∈ H0 , h + ∈ H

form a basis of the free Z[v, v−1 ]-module Uv (Lsln ).

1.1.3 Integral Form Uv (Lsln ) and Its PBWD Bases For k ∈ N, recall [k]v , [k]v ! ∈ Z[v, v−1 ] defined via: [k]v := (vk − v−k )/(v − v−1 ) ,

[k]v ! := [1]v · · · [k]v .

(1.24)

Let us define the divided powers (k) (k) k k := ei,r /[k]v ! and fi,r := f i,r /[k]v ! for i ∈ I , r ∈ Z , k ∈ N . ei,r

(1.25)

< −1 Following [11, §7.8], define integral forms U> v (Lsln ) and Uv (Lsln ) as the Z[v, v ](k) r ∈Z,k∈N (k) r ∈Z,k∈N > < subalgebras of Uv (Lsln ) and Uv (Lsln ) generated by {ei,r }i∈I and {fi,r }i∈I , ∈Z respectively. Recall the PBWD basis elements {eβ (r ), f β (r )}rβ∈ of (1.19), depend+ ing on the choices (1)–(3) used in their definition. We define their divided powers

eβ (r )(k) := eβ (r )k /[k]v !

and

fβ (r )(k) := f β (r )k /[k]v !

(1.26)

for β ∈ + , r ∈ Z, k ∈ N. Proposition 1.2 For any β ∈ + , r ∈ Z, k ∈ N, we have: eβ (r )(k) ∈ U> v (Lsln )

and

fβ (r )(k) ∈ U< v (Lsln ) .

(1.27)

Proof Let UvDJ > (sln ) be the “positive” subalgebra of the Drinfeld-Jimbo quantum group of sln . Explicitly, UvDJ > (sln ) is the associative C(v)-algebra generated by {E i }i∈I subject to the following quantum Serre relations: [E i , E j ] = 0 if ci j = 0 ,

[E i , [E i , E j ]v−1 ]v = 0 if ci j = −1 .

(1.28)

> (sln ) defined as the Z[v, v−1 ]-subalgebra of Recall the Lusztig integral form UDJ v DJ > Uv (sln ) generated by the divided powers E i(k) = E ik /[k]v ! for all i ∈ I, k ∈ N. Then, E α j +···+αi = [· · · [E j , E j+1 ]v , · · · , E i ]v are specific Lusztig root generators of UvDJ > (sln ) (for the convex order on + opposite to (1.17)) and their divided powers > E α(k)j +···+αi := E αk j +···+αi /[k]v ! belong to UDJ (sln ), due to [15, Theorem 6.6]. v

1 Quantum Loop sln , and its Integral Forms

8

For any r ∈ Z and j < i ∈ I , the assignment E ι → eι,r διj gives rise to an algebra > (sln )) ⊂ U> homomorphism σ : UvDJ > (sln ) → Uv> (Lsln ) such that σ (UDJ v v (Lsln ). (k) > This implies that eβ (r ) ∈ Uv (Lsln ) for the particular choice (1.20) of the PBWD basis elements eβ (r ) in (1.19), since eα j +···+αi (r )(k) = σ (E α(k)j +···+αi ) ∈ U> v (Lsln ). The proof of fβ (r )(k) ∈ U< (Lsl ) for the particular choice (1.20) is similar. n v The proof of the inclusions (1.27) for a more general construction (1.19) of eβ (r )  (and similarly also of f β (r )) is derived in Sect. 1.2.4 below, see Lemma 1.13. The monomials eh :=

→ 

← 

eβ (r )(h(β,r )) and fh :=

(β,r )∈+ ×Z

fβ (r )(h(β,r )) for all h ∈ H

(β,r )∈+ ×Z

(1.29) < will be called the ordered PBWD monomials of U> v (Lsln ) and Uv (Lsln ), respectively. Here, the arrows → and ← over the product signs indicate that the products are ordered with respect to (1.18) or its opposite, respectively. The following result < provides the PBWD bases for the above integral forms U> v (Lsln ) and Uv (Lsln ): Theorem 1.6 The ordered PBWD monomials {eh }h∈H and {fh }h∈H form bases of the < free Z[v, v−1 ]-modules U> v (Lsln ) and Uv (Lsln ), respectively. + For i ∈ I, r ∈ Z, k > 0, recall {h i,±k } of (1.9) and define ψi,0k ;r ∈ Uv0 (Lsln ) via:  +  k + r −l+1 − −r +l−1  ψi,0 v − ψi,0 v ψi,0 ; r := . l −l v −v k l=1

(1.30)

Following [2, §3], we define U0v (Lsln ) as the Z[v, v−1 ]-subalgebra of Uv0 (Lsln ) gen + r ∈Z,k>0 ± h i,±k , [k]v , ψi,0k ;r . Finally, we define the integral form Uv (Lsln ) as erated by ψi,0 i∈I

0 > the Z[v, v−1 ]-subalgebra of Uv (Lsln ) generated by U< v (Lsln ), Uv (Lsln ), Uv (Lsln ).

Remark 1.4 Identifying Uv (Lsln ) with the Drinfeld-Jimbo quantum loop algebra I = I ∪ {0} UvDJ (Lsln ) of [3] (a C(v)-algebra generated by {E i , Fi , K i±1 }i∈I where  is the vertex set of the extended Dynkin diagram, cf. Remark 3.3(d)), the integral form Uv (Lsln ) gets identified with the Lusztig form of UvDJ (Lsln ) defined as the k∈N Z[v, v−1 ]-subalgebra generated by {K i±1 , E i(k) , Fi(k) }i∈ , due to [2].  I The following triangular decomposition of Uv (Lsln ) is due to [2, Proposition 6.1]: Theorem 1.7 ([2, Proposition 6.1]) The multiplication map 0 > m : U< v (Lsln ) ⊗Z[v,v−1 ] Uv (Lsln ) ⊗Z[v,v−1 ] Uv (Lsln ) −→ Uv (Lsln )

is an isomorphism of free Z[v, v−1 ]-modules. Combining Theorems 1.6–1.7, we thus obtain (similar to Theorems 1.2 and 1.5) a family of PBWD bases for Uv (Lsln ).

1.2 Shuffle Realizations and Applications to the PBWD Bases

9

1.2 Shuffle Realizations and Applications to the PBWD Bases In this section, we establish the shuffle algebra realization of Uv> (Lsln ) and use it to prove Theorem 1.1. Likewise, we provide the shuffle realizations of both integral > forms U> v (Lsln ) and Uv (Lsln ) and use those to prove Theorems 1.3 and 1.6.

1.2.1 Shuffle Algebra S(n) Let k denote the symmetric group in k elements, and set (k1 ,...,kn−1 ) := k1 × · · · × kn−1 for k1 , . . . , kn−1 ∈ N. Consider an N I -graded C(v)-vector space 

S(n) =

S(n) k ,

k=(k1 ,...,kn−1 )∈N I 1≤r ≤ki . where S(n) k consists of k -symmetric rational functions in the variables {x i,r }i∈I We fix an I × I matrix of rational functions (ζi, j (z))i, j∈I ∈ Mat I ×I (C(v)(z)) via:

ζi, j (z) :=

z − v−ci j . z−1

(1.31)

Let us now introduce the bilinear shuffle product  on S(n) : given F ∈ S(n) k and (n) G ∈ S(n)  , we define F  G ∈ Sk+ via

1 (1.32) (F  G)(x1,1 , . . . , x1,k1 +1 ; . . . ; xn−1,1 , . . . , xn−1,kn−1 +n−1 ) := k! · ! ⎛ ⎞    >ki  i  ∈I r       x i,r k (Lsln ) via the following result2 :

1 2

Following [6]–[8], the role of the wheel conditions is to replace intricate quantum Serre relations. When working over C[[]] rather than over C(v), this construction goes back to [5, Corollary 1.4].

1.2 Shuffle Realizations and Applications to the PBWD Bases

11

Proposition 1.3 There exists an injective C(v)-algebra homomorphism  : Uv> (Lsln ) −→ S (n)

(1.36)

r for any i ∈ I, r ∈ Z. such that ei,r → xi,1 r (i ∈ I, r ∈ Z) gives rise to Proof Let us show that the assignment  : ei,r → xi,1 an algebra homomorphism  : Uv> (Lsln ) → S (n) . To do so, it suffices to verify that (ei (z)) = δ( xzi,1 ) satisfy the defining relations (1.2), (1.7) of Uv> (Lsln ), see Proposition 1.1(b). To this end, let us first recall the key property of the δ-function:

δ(z/w) · γ (z) = δ(z/w) · γ (w)

for any rational function γ (z) .

(1.37)

• Compatibility with (1.2) for i = j. x x (z − vci j w)δ( xzi,1 )  δ( wj,1 ) = δ( xzi,1 )δ( wj,1 ) · (xi,1 − vci j x j,1 ) · δ(

x j,1 )δ( xzi,1 ) w

· (vci j xi,1 − x j,1 ) ·

x j,1 −v−ci j xi,1 x j,1 −xi,1

= (vci j z −

xi,1 −v−ci j x j,1 = xi,1 −x j,1 x j,1 xi,1 w)δ( w )  δ( z ).

• Compatibility with (1.2) for i = j.  −cii xi,2 = (z − vcii w)δ( xzi,1 )  δ( xwi,1 ) = Sym δ( xzi,1 )δ( xwi,2 ) · (xi,1 − vcii xi,2 ) xi,1x−v i,1 −x i,2   −cii xi,2 xi,1 x −v x x x Sym δ( w )δ( z ) · (vcii xi,1 − xi,2 ) i,2xi,2 −xi,1 i,1 = (vcii z − w)δ( wi,1 )  δ( zi,1 ). • Compatibility with (1.7) for ci j = 0. x x x δ( xzi,1 )  δ( wj,1 ) = δ( xzi,1 )δ( wj,1 ) = δ( wj,1 )  δ( xzi,1 ). • Compatibility with (1.7) for ci j = −1. [δ( xzi,11 ), [δ( xzi,12 ), δ(

x j,1 )]v−1 ]v w

  = A(z 1 , z 2 , w) · δ( xzi,11 )δ( xzi,22 ) + δ( xzi,12 )δ( xzi,21 ) ·

δ( wj,1 ) with A(z 1 , z 2 , w) = ζi,i ( zz21 )ζi, j ( zw1 )ζi, j ( zw2 ) − v−1 ζi,i ( zz21 )ζi, j ( zw1 )ζ j,i ( zw2 ) − vζi,i ( zz21 )ζi, j ( zw2 )ζ j,i ( zw1 ) + ζi,i ( zz21 )ζ j,i ( zw1 )ζ j,i ( zw2 ). Therefore, the compatibility with the cubic quantum Serre relations (1.7) follows from a straightforward equality A(z 1 , z 2 , w) = −A(z 2 , z 1 , w). x

The quickest way to prove the injectivity of  is based on the analogue of (3.102): (X )(z 1 , . . . , z k ) , ϕ(X, f i1 (z 1 ) . . . f ik (z k )) = (v − v−1 )−k  1≤a (Lsln ) are naturally isomorwe note that the “positive” subalgebras Uv> ( phic. Thus, the injectivity of  is an immediate consequence of the nondegeneracy of ϕ.  The next result follows from its much harder counterpart [16, Theorem 1.1], see Theorem 3.6 below, but we will provide its alternative simpler proof3 in Sect. 1.2.2: 3 In particular, our proof will be directly generalized to establish the isomorphisms of Theorems 1.18 and 1.21 below, for which no analogue of Theorem 3.6 ([16, Theorem 1.1]) is presently known.

1 Quantum Loop sln , and its Integral Forms

12 ∼

Theorem 1.8  : Uv> (Lsln ) −→ S (n) of (1.36) is a C(v)-algebra isomorphism. Before proceeding to the simultaneous proofs of Theorem 1.1 and Theorem 1.8, let us introduce the key tool in our study of the shuffle algebra: the specialization maps. For any positive root β = α j + α j+1 + · · · + αi , define j (β) := j, i(β) := i, and let [β] denote the integer interval [ j (β); i(β)]. Consider a collection of the intervals {[β]}β∈+ each taken with a multiplicity dβ ∈ N and ordered with respect to the total + order (1.17) on + (the order inside each group is irrelevant). Let d ∈ N denote the collection {dβ }β∈+ . Define  = (1 , . . . , n−1 ) ∈ N I , the N I -degree of d, via: 

i αi =

i∈I



dβ β .

(1.39)

β∈+

Let us now define the specialization map φd on the shuffle algebra S (n) , which vanishes on degree =  components, and is thus determined by  d  ±1 1≤s≤dβ }β∈+ . φd : S(n) −→ C(v) {yβ,s

(1.40)

 1≤r ≤i To do so, we split the variables {xi,r }i∈I into β∈+ dβ groups corresponding to the above intervals, and specialize those in the s-th copy of [β] in the natural order to v− j (β) · yβ,s , . . . , v−i(β) · yβ,s so that xi,r gets specialized to v−i yβ,s . For F =

f (x1,1 ,...,xn−1,n−1 ) n−2 1≤r  ≤i+1 (xi,r −xi+1,r  ) i=1 1≤r ≤

∈ S(n) , we

i

define φd (F) as the corresponding specialization of f . Let us note right away that: 1≤r ≤i into groups, • φd (F) is independent of our splitting of the variables {xi,r }i∈I dβ + • φd (F) is symmetric in {yβ,s }s=1 for any β ∈  .

The key properties of the specialization maps φd and their relevance to (eh ), the images of the ordered PBWD monomials, are discussed in Lemmas 1.5–1.7 below. xa xb xc

2,1 3,1 ∈ S(n) , where  = (1, 1, 1, 0, . . . , 0) ∈ N I Example 1.9 Let F = (x1,1 −x1,12,1 )(x 2,1 −x 3,1 ) and a, b, c ∈ Z. Let us compute the images of F under all possible specialization maps:

(i) If d encodes a single positive root β = α1 + α2 + α3 , then φd (F) is a Laurent polynomial in a single variable yβ,1 and equals (v−1 yβ,1 )a (v−2 yβ,1 )b (v−3 yβ,1 )c . (ii) If d encodes two positive roots β1 = α1 , β2 = α2 + α3 , then φd (F) is a Laurent polynomial in two variables yβ1 ,1 , yβ2 ,1 and equals (v−1 yβ1 ,1 )a (v−2 yβ2 ,1 )b (v−3 yβ2 ,1 )c . (iii) If d encodes two positive roots β1 = α1 + α2 , β2 = α3 , then φd (F) is a Laurent polynomial in two variables yβ1 ,1 , yβ2 ,1 and equals (v−1 yβ1 ,1 )a (v−2 yβ1 ,1 )b (v−3 yβ2 ,1 )c .

1.2 Shuffle Realizations and Applications to the PBWD Bases

13

(iv) If d encodes roots β1 = α1 , β2 = α2 , β3 = α3 , then φd (F) is a Laurent polynomial in three variables yβ1 ,1 , yβ2 ,1 , yβ3 ,1 and equals (v−1 yβ1 ,1 )a (v−2 yβ2 ,1 )b (v−3 yβ3 ,1 )c .

1.2.2 Proofs of Theorems 1.1 and 1.8 Our proof of Theorem 1.1 will proceed in two steps: first, we shall establish the linear independence of the ordered PBWD monomials {eh }h∈H in Sect. 1.2.2.2, and then we will verify that they linearly span the entire algebra Uv> (Lsln ) in Sect. 1.2.2.3. The part of Theorem 1.1 for the “negative” subalgebra Uv< (Lsln ) can either be proved analogously, or can rather be deduced by utilizing a C(v)-algebra anti-isomorphism ς : Uv> (Lsln ) −→ Uv< (Lsln ) defined by ei,r → f i,r ∀ i ∈ I , r ∈ Z , (1.41) which maps the ordered PBWD monomials to the ordered PBWD monomials multiplied by constants in ±vZ . Notably, we shall also prove Theorem 1.8 along the way. First, we shall prove both theorems for n = 2 (the “rank 1” case) in Sect. 1.2.2.1.

1.2.2.1

n = 2 Case

For n = 2, we have I = {1}, hence, we shall denote the variables x1,r simply by xr . We start from the following simple computation in the shuffle algebra S (2) : Lemma 1.3 For any k ≥ 1 and r ∈ Z, the k-th power of x r ∈ S1(2) equals x r  ·!" · ·  x#r = v−

k(k−1) 2

[k]v ! · (x1 · · · xk )r .

(1.42)

k times

Proof We prove this Lemma by induction on k, the base case k = 1 being trivial. Applying the induction assumption to the (k − 1)-st power of x r , the proof of (1.42) boils down to the verification of the following equality: s= p k   xs − v−2 x p = v1−k [k]v . x − x s p p=1 1≤s≤k

(1.43)

The left-hand side of (1.43) is a rational function in {x p }kp=1 of degree 0 and has no poles (as it is symmetric and could only have simple poles along the diagonals), hence, must be a constant. To evaluate this constant, let xk → ∞: the last summand (corresponding to p = k) tends to v−2(k−1) , while the sum of the first k − 1 summands (corresponding to 1 ≤ p ≤ k − 1) tends to 1 + v−2 + · · · + v−2(k−2) by the induction assumption, thus, resulting in 1 + v−2 + v−4 + · · · + v−2(k−1) = v1−k [k]v , as claimed. 

1 Quantum Loop sln , and its Integral Forms

14

The following result simultaneously implies both Theorems 1.1 and 1.8 for n = 2: Proposition 1.4 Fix any total order  on Z. Then: 1 ···rk (a) the ordered monomials {x r1  x r2  · · ·  x rk }rk∈N form a C(v)-basis of S (2) . 1 ···rk (b) the ordered monomials {er1 er2 · · · erk }rk∈N form a C(v)-basis of Uv> (Lsl2 ).

Proof For r1 = · · · = rk1 ≺ rk1 +1 = · · · = rk1 +k2 ≺ · · · ≺ rk1 +···+k−1 +1 = · · · = rk1 +···+k , set k := k1 + · · · + k and choose σ ∈ k so that rσ (1) ≤ · · · ≤ rσ (k) . Then, the shuffle product x r1  · · ·  x rk is a symmetric Laurent polynomial of the form νr m (rσ (1) ,...,rσ (k) ) (x1 , . . . , xk ) +



νr  m r  (x1 , . . . , xk ) ,

(1.44)

where • m (s1 ,...,sk ) (x1 , . . . , xk ) (with s1 ≤ · · · ≤ sk ) are the monomial symmetric polynomials (that is, the sums of all monomials x1t1 · · · xktk as (t1 , . . . , tk ) ranges over all distinct permutations of (s1 , . . . , sk )), • the sum in (1.44) is over r  = (r1 ≤ · · · ≤ rk ) ∈ Zk distinct from (rσ (1) , . . . , rσ (k) ) and satisfying rσ (1) ≤ r1 ≤ rk ≤ rσ (k) as well as r1 + · · · + rk = r1 + · · · + rk , • the coefficients νr  in (1.44) are Laurent polynomials in v, that is, νr  ∈ Z[v, v−1 ], • due to Lemma 1.3, the coefficient νr is explicitly given by νr =

    k p (k p −1) v− 2 [k p ]v ! .

(1.45)

p=1 k k as vector spaces and {m (s1 ,...,sk ) (x1 , . . . , xk )}s1 ≤···≤sk Since Sk(2) C(v)[{x ±1 p } p=1 ] ±1 k form a C(v)-basis of C(v)[{x p } p=1 ] k , we conclude that the ordered shuffle products 1 ···rk form a C(v)-basis of S (2) . This proves part (a). {x r1  x r2  · · ·  x rk }rk∈N As the homomorphism  : Uv> (Lsl2 ) → S (2) , er → x r , of (1.36) is injective by Proposition 1.3 and (er1 er2 · · · erk ) = x r1  x r2  · · ·  x rk , part (a) implies that 1 ···rk form a C(v)-basis of Uv> (Lsl2 ). This proves part (b).  {er1 er2 · · · erk }rk∈N

1.2.2.2

Linear Independence of eh and Key Properties of φd

For an ordered PBWD monomial eh (h ∈ H) of (1.21), we define its degree deg(eh ) = deg(h) ∈ N

+

(1.46)

 as a collection of dβ := r ∈Z h(β, r ) ∈ N (β ∈ + ) ordered with respect to the total + order (1.17) on + . Let us further consider the anti-lexicographical order on N :

1.2 Shuffle Realizations and Applications to the PBWD Bases

15

{dβ }β∈+ < {dβ }β∈+ iff ∃ γ ∈ + s.t. dγ > dγ and dβ = dβ ∀ β < γ . (1.47) In what follows, we shall need an explicit formula for the image (eβ (r )) ∈ S (n) : Lemma 1.4 For any 1 ≤ j < i < n and r ∈ Z, we have: (eα j +α j+1 +···+αi (r )) = (1 − v2 )i− j

p(x j,1 , . . . , xi,1 ) , (x j,1 − x j+1,1 ) · · · (xi−1,1 − xi,1 )

(1.48)

where p(x j,1 , . . . , xi,1 ) is a degree r + i − j monomial, up to a sign and an integer power of v. 

Proof Straightforward computation.

Example 1.10 For the particular choice (1.20) of the elements eα j +α j+1 +···+αi (r ), we +1 have p(x j,1 , . . . , xi,1 ) = x rj,1 x j+1,1 · · · xi−1,1 in the formula (1.48). Our proof of the linear independence of {eh }h∈H is crucially based on the following key properties of the specialization maps φd from (1.40): Lemma 1.5 If deg(h) < d while N I -degrees of d and deg(h) coincide, then φd ((eh )) = 0 . Proof The condition deg(h) < d guarantees that φd -specialization of any summand from the symmetrization appearing in (eh ) contains among all the ζ -factors at least  one factor of the form ζi,i+1 (v) = 0, hence, must vanish. The result follows.

deg(h)=d + Lemma 1.6 For any d ∈ N , the specializations φd ((eh )) h∈H are linearly independent. Proof Consider the image (eh ) ∈ Sk(n) (here, k is the N I -degree of d = deg(h)). It is a sum of k! terms, but most of them specialize to zero under φd , as in the proof of Lemma 1.5.  The summands which do not specialize to zero are parametrized by d := β∈+ dβ . More precisely, given (σβ )β∈+ ∈ d , the associatedsummand corresponds to the case when for all β ∈ + and 1 ≤ s ≤ dβ , the ( β  d, then there exists Fd ∈ M ∩ Sk(n) such that φd (F) = φd (Fd )

and

φd  (Fd ) = 0 for all d  > d .

Proof Consider the following total order on the set {(β, s)|β ∈ + , 1 ≤ s ≤ dβ }: (β, s) ≤ (β  , s  ) iff β < β  or β = β  and s ≤ s  .

(1.51)

First, we note that the wheel conditions (1.35) for F guarantee that φd (F) (which is s≤d a Laurent polynomial in the variables {yβ,s }β∈β+ ) vanishes, up to appropriate orders, under the following specializations: (i) yβ,s = v−2 yβ  ,s  for (β, s) < (β  , s  ), (ii) yβ,s = v2 yβ  ,s  for (β, s) < (β  , s  ). To evaluate the aforementioned orders of vanishing, let us view the specialization appearing in the definition of φd as a step-by-step specialization in each interval [β], ordered first in the non-increasing length order, while the intervals of the same length are ordered in the non-decreasing order of j (β). As we specialize the variables in the  s-th interval (1 ≤ s ≤ β∈+ dβ ), we count only those wheel conditions that arise from the non-specialized yet variables. A straightforward case-by-case verification (treating each of the following cases separately: j = j  = i = i  , j = j  = i < i  , j = j  < i = i  , j = j  < i < i  , j < j  ≤ i  < i, j < j  < i  = i, j < j  < i < i  , j = i < j  = i , j = i < j  < i , j < j  = i = i , j < j  = i < i , j < i < j  ≤ i , where we set j := j (β), j  := j (β  ), i := i(β), i  := i(β  )) shows that the corresponding orders of vanishing under the specializations (i) and (ii) are respectively equal to:



# ( j, j  ) ∈ [β] × [β  ] | j = j  − δββ  and # ( j, j  ) ∈ [β] × [β  ] | j = j  + 1 . (1.52) Second, we claim that φd (F) vanishes under the following specializations: (iii) yβ,s = yβ  ,s  for (β, s) < (β  , s  ) such that j (β) < j (β  ) and i(β) + 1 ∈ [β  ]. Indeed, if j (β) < j (β  ) and i(β) + 1 ∈ [β  ], then there are positive roots γ , γ  ∈ + such that j (γ ) = j (β), i(γ ) = i(β  ), j (γ  ) = j (β  ), i(γ  ) = i(β). Consider the

1 Quantum Loop sln , and its Integral Forms

18

degree vector d  ∈ Tk given by dα = dα + δαγ + δαγ  − δαβ − δαβ  . As β < γ < γ  < β  , we have d  > d and therefore φd  (F) = 0 by the assumption. The result follows. Combining the above vanishing conditions (i)–(iii)  and (1.52), we see that φd (F)  is divisible precisely by the product β (Lsln ): Proposition 1.7 The ordered PBWD monomials {eh }h∈H linearly span Uv> (Lsln ). Proof Invoking the injectivity of the homomorphism  : Uv> (Lsln ) → S (n) , Propo  sition 1.6 implies that the ordered PBWD monomials {eh }h∈H span Uv> (Lsln ). Combining Propositions 1.5 and 1.7, we finally obtain: Corollary 1.2 {eh }h∈H form a C(v)-basis of Uv> (Lsln ). This completes our proof of Theorem 1.1 (as noted earlier, the claim for Uv< (Lsln ) can be easily deduced from Corollary 1.2 by using the anti-isomorphism (1.41)). Remark 1.5 We note that the above proof of Theorem 1.1 actually provides a much larger class of the PBWD bases for Uv> (Lsln ) and Uv< (Lsln ), with the PBWD basis elements given explicitly in the shuffle form (rather than long expressions in ei,r , f i,r ).

1.2 Shuffle Realizations and Applications to the PBWD Bases

19

1.2.3 Integral Form S(n) and a Proof of Theorem 1.3 Inspired by the above proof of Theorem 1.1 through the shuffle algebra realization of Uv> (Lsln ), let us now prove Theorem 1.3 by simultaneously providing the shuffle algebra realization of the integral form U> v (Lsln ), which is of independent interest. 1.2.3.1

Integral Form S(n)

 (n) of S (n) consisting of those F ∈ S (n) For k ∈ N I , consider a Z[v, v−1 ]-submodule S k k k for which the numerator f in (1.34) has the following form:  k  ±1 1≤r ≤ki f ∈ (v − v−1 )|k| · Z[v, v−1 ] {xi,r }i∈I , where |k| :=

 i∈I

(1.55)

ki . We define  (n) := S



 (n) , S k

k∈N I

which is easily seen to be a Z[v, v−1 ]-subalgebra of S (n) . To this end, let us note 1 that the shuffle product F  G of (1.32) still makes sense as k!·! · Sym(· · · ) may be simplified as a sum over so-called (k, )-shuffles, since F and G are symmetric.  (n) Moreover, we note that (U> v (Lsln )) ⊂ S , due to the formula (1.48). The key (n) goal of this subsection is to explicitly describe the image (U> v (Lsln )) inside S .  (2) with r ∈ Z. Example 1.11 Consider F(x1 , x2 ) = (v − v−1 )2 (x1 x2 )r ∈ S 2 Then, due to Lemmas 1.8–1.9 below, we have [2]v · F ∈ (U> v (Lsl2 )), but (Lsl )). F∈ / (U> 2 v  (n) . For any We shall now introduce some more specialization maps. Pick F ∈ S k   + d = {dβ }β∈+ ∈ N such that β∈+ dβ β = i∈I ki αi , consider the specialization

±1 }β∈+ β ] d of (1.40). First, we note that φd (F) is divisible by φd (F) ∈ Z[v, v−1 ][{yβ,s 1≤s≤d

Ad := (v − v−1 )|k| .

(1.56)

Second, following the first part of the proof of Lemma 1.7 (see (i), (ii), and (1.52), which are consequences of the wheel conditions (1.35)), φd (F) is also divisible by Bd :=









(yβ,s − v−2 yβ  ,s  )#{( j, j )∈[β]×[β ]| j= j }−δββ 

(β,s) (Lsln ) −→ S (n) of Theo∼ (n) rem 1.8 gives rise to a Z[v, v−1 ]-algebra isomorphism  : U> v (Lsln ) −→ S . Before proceeding to the proof of this result, let us record the following corollary: Corollary 1.3 (a) S(n) is a Z[v, v−1 ]-subalgebra of S (n) . (b) The subalgebra U> v (Lsln ) is independent of the choices (1)–(3) preceding (1.19). 1.2.3.2

Proofs of Theorems 1.3 and 1.13

Both Theorem 1.3(b) and Theorem 1.13 follow from the following two statements: (I) For any k ≥ 1, {β p }kp=1 ⊂ + , {r p }kp=1 ⊂ Z, we have ( eβ1 (r1 ) · · · eβk (rk )) ∈ S(n) . eh )}h∈H . (II) Any F ∈ S(n) may be written as a Z[v, v−1 ]-linear combination of {( The proof of (I) shall be easily deduced from our definition of S(n) , while the proof of (II) will follow from Lemma 1.7 and the validity of (II) for n = 2, see Lemma 1.9. We start by establishing both (I), (II) in the simplest “rank 1” case of n = 2, for which the integral form S(2) has been already described in the Example above. Set  er := (v − v−1 )er ∈ Uv> (Lsl2 ) for r ∈ Z. The following lemma implies (I) for n = 2: er1 · · · erk ) ∈ S(2) Lemma 1.8 For any k ≥ 1 and r1 , . . . , rk ∈ Z, we have ( k . Proof Pick any decomposition k = k1 + · · · + k with all k p ≥ 1. According to the above Example, it suffices to prove that as we specialize the variables x1 , . . . , xk to 1≤r ≤k {v−2r z p }1≤ p≤p , the image of any summand from the symmetrization appearing in  (er1 · · · erk ) ∈ Sk(2) will be divisible by p=1 [k p ]v !, as required in (1.62). To this end, let us fix 1 ≤ p ≤  and consider the relative position of the variables v−2 z p , v−4 z p , . . . , v−2k p z p . If there is an index 1 ≤ r < k p such that v−2(r +1) z p is placed to the left of v−2r z p (that is, (era ) is evaluated at v−2(r +1) z p and (erb ) is evaluated at v−2r z p for a < b), then the above specialization of the corresponding ζ -factor equals

v−2(r +1) z p −v−2 ·v−2r z p v−2(r +1) z p −v−2r z p

= 0. However, if v−2r z p is placed to the left of

v−2(r +1) z p for all 1 ≤ r < k p , then the total contribution of the specializations of the corresponding ζ -factors equals  1≤r p . Let us present F¯ as a linear combination of the monomial symmetric polynomials: ¯ 1 , . . . , xk ) = F(x



μr m r (x1 , . . . , xk ) with μr ∈ Z[v, v−1 ] .

r ∈VN

¯ := {r ∈ VN |μr = 0} Pick the maximal r max = (r1 , . . . , rk ) from the finite set VN ( F) and consider a decomposition k = k1 + · · · + k such that r1 = · · · = rk1 < rk1 +1 = · · · = rk1 +k2 < · · · < rk1 +···+k−1 +1 = · · · = rk . Evaluating F¯ at the resulting specialization {v−2s z p }1≤ p≤p , we see that the coefficient of the lexicographically largest monomial in the variables {z p }p=1 equals μr max , up to an integer power of v. Therefore, the divisibility condition (1.62) implies that 1≤s≤k



μr max

p=1 [k p ]v !

∈ Z[v, v−1 ] .

(1.64)

k k Set F¯ (0) := F¯ and define F¯ (1) ∈ Z[v, v−1 ][{x ±1 via p } p=1 ]

F¯ (1) := F¯ (0) − v

 p=1

k p (k p −1)/2



μr max

p=1 [k p ]v !

(er1 · · · erk ) .

(1.65)

We note that F¯ (1) satisfies the divisibility condition (1.62), due to (1.64) and Lemma 1.8. Applying the same argument to F¯ (1) in place of F = F¯ (0) , we obtain F¯ (2) also satisfying (1.62). Iterating this process, we thus construct a sequence of symmetric Laurent polynomials { F¯ (s) }s∈N satisfying the divisibility condition (1.62). However, invoking our proof of Proposition 1.4, specifically the formula (1.45), ¯ (s) we note that the sequence r (s) max ∈ VN of the maximal elements of VN ( F ) strictly

1.2 Shuffle Realizations and Applications to the PBWD Bases

23

decreases. Meanwhile, the sequence of the minimal powers of any variable in F¯ (s) is a non-decreasing sequence. Hence, F¯ (s) = 0 for some s ∈ N. This completes our proof of Lemma 1.9.  Let us now generalize the above arguments to prove (I) and (II) for any n > 2. The proof of the former is quite similar (though is more tedious) to that of Lemma 1.8: Lemma 1.10 For any m ≥ 1, {β p }mp=1 ⊂ + , and {r p }mp=1 ⊂ Z, we have:   eβm (rm ) ∈ S(n) .   eβ1 (r1 ) · · ·     eβ1 (r1 ) · · · eβm (rm ) Proof Define k ∈ N I via i∈I ki αi = mp=1 β p , so that F :=    (n) as (v − v−1 )|k| divides F, due to Lemma 1.4. is in Sk(n) . First, we note that F ∈ S k It remains to show that ϒd,t (F) satisfies the divisibility condition of Definition 1.1 + 1≤s≤ for any collections d = {dβ }β∈+ ∈ N and t = {tβ,s }β∈+ β satisfying (1.60). To this end, we recall that the cross specialization ϒd,t (F) is computed in three steps: • first, we specialize x∗,∗ ’s in f of (1.34) to v? -multiples of y∗,∗ ’s as in (1.40); • second, we divide that specialization by the product of appropriate powers of v − v−1 and linear terms in y∗,∗ ’s due to the wheel conditions, see (1.56)–(1.58); • finally, we specialize y∗,∗ -variables to v? -multiples of z ∗,∗ -variables as in (1.59). Fix β ∈ + and 1 ≤ s ≤ β , and consider those x∗,∗ ’s that eventually got special1≤r ≤tβ,s ized to v? z β,s . Without loss of generality, we may assume those are {xi,r } j (β)≤i≤i(β) . −i We may also assume that xi,r got specialized to v yβ,r under the first specialization (1.40), while yβ,r got further specialized to v−2r z β,s under the second specialization (1.59), for any j (β) ≤ i ≤ i(β) and 1 ≤ r ≤ tβ,s . For j (β) ≤ i < i(β) and 1 ≤ r = r  ≤ tβ,s , consider the relative position of the variables xi,r , xi,r  , xi+1,r  in any summand from the symmetrization appearing in F. e∗ (∗)), xi,r is placed either to the left of As xi,r , xi,r  cannot enter the same function ( xi,r  or to the right. In the former case, we gain the factor ζi,i (xi,r /xi,r  ), which upon the specialization φd contributes the factor (yβ,r − v−2 yβ,r  ). Likewise, if xi+1,r  is placed to the left of xi,r , we gain the factor ζi+1,i (xi+1,r  /xi,r ), which upon the specialization φd contributes the factor (yβ,r − v−2 yβ,r  ) as well. In the remaining case, when xi,r  is placed to the left of xi,r while xi+1,r  is not, we gain the factor ζi,i+1 (xi,r  /xi+1,r  ), which specializes to 0 under the specialization φd . As i ranges from j (β) up to i(β) − 1, we thus gain the (i(β) − j (β))-th power of (yβ,r − v−2 yβ,r  ). The latter precisely coincides with the power of (yβ,r − v−2 yβ,r  ) in Bd of (1.57), by which we divide φd (F) to obtain ϕd (F) of (1.58). However, we have not used yet ζ -factors ζi(β),i(β) (xi(β),r /xi(β),r  ) for xi(β),r placed to the left of xi(β),r  . If xi(β),r +1 is placed to the left of xi(β),r for some 1 ≤ r < tβ,s , then ζi(β),i(β) (xi(β),r +1 /xi(β),r ) specializes to 0 under (1.59). In the remaining case, when each xi(β),r is placed to the left of xi(β),r +1 , the total contribution of the specializations of the corresponding ζ -factors equals v−tβ,s (tβ,s −1)/2 [tβ,s ]v !, cf. (1.63). This completes the proof of Lemma 1.10. 

1 Quantum Loop sln , and its Integral Forms

24

 ⊂ S (n) denote the Z[v, v−1 ]-span of {( Let M eh )}h∈H , the images of the ordered PBWD monomials, and define Tk as before. Generalizing Lemma 1.7, we have:  Lemma 1.11 Consider F ∈ S(n) k and a degree vector d ∈ Tk . If φd (F) = 0 for all (n)    ∩ Sk such that d ∈ Tk such that d > d, then there exists an element Fd ∈ M

φd (F) = φd (Fd )

and

φd  (Fd ) = 0 for all d  > d .

Proof The proof is completely analogous to that of Lemma 1.7. More precisely, combining the formulas (1.53), (1.54), the condition F ∈ S(n) k together with the formula (1.50) and the discussion of a “rank 1 reduction” after it, the result follows from its n = 2 counterpart. The latter has been already established in Lemma 1.9. Combining Lemmas 1.10–1.11, we obtain (II) similarly to Proposition 1.6: (n)  = S(n) and  : U> Proposition 1.8 We have M is surjective. v (Lsln ) → S

Combining this with the injectivity of  (Proposition 1.3) implies Theorem 1.13. Corollary 1.4 { eh }h∈H is a basis of the free Z[v, v−1 ]-module U> v (Lsln ). This completes our proof of Theorem 1.3(b) (as before, the claim for U< v (Lsln ) follows from Corollary 1.4 by using the algebra anti-isomorphism (1.41)). Finally, the result of Theorems 1.3(a) has been already established in Corollary 1.3(b).

1.2.4 Integral Form S(n) and a Proof of Theorem 1.6 Likewise, let us now prove Theorem 1.6 by simultaneously providing the shuffle algebra realization of the integral form U> v (Lsln ), which is of independent interest. Definition 1.2 We call F ∈ Sk(n) good if it satisfies the following two properties:  k  f (x1,1 ,...,xn−1,k ) ±1 1≤r ≤ki (i) F = n−2 r  ≤ki+1 n−1 with f ∈ Z[v, v−1 ] {xi,r }i∈I ; i=1

r ≤ki

(xi,r −xi+1,r  )



(ii) the specialization φd (F) of (1.40) is divisible by (v − v−1 ) β∈+ dβ (i(β)− j (β)) for   + any degree vector d = {dβ }β∈+ ∈ N satisfying β∈+ dβ β = i∈I ki αi . Example 1.14 In the simplest “rank 1” case of n = 2, we have: k k F ∈ Sk(2) is good ⇐⇒ F ∈ Z[v, v−1 ][{x ±1 p } p=1 ] . (n) −1 Let S(n) k ⊂ Sk denote the Z[v, v ]-submodule of all good elements and set

S(n) :=

 k∈N I

S(n) k .

1.2 Shuffle Realizations and Applications to the PBWD Bases

25

The following is the key result of this section: ∼

Theorem 1.15 The C(v)-algebra isomorphism  : Uv> (Lsln ) −→ S (n) of Theo∼ (n) rem 1.8 gives rise to a Z[v, v−1 ]-algebra isomorphism  : U> v (Lsln ) −→ S . Corollary 1.5 S(n) is a Z[v, v−1 ]-subalgebra of S (n) . Both Theorem 1.6 and Theorem 1.15 follow from the following three results: (I) For any N ∈ N, {(i p , r p , k p )} Np=1 ⊂ I ×Z×N, we have (ei(k1 ,r1 )1 · · · ei(kN N,r)N ) ∈ S(n) . (II) For any h ∈ H and eh defined in (1.29), we have (eh ) ∈ S(n) . (III) Any F ∈ S(n) may be written as a Z[v, v−1 ]-linear combination of {(eh )}h∈H . The proof of the first statement (I) is straightforward: Lemma 1.12 For any N ∈ N and {(i p , r p , k p )} Np=1 ⊂ I × Z × N, we have:    ei(k1 ,r1 )1 · · · ei(kN N,r)N ∈ S(n) . Proof The validity of (i) for F = (ei(k1 ,r1 )1 · · · ei(kN N,r)N ) follows immediately from the

(k) ) = v− 2 (x1 · · · xk )r , due to Lemma 1.3. To verify (ii), pick any equality (ei,r   + degree vector d = {dβ }β∈+ ∈ N such that β∈+ dβ β = Np=1 k p αi p . For any β ∈ + and 1 ≤ s ≤ dβ , consider ζ -factors between those pairs of x∗,∗ -variables that are specialized to v− yβ,s and v−−1 yβ,s with j (β) ≤  < i(β) under the special−1 ization map (1.40). Each of them contributes a multiple of (v − v ) into φd (F), and there are exactly β∈+ dβ (i(β) − j (β)) of such pairs. Hence, (ii) indeed holds.  k(k−1)

The above lemma can be rephrased as:   (n)  U> . v (Lsln ) ⊆ S

(1.66)

Thus, the second result (II) follows from (previously stated Proposition 1.2): Lemma 1.13 For any β ∈ + , r ∈ Z, k ∈ N, we have eβ (r )(k) ∈ U> v (Lsln ). Proof For β = α j + α j+1 + · · · + αi and r ∈ Z, consider the PBWD basis element eβ (r ) of (1.19), and let e¯β (r ) denote the particular choice of (1.20). Combining Lemma 1.4 with the Example following after it, we get (eβ (r )) = m m j+1 mi (e¯β (r )) · c · x j,1j x j+1,1 · · · xi,1 for some m = (m j , m j+1 , . . . , m i ) ∈ Zi− j+1 satisfying m j + · · · + m i = 0 and c ∈ ±vZ . Thus, we have: i       eβ (r )(k) =  e¯ β (r )(k) · ck · (xι,1 · · · xι,k )m ι .

(1.67)

ι= j

Consider an algebra automorphism σm of Uv> (Lsln ) that maps eι,k → eι,k+m ι if j ≤ (k) (k) ι ≤ i and eι,k → eι,k otherwise. Since σm maps e(k) ι,r to eι,r +m ι or eι,r , it does induce the

1 Quantum Loop sln , and its Integral Forms

26

same-named automorphism of U> v (Lsln ). Then, combining (1.67) and Theorem 1.8, we see that eβ (r )(k) = ck σm (e¯ β (r )(k) ). However, the inclusion e¯ β (r )(k) ∈ U> v (Lsln ) has been already established in our proof of Proposition 1.2. Therefore, we get eβ (r )(k) ∈ U> v (Lsln ), which completes our proof of Lemma 1.13 (and hence also of Proposition 1.2).  Let M ⊂ S (n) denote the Z[v, v−1 ]-span of {(eh )}h∈H and define Tk as before. Generalizing Lemma 1.7, we have:  Lemma 1.14 Consider F ∈ S(n) k and a degree vector d ∈ Tk . If φd (F) = 0 for all

d  ∈ Tk such that d  > d, then there exists an element Fd ∈ M ∩ Sk(n) such that φd (F) = φd (Fd )

and

φd  (Fd ) = 0 for all d  > d .

Proof The proof is completely analogous to that of Lemma 1.7. More precisely, combining the formula (1.49) with the condition F ∈ S(n) k , we obtain the formula (1.53)  d  ±1 1≤s≤dβ }β∈+ . To this end, with the factor G necessarily of the form G ∈ Z[v, v−1 ] {yβ,s

it suffices to note that the power of (v − v−1 ) in the condition (ii) of Definition 1.2 precisely coincides with the overall power of (v − v−1 ) arising from the factors G β in (1.49). Therefore, the result follows from its “rank 1” n = 2 counterpart, which r1 rN N is a basis of the free Z[v, v−1 ]-module S(2) . states that { [rx1 ]v !  · · ·  [rxN ]v ! }rN1 ≺···≺r ∈N The proof of this claim is completely analogous to that of Proposition 1.4(a): the key  N r p (r p −1) point is that the formula (1.45) will get replaced by νr = v− p=1 2 .  Combining Lemmas 1.12–1.14, we obtain (III) similarly to Proposition 1.6: (n) Proposition 1.9 We have M = S(n) and  : U> is surjective. v (Lsln ) → S

This completes our proof of Theorem 1.15 and Theorem 1.6 (as before, the claim for U< v (Lsln ) follows by using the algebra anti-isomorphism (1.41)). Remark 1.6 We note that an interesting duality between the integral forms U> v (Lsln ) < > and U< v (Lsln ) (similarly, between Uv (Lsln ) and Uv (Lsln )) was established in [18] using the shuffle realizations of these integral forms from Theorems 1.13 and 1.15.

1.3 Generalizations to Two-Parameter Quantum Loop Algebras The two-parameter quantum loop algebra Uv1 ,v2 (Lsln ) was first introduced in [13]4 as a generalization of Uv (Lsln ), the latter recovered by setting v1 = v−1 2 = v and identifying some Cartan elements, see [13, Remark 3.3(4)]. The key results of [13] are: 4

To be more precise, this recovers the algebra of loc.cit. with the trivial central charges. However, we can ignore this since their “positive” subalgebras are naturally isomorphic.

1.3 Generalizations to Two-Parameter Quantum Loop Algebras

27

• [13, Theorem 3.12]: the Drinfeld-Jimbo type realization of Uv1 ,v2 (Lsln ), • [13, Theorem 3.11]: the PBW basis of its subalgebras Uv>1 ,v2 (Lsln ), Uv1 ,v2 (Lsln ), thus finally proving [13, Theorem 3.11]. Along the way, we also generalize Theorem 1.8 by providing the shuffle algebra realization of Uv>1 ,v2 (Lsln ), which is of independent interest. The latter is also used to construct the PBWD bases for > the integral form U> v1 ,v2 (Lsln ) of Uv1 ,v2 (Lsln ), thus generalizing Theorem 1.3.

1.3.1 Quantum Loop Algebra Uv>1 ,v2 (Lsln ) and Its PBWD Bases To simplify our exposition, we shall consider only the “positive” subalgebra Uv>1 ,v2 (Lsln ) of Uv1 ,v2 (Lsln ). Let v1 , v2 be two independent formal variables and 1/2 1/2 set K := C(v1 , v2 ). Following [13, Definition 3.1], we define Uv>1 ,v2 (Lsln ) as the ∈Z associative K-algebra generated by {ei,r }ri∈I with the following defining relations:    z − ( j, ii, j)1/2 w ei (z)e j (w) =  j, iz − ( j, ii, j−1 )1/2 w e j (w)ei (z) (1.68) as well as quantum Serre relations: 

ei (z)e j (w) = e j (w)ei (z) if ci j = 0 , [ei (z 1 ), [ei (z 2 ), ei+1 (w)]v2 ]v1 + [ei (z 2 ), [ei (z 1 ), ei+1 (w)]v2 ]v1 = 0 , (1.69) −1 + [ei (z 2 ), [ei (z 1 ), ei−1 (w)] −1 ] −1 = 0 , [ei (z 1 ), [ei (z 2 ), ei−1 (w)]v−1 ] v1 v2 v1 2 where ei (z) =



r ∈Z ei,r z

−r

and i, j ∈ K is defined via δ −δi+1, j δi−1, j −δi j v2

i, j := v1i j

.

We follow our previous notations, except that now (λ1 , . . . , λ p−1 ) ∈ {v1 , v2 } p−1 . Similarly to (1.19), we define the PBWD basis elements eβ (r ) ∈ Uv>1 ,v2 (Lsln ) via: eβ (r ) := [· · · [[ei1 ,r1 , ei2 ,r2 ]λ1 , ei3 ,r3 ]λ2 , · · · , ei p ,r p ]λ p−1 .

(1.70)

Following (1.21), the monomials eh :=

→  (β,r )∈+ ×Z

eβ (r )h(β,r )

with

h∈H

(1.71)

1 Quantum Loop sln , and its Integral Forms

28

will be called the ordered PBWD monomials of Uv>1 ,v2 (Lsln ), where the arrow → over the product sign indicates that the product is ordered with respect to (1.18). Our first main result establishes the PBWD property of Uv>1 ,v2 (Lsln ) (the proof is outlined in Sect. 1.3.4 and is based on the shuffle approach): Theorem 1.16 {eh }h∈H of (1.71) form a K-basis of Uv>1 ,v2 (Lsln ). Remark 1.7 We note that the PBWD basis elements introduced in [13, (3.14)] are eα j +α j+1 +···+αi (r ) := [· · · [[e j,r , e j+1,0 ]v1 , e j+2,0 ]v1 , · · · , ei,0 ]v1 .

(1.72)

In this setup, Theorem 1.16 recovers Theorem 3.11 of [13] presented without a proof. Remark 1.8 We note that the entire two-parameter quantum loop algebra Uv1 ,v2 (Lsln ) admits a triangular decomposition similar to that of Uv (Lsln ) from Proposition 1.1(a), cf. [13, §3.2]. Hence, a natural analogue of Theorem 1.2 holds for Uv1 ,v2 (Lsln ) as well, thus providing a family of PBWD K-bases for Uv1 ,v2 (Lsln ).

1.3.2 Integral Form U> v1 ,v2 (Lsln ) and Its PBWD Bases Following (1.23), for any (β, r ) ∈ + × Z, we define  eβ (r ) ∈ Uv>1 ,v2 (Lsln ) via:   1/2 −1/2 −1/2 1/2  eβ (r ) := v1 v2 − v1 v2 eβ (r ) .

(1.73)

eβ (r ) of (1.73) instead of eβ (r ). We also define eh ∈ Uv>1 ,v2 (Lsln ) via (1.71) but using 1/2 1/2 −1/2 −1/2 > Finally, let us define an integral form Uv1 ,v2 (Lsln ) as the Z[v1 , v2 , v1 , v2 ]∈Z subalgebra of Uv>1 ,v2 (Lsln ) generated by { eβ (r )}rβ∈ +. The following natural counterpart of Theorem 1.3 strengthens our Theorem 1.16: Theorem 1.17 (a) The subalgebra U> v1 ,v2 (Lsln ) is independent of all our choices. 1/2

1/2

−1/2

(b) { eh }h∈H form a basis of the free C[v1 , v2 , v1

−1/2

, v2

]-module U> v1 ,v2 (Lsln ).

The proof of Theorem 1.17 follows easily from that of Theorem 1.16 sketched below in the same way as we deduced the proof of Theorem 1.3 in Sect. 1.2.3.2 from that of Theorem 1.1; we leave details to the interested reader. Remark 1.9 Similarly to Remark 1.3, let us point out that it is often more convenient to work with the two-parameter quantum loop algebra Uv1 ,v2 (Lgln ) and its similarly defined integral form Uv1 ,v2 (Lgln ). Following the arguments of [10, Proposition 3.11], Uv1 ,v2 (Lgln ) is identified with the RTT integral form Urtt v1 ,v2 (Lgln ) under the K-algebra 1/2 1/2 −1/2 −1/2 K of [14]. Therefore, (Lgl ) ⊗ isomorphism Uv1 ,v2 (Lgln ) Urtt n v1 ,v2 Z[v1 ,v2 ,v1 ,v2 ] the analogue of [10, Theorem 3.24] (cf. our Theorem 1.4) provides a family of Poincaré-Birkhoff-Witt-Drinfeld bases for Uv1 ,v2 (Lgln ), cf. Theorem 1.5.

1.3 Generalizations to Two-Parameter Quantum Loop Algebras

29

(n) 1.3.3 Shuffle Algebra S We define the shuffle algebra ( S (n) , ) similarly to (S (n) , ) with three modifications: (1) all vector spaces are now defined over K (rather than over C(v)); (2) the shuffle factors (1.31) used in the shuffle product (1.32) are now replaced with:  ζi, j (z) =

1/2 −1/2

z − v1 v2 z−1

δ j,i−1 

z − v−1 1 v2 z−1

δ ji  1/2 1/2 v1 v2

1/2 −1/2

z − v1 v2 · z−1

δ j,i+1 ;

(3) the wheel conditions (1.35) for F are replaced with   1/2 −1/2 F {xi,r } = 0 once xi,r1 = v1 v2 xi+,s = v1 v−1 2 x i,r2 for some  ∈ {±1}, i, r1 = r2 , s. The following result relates ( S (n) , ) to Uv>1 ,v2 (Lsln ), generalizing Proposition 1.3: Proposition 1.10 There exists an injective K-algebra homomorphism S (n)  : Uv>1 ,v2 (Lsln ) −→ 

(1.74)

r for any i ∈ I, r ∈ Z. such that ei,r → xi,1

Our proof of Theorem 1.16 also implies the following counterpart of Theorem 1.8: ∼

Theorem 1.18  : Uv>1 ,v2 (Lsln ) −→  S (n) of (1.74) is a K-algebra isomorphism.

1.3.4 Proofs of Theorems 1.16 and 1.18 The proofs of Theorems 1.16, 1.18 are completely analogous to our proofs of TheoS (n) of Proposition 1.10. rems 1.1, 1.8, and utilize the embedding  : Uv>1 ,v2 (Lsln ) →  To this end, we note that: • The linear independence of {eh }h∈H is deduced exactly as in Proposition 1.5 with ±1 1≤s≤dβ d S(n) → K[{yβ,s }β∈+ ] the only modification of the specialization maps φd :  1/2 −1/2

of (1.40) by replacing v−i with (v1 v2 )−i everywhere. Then, the results of Lemmas 1.5 and 1.6 still hold, thus implying the linear independence of {eh }h∈H .

• The fact that {eh }h∈H span Uv>1 ,v2 (Lsln ) is deduced exactly as in Proposition 1.7. To be more precise, Lemma 1.7 still holds and its iterative application immediately implies that any shuffle element F ∈  S (n) belongs to the K-span of {(eh )}h∈H . • The latter observation also implies the surjectivity of the algebra embedding  from Proposition 1.10, thus proving Theorem 1.18.

30

1 Quantum Loop sln , and its Integral Forms

1.4 Generalizations to Quantum Loop Superalgebras The quantum loop superalgebra Uv (Lsl(m|n))5 was introduced in [20], both in the Drinfeld-Jimbo and new Drinfeld realizations, see [20, Theorem 8.5.1] for an identification of those. The representation theory of these algebras was partially studied in [21] by crucially utilizing a weak version of the PBW theorem for Uv> (Lsl(m|n)), see [21, Theorem 3.12]. Inspired by [13, Theorem 3.11] discussed above, the author also conjectured the PBW theorem for Uv> (Lsl(m|n)), see [21, Remark 3.13(2)]. The primary goal of this section is to generalize Theorem 1.1 to the case of Uv> (Lsl(m|n)), thus proving the aforementioned conjecture of [21]. Along the way, we also generalize Theorem 1.8 by providing the shuffle realization of Uv> (Lsl(m|n)), which is of independent interest. The latter is also used to construct > the PBWD bases for the integral form U> v (Lsl(m|n)) of Uv (Lsl(m|n)), thus generalizing Theorem 1.3. Remark 1.10 (a) Let us emphasize right away that in our exposition below we consider the distinguished Dynkin diagram with a single simple positive odd root. The generalization of all our results to an arbitrary Dynkin diagram is carried out in [17]. (b) The new Drinfeld realization of quantum loop superalgebras surprisingly has not been properly developed yet. Thus, the shuffle approach, involving both symmetric and skew-symmetric functions, provides a promising new toolkit for such a treatment.

1.4.1 Quantum Loop Superalgebra Uv> (Lsl(m|n)) To simplify our exposition, we shall only consider the “positive” subalgebra Uv> (Lsl(m|n)) of Uv (Lsl(m|n)). From &m+nnow on, we redefine I = {1, . . . , m + n − 1}. Zi with the bilinear form (·, ·) determined Let us consider a free Z-module i=1 by (i ,  j ) = (−1)δi>m δi j . Let v be a formal variable and define {vi }i∈I ⊂ {v, v−1 } via vi := v(i ,i ) . For i, j ∈ I , we set c¯i j := (αi , α j ) with αi := i − i+1 . Following [20] (cf. [21, Theorem 3.3]), we define Uv> (Lsl(m|n)) as the associative ∈Z , with the Z2 -grading defined by: C(v)-superalgebra generated by {ei,r }ri∈I |em,r | = 1¯ , |ei,r | = 0¯ for all i = m , r ∈ Z ,

(1.75)

and subject to the following defining relations:

Here, one actually needs to use the classical Lie superalgebra A(m − 1, n − 1) in place of sl(m|n), which do coincide only for m = n. However, we shall ignore this difference, since we will consider only the “positive” subalgebras and those are isomorphic: Uv> (Lsl(m|n)) Uv> (L A(m −1, n −1)). 5

1.4 Generalizations to Quantum Loop Superalgebras

31

(z − vc¯i j w)ei (z)e j (w) = (vc¯i j z − w)e j (w)ei (z) if c¯i j = 0 ,

(1.76)

quadratic and cubic quantum Serre relations: [ei (z), e j (w)] = 0 if c¯i j = 0 , [ei (z 1 ), [ei (z 2 ), e j (w)]v−1 ]v + [ei (z 2 ), [ei (z 1 ), e j (w)]v−1 ]v = 0 if c¯i j = ±1 and i  = m ,

(1.77) as well as quartic quantum Serre relations: [[[em−1 (w), em (z 1 )]v−1 , em+1 (u)]v , em (z 2 )] + [[[em−1 (w), em (z 2 )]v−1 , em+1 (u)]v , em (z 1 )] = 0 . Here, ei (z) =



r ∈Z ei,r z

−r

(1.78)

and we use the super-bracket notations:

[a, b]x := ab − (−1)|a||b| x · ba ,

(1.79)

[a, b] := [a, b]1 ¯

¯

for Z2 -homogeneous elements a and b (with conventions (−1)0 = 1, (−1)1 = −1).

1.4.2 PBWD Bases of Uv> (Lsl(m|n)) Let + = {α j + α j+1 + · · · + αi }1≤ j≤i (Lsl(m|n)) via (1.19), but with [·, ·]λk denoting the super-bracket (1.79). ¯ be the set of all functions h : + × Z → N with finite support and such Let H that h(β, r ) ≤ 1 if |β| = 1¯ . Following (1.21), the monomials eh :=

→ 

eβ (r )h(β,r )

with

¯ h∈H

(1.81)

(β,r )∈+ ×Z

will be called the ordered PBWD monomials of Uv> (Lsl(m|n)), where the arrow → over the product sign indicates that the product is ordered with respect to (1.18).

32

1 Quantum Loop sln , and its Integral Forms

Our first main result establishes the PBWD property of Uv> (Lsl(m|n)) (the proof is outlined in Sect. 1.4.5 and is based on the shuffle approach): Theorem 1.19 {eh }h∈H¯ of (1.81) form a C(v)-basis of Uv> (Lsl(m|n)). Remark 1.11 We note that the PBWD basis elements introduced in [21, (3.12)] are eα j +α j+1 +···+αi (r ) := [· · · [[e j,r , e j+1,0 ]v j+1 , e j+2,0 ]v j+2 , · · · , ei,0 ]vi .

(1.82)

In this setup, Theorem 1.19 recovers the conjecture from Remark 3.13(2) of [21]. Remark 1.12 We note that the entire quantum loop superalgebra Uv (Lsl(m|n)) admits a triangular decomposition similar to that of Uv (Lsln ) from Proposition 1.1(a), see [21, Theorem 3.3]. Hence, a natural analogue of Theorem 1.2 holds for Uv (Lsl(m|n)) as well, thus providing a family of PBWD C(v)-bases for Uv (Lsl(m|n)).

1.4.3 Integral Form U> v (Lsl(m|n)) and Its PBWD Bases Following (1.23), for any (β, r ) ∈ + × Z, we define  eβ (r ) ∈ Uv> (Lsl(m|n)) via:  eβ (r ) := (v − v−1 )eβ (r ) .

(1.83)

¯ we also define  For h ∈ H, eh ∈ Uv> (Lsl(m|n)) via (1.81) but using  eβ (r ) of (1.83) (Lsl(m|n)) as the Z[v, v−1 ]instead of eβ (r ). Finally, let us define an integral form U> v ∈Z eβ (r )}rβ∈ . subalgebra of Uv> (Lsl(m|n)) generated by { + The following natural counterpart of Theorem 1.3 strengthens our Theorem 1.19: Theorem 1.20 (a) The subalgebra U> v (Lsl(m|n)) is independent of all our choices. (b) The elements { eh }h∈H¯ form a basis of the free Z[v, v−1 ]-module U> v (Lsl(m|n)). The proof of Theorem 1.20 follows easily from that of Theorem 1.19 outlined below in the same way as we deduced the proof of Theorem 1.3 in Sect. 1.2.3.2 from that of Theorem 1.1; we leave details to the interested reader. Remark 1.13 Similarly to Remarks 1.3 and 1.9, let us note that it is often more convenient to work with the quantum loop superalgebra Uv (Lgl(m|n)) and its similarly defined integral form Uv (Lgl(m|n)). Following the arguments of [10, Proposition 3.11], Uv (Lgl(m|n)) is identified with the RTT integral form Urtt v (Lgl(m|n)), under the isomorphism Uv (Lgl(m|n)) Urtt v (Lgl(m|n)) ⊗Z[v,v−1 ] C(v), see [22, Definition 3.1]. Hence, the analogue of [10, Theorem 3.24] (cf. our Theorem 1.4) provides a family of PBWD bases for Uv (Lgl(m|n)), cf. Theorem 1.5.

1.4 Generalizations to Quantum Loop Superalgebras

33

1.4.4 Shuffle Algebra S(m|n) Consider an N I -graded C(v)-vector superspace 

S(m|n) =

(m|n)

Sk

,

k=(k1 ,...,km+n−1 )∈N I (m|n)

Z2 -graded by the parity of km , where Sk

consists of supersymmetric rational

1≤r ≤ki functions in {xi,r }i∈I , that is, symmetric in {xi,r }rki=1 for every i = m and skewkm symmetric in {xm,r }r =1 . We also fix an I × I matrix of rational functions ζi, j (z) via:

ζi, j (z) =

z − v−c¯i j . z−1

(1.84)

We equip S(m|n) with a structure of an associative algebra with the shuffle product defined similar to (1.32), but Sym of (1.33) being replaced with a supersymmetrization SSym. Here, the supersymmetrization of f ∈ C({xi,1 , . . . , xi,si }i∈I ) is defined via:   SSym( f ) {xi,1 , . . . , xi,si }i∈I    := sgn(σm ) f {xi,σi (1) , . . . , xi,σi (si ) }i∈I .

(1.85)

(σ1 ,...,σm+n−1 )∈ s

As before, let us consider the subspace of S(m|n) defined by the pole and wheel conditions (though now there will be two kinds of the wheel conditions): (m|n)

• We say that F ∈ Sk

satisfies the pole conditions if

f (x1,1 , . . . , xm+n−1,km+n−1 ) F = m+n−2 r  ≤ki+1 , i=1 r ≤ki (x i,r − x i+1,r  )

(1.86)

±1 1≤r ≤ki where f ∈ C(v)[{xi,r }i∈I ] is a supersymmetric Laurent polynomial, that is, ki symmetric in {xi,r }r =1 for every i = m and skew-symmetric in {xm,r }rkm=1 . (m|n)

• We say that F ∈ Sk

satisfies the first kind wheel conditions if

  F {xi,r } = 0 once xi,r1 = vi xi+,s = vi2 xi,r2

(1.87)

for some  ∈ {±1}, i ∈ I \{m}, 1 ≤ r1 = r2 ≤ ki , 1 ≤ s ≤ ki+ . (m|n)

• We say that F ∈ Sk

satisfies the second kind wheel conditions if

  F {xi,r } = 0 once xm−1,s = vxm,r1 = xm+1,s  = v−1 xm,r2

(1.88)

1 Quantum Loop sln , and its Integral Forms

34

for some 1 ≤ r1 = r2 ≤ km , 1 ≤ s ≤ km−1 , 1 ≤ s  ≤ km+1 . (m|n)

Let Sk

(m|n)

⊂ Sk

denote the subspace of all F satisfying these conditions and set S (m|n) :=



(m|n)

Sk

.

k∈N I

It is straightforward to check that S (m|n) ⊂ S(m|n) is -closed, cf. Lemma 1.2. Similar to Proposition 1.3, the shuffle algebra S (m|n) ,  is related to Uv> (Lsl(m|n)) via: Proposition 1.11 There exists an injective C(v)-superalgebra homomorphism  : Uv> (Lsl(m|n)) −→ S (m|n)

(1.89)

r for any i ∈ I, r ∈ Z. such that ei,r → xi,1

In fact, we have the following generalization of Theorem 1.8: ∼

Theorem 1.21  : Uv> (Lsl(m|n)) −→ S (m|n) of (1.89) is a C(v)-superalgebra isomorphism.

1.4.5 Proofs of Theorems 1.19 and 1.21 The proofs of Theorems 1.19, 1.21 are completely analogous to those of Theorems 1.1, 1.8, and utilize the embedding  : Uv> (Lsl(m|n)) → S (m|n) of Proposition 1.11. Let us first establish both theorems in the simplest “rank 1” m = n = 1 case: Proposition 1.12 Fix any total order  on Z. Then: 1 ≺···≺rk (a) the ordered monomials {x r1  x r2  · · ·  x rk }rk∈N form a C(v)-basis of S (1|1) . 1 ≺···≺rk (b) the ordered monomials {er1 er2 · · · erk }rk∈N form a C(v)-basis of Uv> (Lsl(1|1)).

Proof To prove (a), it suffices to note the superalgebra isomorphism S (1|1) ⊕k k , where k denotes the vector space of skew-symmetric Laurent polynomials in k variables, while the algebra structure on ⊕k k arises via the skew-symmetrization maps k ⊗  → k+ . Combining part (a) with Proposition 1.11 implies part (b).  +

Let us now treat the general  m, n ∈ N. For adegree vector d = {dβ }β∈+ ∈ N , I I define its N -degree  ∈ N via β∈+ dβ β = i∈I i αi . The specialization map φd , vanishing on degree =  components, and thus determined by (m|n)

φd : S

  ±1 1≤s≤dβ −→ C(v) {yβ,s }β∈+

(1.90)

1.4 Generalizations to Quantum Loop Superalgebras

35

is defined alike (1.40), with the only modification that the variable xi,r in the s-th copy of [β] is specialized to v−i yβ,s if i ≤ m and to vi−2m yβ,s if i > m. Let us note dβ right away that φd (F) is a supersymmetric Laurent polynomial in {yβ,s }s=1 for any dβ ¯ if |β| = 0¯ and skew-symmetric if |β| = 1. β ∈ + , that is, symmetric in {yβ,s }s=1 ¯ Then, the results of Lemmas 1.5 and 1.6 still hold, with H being used instead of H, ¯ with thus implying the linear independence of {eh }h∈H¯ . Furthermore, for any h ∈ H deg(h) = d, the formula (1.50) still holds but with the factors generalizing (1.49): 1≤s  ≤dβ 



G β,β  =

(yβ,s − v−2 yβ  ,s  )ν

−

(β,β  )

· (yβ,s − v2 yβ  ,s  )ν

+

(β,β  )

×

1≤s≤dβ 1≤s  ≤dβ 



(yβ,s − yβ  ,s  )δ j (β  )> j (β) δi(β)+1∈[β  ] +δm∈[β] δm∈[β  ] ,

1≤s≤dβ

G β = (1 − v2 )dβ (i(β)− j (β)) ·

 1≤s=s  ≤dβ

(σ )

Gβ β



(yβ,s − v2 yβ,s  )i(β)− j (β) ·

⎧ yβ,σβ (s) −v−2 yβ,σβ (s  ) ⎪ ⎪  ⎪ s i(β) dβ ⎨  2 rβ (h,s) = yβ,σβ (s) ·   yβ,σβ (s) −v yβ,σβ (s ) if m < j (β) . s m ,



j  ) ∈ [β] × [β  ] | j = j  > m + # ( j, j  ) ∈ [β] × [β  ] | j = j  + 1 ≤ m .

The key observation is that for any β ∈ + , the sum

 σβ ∈ dβ

(σ )

G β β coincides with

the value of the shuffle product x rβ (h,1)  · · ·  x rβ (h,dβ ) , viewed as an element of S (2|0) if m > i(β), S (0|2) if m < j (β), or S (1|1) if m ∈ [β] (a “rank 1 reduction” dβ feature), evaluated at {yβ,s }s=1 . The latter elements are linearly independent, due to Propositions 1.4(a) and 1.12(a), hence so are {eh }h∈H¯ of (1.81), cf. Proposition 1.5. The fact that the ordered PBWD monomials {eh }h∈H¯ span Uv> (Lsl(m|n)) follows from the validity of Lemma 1.7 in the present setup. The latter result also implies the surjectivity of the homomorphism  from (1.89), thus proving Theorem 1.21.

36

1 Quantum Loop sln , and its Integral Forms

References 1. Beck, J.: Convex bases of PBW type for quantum affine algebras. Commun. Math. Phys. 165(1), 193–199 (1994) 2. Chari, V., Pressley, A.: Quantum affine algebras at roots of unity. Represent. Theory 1, 280–328 (1997) 3. Drinfeld, V.: A New realization of Yangians and quantized affine algebras. Sov. Math. Dokl. 36(2), 212–216 (1988)  4. Ding, J., Frenkel, I.: Isomorphism of two realizations of quantum affine algebra Uq (gl(n)). Commun. Math. Phys. 156(2), 277–300 (1993) 5. Enriquez, B.: PBW and duality theorems for quantum groups and quantum current algebras. J. Lie Theory 13(1), 21–64 (2003) 6. Feigin, B., Odesskii, A.: Sklyanin’s elliptic algebras. (Russian) Funktsional. Anal. i Prilozhen. 23(3), 45–54 (1989). translation in Funct. Anal. Appl. 23(3), 207–214 (1990) 7. Feigin, B., Odesskii, A.: Elliptic deformations of current algebras and their representations by difference operators. (Russian) Funktsional. Anal. i Prilozhen. 31(3), 57–70 (1997). translation in Funct. Anal. Appl. 31(3), 193–203 (1998) 8. Feigin, B., Odesskii, A.: Quantized moduli spaces of the bundles on the elliptic curve and their applications. In: Integrable Structures of Exactly Solvable Two-dimensional Models of Quantum Field Theory (Kiev, 2000), 123–137; NATO Sci. Ser. II Math. Phys. Chem., 35. Kluwer Academic Publishing, Dordrecht (2001) 9. Faddeev, L., Reshetikhin, N., Takhtadzhyan, L.: Quantization of Lie groups and Lie algebras. (Russian) Algebra i Analiz 1(1), 178–206 (1989). translation in Leningrad Math. J. 1(1), 193–225 (1990) 10. Finkelberg, M., Tsymbaliuk, A.: Shifted quantum affine algebras: integral forms in type A (with appendices by A. Tsymbaliuk, A. Weekes). Arnold Math. J. 5(2–3), 197–283 (2019) 11. Grojnowski, I.: Affinizing quantum algebras: from D-modules to K -theory, unpublished manuscript https://www.dpmms.cam.ac.uk/~groj/char.ps (1994) 12. Hernandez, D.: Representations of quantum affinizations and fusion product. Transform. Groups 10(2), 163–200 (2005) 13. Hu, N., Rosso, M., Zhang, H.: Two-parameter quantum affine algebra Ur,s ( sln ), Drinfel’d realization and quantum affine Lyndon basis. Commun. Math. Phys. 278(2), 453–486 (2008) n ). J. 14. Jing, N., Liu, M.: R-matrix realization of two-parameter quantum affine algebra Ur,s (gl Algebra 488, 1–28 (2017) 15. Lusztig, G.: Quantum groups at roots of 1. Geom. Dedicata 35(1–3), 89–113 (1990) 16. Negu¸t, A.: Quantum toroidal and shuffle algebras. Adv. Math. 372, Paper No. 107288 (2020) 17. Tsymbaliuk, A.: Shuffle algebra realizations of type A super Yangians and quantum affine superalgebras for all Cartan data. Lett. Math. Phys. 110(8), 2083–2111 (2020) 18. Tsymbaliuk, A.: Duality of Lusztig and RTT integral forms of Uv (Lsln ). J. Pure Appl. Algebra 225(1), Paper No. 106469 (2021) 19. Tsymbaliuk, A.: PBWD bases and shuffle algebra realizations for Uv (Lsln ), Uv1 ,v2 (Lsln ), Uv (Lsl(m|n)) and their integral forms. Selecta Math. (N.S.) 27(3), Paper No. 35 (2021) 20. Yamane, H.: On defining relations of affine Lie superalgebras and affine quantized universal enveloping superalgebras. Publ. Res. Inst. Math. Sci. 35(3), 321–390 (1999) 21. Zhang, H.: Representations of quantum affine superalgebras. Math. Z. 278(3–4), 663–703 (2014) 22. Zhang, H.: RTT realization of quantum affine superalgebras and tensor products. Int. Math. Res. Not. IMRN (4), 1126–1157 (2016)

Chapter 2

Quantum Toroidal gl1 , Its Representations, and Geometric Realization

Abstract In this chapter, we recall the basic results on the quantum toroidal algebra of gl1 . This algebra has an important interpretation [18, 19] as an extended elliptic Hall algebra of [1], which provides its “90 degree rotation” automorphism  (first discovered in [13]). We also establish the shuffle realization of its “positive” subalgebra and its particular commutative subalgebra, due to [6, 17], respectively. Following [4, 5, 7], we discuss a combinatorial approach to the construction of several important families of representations. We recall the shuffle realization [8, 22] of these combinatorial modules, as well as explain how vertex-type representations of [6] can be naturally related to them via a  -twist. Finally, following [9, 19, 21], we provide some flavor of the applications to the geometry by realizing Fock modules and their tensor products via equivariant K -theory of the Gieseker moduli spaces, as well as evoking the K -theoretic version of the Nakajima’s construction from [15].

2.1 Quantum Toroidal Algebras of gl1 In this section, we recall the definition of the quantum toroidal algebra U¨ q1 ,q2 ,q3 (gl1 ), first studied in [13] and rediscovered later in [4, 9]. We provide the shuffle algebra realization of U¨ q>1 ,q2 ,q3 (gl1 ) established in [17]. Following [18, 19], we evoke the identification of U¨ q1 ,q2 ,q3 (gl1 ) with a (centrally extended) elliptic Hall algebra of [1], and the resulting “90 degree rotation” automorphism of [13] interchanging the horizontal and vertical Heisenberg subalgebras. We also provide the shuffle realization [6] of the commutative “positive half” A of the horizontal Heisenberg.

2.1.1 Quantum Toroidal gl1 We fix q1 , q2 , q3 ∈ C× satisfying the relation q1 q2 q3 = 1. Choosing a square root q of q2 , we can equivalently use q, d ∈ C× related to the above via:

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 A. Tsymbaliuk, Shuffle Approach Towards Quantum Affine and Toroidal Algebras, SpringerBriefs in Mathematical Physics 49, https://doi.org/10.1007/978-981-99-3150-7_2

37

2 Quantum Toroidal gl1 , its Representations and Geometry

38

q1 := q −1 d , q2 := q 2 , q3 := q −1 d −1 .

(2.1)

Throughout this section, we assume that none of q1 , q2 , q3 is a root of 1. We define βr := (1 − q1r )(1 − q2r )(1 − q3r ) ∈ C× for all r > 0 ,

(2.2)

and consider the rational function g(z) :=

(1 − q1 z)(1 − q2 z)(1 − q3 z) (1 − q1−1 z)(1 − q2−1 z)(1 − q3−1 z)

(2.3)

satisfying the equality g(1/z) = 1/g(z). Following [4, 7], we define the quantum toroidal algebra of gl1 , denoted by U¨ q1 ,q2 ,q3 (gl1 ), to be the associative C-algebra generated by {er , fr , ψr , ψ0−1 , γ ±1/2 , q ±d1 , q ±d2 }r ∈Z with the following defining relations: [ψ ± (z), ψ ± (w)] = 0 , γ ±1/2 − central , ψ0±1 · ψ0∓1 = γ ±1/2 · γ ∓1/2 = q ±d1 · q ∓d1 = q ±d2 · q ∓d2 = 1 ,

(t0.1) (t0.2)

q d1 e(z)q −d1 = e(qz) , q d1 f (z)q −d1 = f (qz) , q d1 ψ ± (z)q −d1 = ψ ± (qz) , (t0.3) q d2 e(z)q −d2 = qe(z) , q d2 f (z)q −d2 = q −1 f (z) , q d2 ψ ± (z)q −d2 = ψ ± (z) , (t0.4) g(γ −1 z/w)ψ + (z)ψ − (w) = g(γ z/w)ψ − (w)ψ + (z) ,

(t1)

e(z)e(w) = g(z/w)e(w)e(z) ,

(t2)

f (z) f (w) = g(z/w)−1 f (w) f (z) ,

(t3)

β1 · [e(z), f (w)] = δ(γ w/z)ψ + (γ 1/2 w) − δ(γ z/w)ψ − (γ 1/2 z) ,

(t4)

ψ ± (z)e(w) = g(γ ±1/2 z/w)e(w)ψ ± (z) ,

(t5)

ψ ± (z) f (w) = g(γ ∓1/2 z/w)−1 f (w)ψ ± (z) ,

(t6)

Sym

z2 [e(z 1 ), [e(z 2 ), e(z 3 )]] = 0 , z3

(t7)

2.1 Quantum Toroidal Algebras of gl1

39

z2 [ f (z 1 ), [ f (z 2 ), f (z 3 )]] = 0 , z3

Sym

(t8)

where the generating series are defined via: e(z) :=



er z −r ,

f (z) :=

r ∈Z



fr z −r , ψ ± (z) := ψ0±1 +



ψ±r z ∓r ,

r >0

r ∈Z

and Sym denotes the symmetrization with respect to all permutations of z 1 , z 2 , z 3 . This construction is manifestly symmetric with respect to the parameters q1 , q2 , q3 . Remark 2.1 In [4, 5], this algebra without q ±d1 , q ±d2 and with γ ±1/2 = 1 was referred to as the “quantum continuous gl∞ ”. A version of the latter algebra without the Serre relations (t7), (t8) was referred to as the “Ding-Iohara algebra” in [9]. It is convenient to use the generators {h ±r }r >0 instead of {ψ±r }r >0 , defined via:  exp ∓

 βr r >0

r

 h ±r z

∓r

= ψ¯ ± (z) := ψ0∓1 ψ ± (z)

(2.4)

with βr ∈ C× introduced in (2.2). Then, the relations (t5, t6) are equivalent to: ψ0 er = er ψ0 ,

[h k , er ] = γ −|k|/2 er +k (k = 0) ,

ψ0 fr = fr ψ0 ,

[h k , fr ] = −γ |k|/2 fr +k (k = 0) ,

(t5 ) (t6 )

as follows from the equality  log

(z − q1−1 γ −1/2 w)(z − q2−1 γ −1/2 w)(z − q3−1 γ −1/2 w)



(z − q1 γ −1/2 w)(z − q2 γ −1/2 w)(z − q3 γ −1/2 w)

=



−

r >0

wr βr · γ −r/2 r , r z

with the left-hand side expanded in w/z. Likewise, the relation (t1) is equivalent to: ψ0 h k = h k ψ0 ,

[h k , h −l ] = −δkl

k(γ k − γ −k ) (k, l = 0) , βk

(t1 )

which we refer to as the Heisenberg algebra relations. Finally, we note that in view of the relations (t0.1)–(t6), the Serre relations (t7), (t8) can be replaced by equivalent simpler ones: Proposition 2.1 If the relations (t5 ), (t6 ) hold, then (t7) and (t8) are equivalent to:   e0 , [e1 , e−1 ] = 0 ,

(t7 )

 f 0 , [ f 1 , f −1 ] = 0 .

(t8 )



40

2 Quantum Toroidal gl1 , its Representations and Geometry

Proof The relation (t7 ) arises by comparing the free terms in (t7). Let us now prove that (t5 ), (t7 ) imply (t7). To this end, we consider endomorphisms Br of U¨ q1 ,q2 ,q3 (gl1 ) given by B0 := 3Id and Br : x → [γ |r |/2 h r , x] for r = 0. Then, combining (t5 ) with the algebraic equality [t, [a, [b, c]]] = [[t, a], [b, c]] + [a, [[t, b], c]] + [a, [b, [t, c]]], we get that Br1 ,r2 ,r3 := Br1 Br2 Br3 − Br1 +r2 Br3 − Br1 +r3 Br2 − Br2 +r3 Br1 + 2Br1 +r2 +r3 maps Br1 ,r2 ,r3 : [e0 , [e1 , e−1 ]] → Sym [er1 , [er2 +1 , er3 −1 ]] , where Sym denotes the symmetrization with respect to all permutations of r1 , r2 , r3 . Thus, (t5 ) and (t7 ) imply: Sym [er1 , [er2 +1 , er3 −1 ]] = 0 .

(2.5)

Multiplying (2.5) by z 1−r1 z 2−r2 z 3−r3 and summing over all r1 , r2 , r3 ∈ Z, we obtain (t7).  The proof of the equivalence of (t8) and (t8 ) modulo (t6 ) is analogous. Let U¨ q1 ,q2 ,q3 , and U¨ q01 ,q2 ,q3 be the C-subalgebras of U¨ q1 ,q2 ,q3 (gl1 ) generated by { fr }r ∈Z , {er }r ∈Z , and {ψr , ψ0−1 , γ ±1/2 , q ±d1 , q ±d2 }r ∈Z , respectively. The following result is completely analogous to Proposition 1.1 (though its proof is slightly more tedious, due to the presence of the generators γ ±1/2 , q ±d1 , q ±d2 ): Proposition 2.2 (a) (Triangular decomposition of U¨ q1 ,q2 ,q3 (gl1 )) The multiplication map m : U¨ q1 ,q2 ,q3 −→ U¨ q1 ,q2 ,q3 (gl1 ) is an isomorphism of C-vector spaces. (b) The algebra U¨ q>1 ,q2 ,q3 (resp. U¨ q01 ,q2 ,q3 and U¨ q 0 U¨ q01 ,q2 ,q3 , U¨ q1 ,q2 ,q3 , U¨ q01 ,q2 ,q3 , and Uqtor , Uqtor , Uqtor , respectively. 1 ,q2 ,q3 1 ,q2 ,q3 1 ,q2 ,q3 In particular, their “positive” subalgebras are naturally isomorphic:   > . U¨ q>1 ,q2 ,q3 U¨ q>1 ,q2 ,q3 U¨ q>1 ,q2 ,q3 Uqtor 1 ,q2 ,q3

(2.7)

Finally, let us note that the algebra U¨ q1 ,q2 ,q3 (gl1 ) is endowed with a topological Hopf algebra structure given by the same formulas (H1)–(H13) as in Theorem 3.3, see [3], and can be realized as a Drinfeld double similarly to Theorem 3.4(b). However, we do not need these features in the present chapter besides for the discussion of  U¨ q1 ,q2 ,q3 (gl1 )-representations for which a simplified coproduct is recalled in (2.41).

2.1.2 Elliptic Hall Algebra We recall the (centrally extended) elliptic Hall algebra of [1]. First, let us introduce the following notation: Define (Z2 )∗ := Z2 \{(0, 0)} and (Z2 )± := {(a, b)| ± a > 0 or a = 0, ±b > 0}. For x = (a, b) ∈ (Z2 )∗ , we define deg(x) := gcd(a, b) ∈ Z≥1 . For x ∈ (Z2 )∗ , we define x := 1 if x ∈ (Z2 )+ and x := −1 if x ∈ (Z2 )− . For noncollinear x, y ∈ (Z2 )∗ , we define x,y := sign(det(x, y)) ∈ {±1}. For noncollinear x, y ∈ (Z2 )∗ , we use x,y to denote the triangle with vertices {(0, 0), x, x + y}. • We call the triangle x,y empty if it has no interior lattice points.

• • • • •

• For r ≥ 1, we define αr := − βrr =

(1−q1−r )(1−q2−r )(1−q3−r ) . r

Definition 2.1 [1, Definition 6.4] The (central extension of) elliptic Hall algebra E is the associative algebra generated by {u x , κy | x ∈ (Z2 )∗ , y ∈ Z2 } with the following defining relations: κx − central , κx κy = κx+y , κ0,0 = 1 , [u y , u x ] = δx,−y · [u y , u x ] = x,y κα(x,y) 

where α(x, y) =

κx − κx−1 if x, y are collinear , αdeg(x)

θx+y if x,y is empty and deg(x) = 1 , α1

x (x x + y y − x+y (x + y))/2 if x,y = 1 y (x x + y y − x+y (x + y))/2 if x,y = −1

(2.8) (2.9) (2.10)

2 Quantum Toroidal gl1 , its Representations and Geometry

42

and the elements {θx | x ∈ Z2 } are defined via 

θr x0 z = exp r

 

r ≥0

 αr u r x0 z

for x0 ∈ (Z2 )∗ with deg(x0 ) = 1 . (2.11)

r

r >0

This algebra is related to the quantum toroidal of gl1 via the following result: Theorem 2.1 ([18, 19]) The assignment u 1,r → er , u −1,r → fr , θ0,±k → γ ±k/2 ψ±k ψ0∓1 , κ1,0 → ψ0 , κ0,1 → γ , for all r ∈ Z and k ≥ 1, gives rise to an algebra isomorphism ∼  ±1/2 ] −→ Uqtor (gl1 ) .  : E[κ 0,1 1 ,q2 ,q3

(2.12)

Remark 2.3 Since the algebra E is naturally Z2 -graded via deg(u x ) = x, deg(κy ) = y, one can enlarge it to E¨ by adjoining degree generators q ±d1 and q ±d2 satisfying: q d1 u a,b q −d1 = q −b u a,b , q d2 u a,b q −d2 = q a u a,b , q ±dr · q ∓dr = 1 , q dr κa,b q −dr = κa,b for r ∈ {1, 2} , a, b ∈ Z .

(2.13)

∼ ¨ ±1/2 ]−→ U¨ q1 ,q2 ,q3 (gl1 ). Then, (2.12) naturally extends to an algebra isomorphism E[κ 0,1

2.1.3 “90 Degree Rotation” Automorphism ±1/2

In this section, we shall enlarge the algebra Uqtor (gl1 ) by formally adjoining ψ0 , 1 ,q2 ,q3 (gl1 ) a square root of the central element ψ0 . After this upgrade, the algebra Uqtor 1 ,q2 ,q3 admits an important automorphism constructed first in [13]: Theorem 2.2 ([13, Proposition 3.2]) (a) The assignment ±1/2

ψ0

→ γ ±1/2 ,

e0 → −h 1 γ

−1/2

,

∓1/2

γ ±1/2 → ψ0 f 0 → h −1 γ

1/2

, −1/2

1/2

, h 1 → − f 0 ψ0

h −1 → e0 ψ0 ,

(2.14)

(gl1 ): gives rise to an order four automorphism of Uqtor 1 ,q2 ,q3 ∼

(gl1 ) −→ Uqtor (gl1 ) ,  : Uqtor 1 ,q2 ,q3 1 ,q2 ,q3

 4 = Id .

(b) If deg(x) = (a, b) ∈ Z2 , see (2.6), then deg( (x)) = (−b, a) ∈ Z2 . (c) For k ≥ 1, we have:

(2.15)

2.1 Quantum Toroidal Algebras of gl1

43

 : Yk+ → β1−1 γ −k/2 · ψk ψ0−1 , Yk− → β1−1 γ k/2 · ψ−k ψ0 ,

(2.16)

where we set Yk+ := [e−1 , [e0 , · · · , [e0 , e1 ] · · · ]] ,



Yk− := [ f 1 , [ f 0 , · · · , [ f 0 , f −1 ] · · · ]]



k factors

k factors

(2.17)

for k ≥ 2, while Y1+ := e0 and Y1− := f 0 .

(gl1 ) and Efrom Theorem 2.1, we shall refer Evoking the identification of Uqtor 1 ,q2 ,q3 to  of (2.15) as (counterclockwise) “90 degree rotation” automorphism. We note that Yk± from (2.17) get identified with nonzero multiples of θ(±k,0) from (2.11). 1/2

Remark 2.4 (a) In [13], the parameters q, γ are related to our q1 , q2 via q ↔ q1 , 1/2 ≥1 of Uq,γ are related to γ ↔ q2 , while the generators {a±r , X k± , C ±1 , C ±1 }rk∈Z ours via: C ±1 ↔ γ ±1 , C ±1 ↔ ψ0±1 ,

−1/2

X k+ ↔ (q3

1/2

− q3 )ek ,

−1/2

X k− ↔ (q3

1/2

− q3 ) f k ,

a±r ↔ ∓q1 q2 (1 − q1r )(1 − q3r )/r (1 − q1 ) · h ±r γ ∓r/2 . 1/2 r/2

(b) The automorphism ψ from [13, Proposition 3.2] is the inverse of  from (2.15). In the next section we will need a version of  , the algebra isomorphism:   ∼  : U¨ q1 ,q2 ,q3 (gl1 ) −→ U¨ q1 ,q2 ,q3 (gl1 )

(2.18)

determined by (2.14) and the assignment q d2 → q −d1 , which is compatible with  Theorem 2.2(b). According to Theorem 2.2(c), the  -preimage of U¨ q01 ,q2 ,q3 provides  ±1/2 a horizontal Heisenberg subalgebra of U¨ q01 ,q2 ,q3 generated by {Yk± , ψ0 }k≥1 .

2.1.4 Shuffle Algebra Realization

Consider an N-graded C-vector space S = k∈N Sk , where Sk consists of k symmetric rational functions in the variables {xr }rk=1 . We also fix a rational function ζ (z) :=

(q1 z − 1)(q2 z − 1)(q3 z − 1) . (z − 1)3

(2.19)

Similarly to (1.32), let us now introduce the bilinear shuffle product  on S: given F ∈ Sk and G ∈ S , we define F  G ∈ Sk+ via ⎞ ⎛  >k   r 1 x r ⎠. (F  G)(x1 , . . . , xk+ ) := ζ Sym ⎝ F(x1 , . . . , xk )G(xk+1 , . . . , xk+ ) k!! xr  r ≤k

2 Quantum Toroidal gl1 , its Representations and Geometry

44

This endows S with a structure of an associative C-algebra with the unit 1 ∈ S0 . As before, we consider the subspace of S defined by the pole and wheel conditions: • We say that F ∈ Sk satisfies the pole conditions if f (x1 , . . . , xk ) F(x1 , . . . , xk ) =  , where r =r  (xr − xr  )

f ∈ C[x1±1 , . . . , xk±1 ] k . (2.20)

• We say that F ∈ Sk satisfies the wheel conditions if  F(x1 , . . . , xk ) = 0 once

xr1 xr2 xr3 , , xr2 xr3 xr1

 = {q1 , q2 , q3 } for some r1 , r2 , r3 . (2.21)

Let Sk ⊂ Sk denote the subspace of all F satisfying these two conditions and set S :=



Sk .

k∈N

Similarly to Lemma 1.2, it is straightforward to check that S ⊂ S is actually -closed: Lemma 2.1 For any F ∈ Sk and G ∈ S , we have F  G ∈ Sk+ . 1 ). It is Z2 -graded via: The algebra (S, ) is called the shuffle algebra (of type gl S =



Sk,d

    Sk,d := F ∈ Sk  tot.deg(F) = d .

with

(2.22)

(k,d)∈N×Z

Similarly to Proposition 1.3, the shuffle algebra (S, ) is linked to the “positive” subalgebra of the quantum toroidal gl1 , cf. (2.7), via: Proposition 2.3 There exists an injective C-algebra homomorphism  : U¨ q>1 ,q2 ,q3 −→ S

(2.23)

such that er → x1r for any r ∈ Z. Let E> be the “positive” subalgebra of E generated by {u k,d }d∈Z k≥1 . Then, composing (2.23) with the restriction of (2.12) to the “positive” subalgebras, we obtain: ϒ : E> → S .

(2.24)

The following result was conjectured in [9, 19] and proved in [17, Proposition 3.2]: ∼ Theorem 2.3 ([17]) ϒ : E> −→ S is an isomorphism of N × Z-graded C-algebras.

Thus, the shuffle algebra S is generated by its first component S1 , and we have: E> U¨ q>1 ,q2 ,q3 S .

(2.25)

2.1 Quantum Toroidal Algebras of gl1

45

2.1.5 Negu¸t’s Proof of Theorem 2.3 In this section, we provide a proof of Theorem 2.3 following the original argument of [17] that crucially relies on the notion of slope ≤ μ subalgebras:

μ with Definition 2.2 For μ ∈ R, define S μ = (k,d)∈N×Z Sk,d    F(ξ · x1 , . . . , ξ · x , x+1 , . . . , xk )  μ exists ∀ 0 ≤  ≤ k . Sk,d = F ∈ Sk,d  lim μ ξ →∞ ξ (2.26) Using the equalities lim ζ (ξ ) = lim ζ (1/ξ ) = 1, we immediately get: ξ →∞

ξ →∞

Lemma 2.2 ([17, Proposition 2.3]) The subspace S μ is a subalgebra of S. μ

The subspaces Sk,d give an increasing filtration of (an infinite dimensional) Sk,d :  μ μ μ μ Sk,d = μ∈R Sk,d and Sk,d ⊂ Sk,d for μ < μ . The key point is that Sk,d are actually finite dimensional, with an explicit upper bound on their dimensions: Proposition 2.4 ([17, Proposition 2.4]) collections

μ

dim(Sk,d ) ≤ number of unordered

(k1 , d1 ), . . . , (kt , dt ) with k1 + · · · + kt = k , d1 + · · · + dt = d , dr ≤ μkr , (2.27) where t ≥ 1, kr ≥ 1, and dr ∈ Z for 1 ≤ r ≤ t. Proof Similarly to (1.40), for any partition π = {k1 ≥ k2 ≥ · · · ≥ kt > 0} of k let us define the specialization map μ

φπ : Sk,d −→ C[y1±1 , . . . , yt±1 ] .

(2.28)

Explicitly, for F of the form (2.20), we define φπ (F) as a specialization of f under: xk1 +···+kr −1 +a → q2a yr for 1 ≤ r ≤ t , 1 ≤ a ≤ kr . μ

This gives rise to the filtration (called the Gordon filtration in [6]) on Sk,d via: μ,π

Sk,d :=



Ker(φ π) ,

(2.29)

 π >π

where > denotes the dominance partial order on the set of all partitions of k. To μ prove the desired upper bound on dim(Sk,d ), it thus suffices to show:     μ,π dim φπ Sk,d ≤ # (d1 , . . . , dt ) ∈ Zt | d1 + · · · + dt = d, dr ≤ μkr ∀ r , (2.30) where the order of dr and dr  is disregarded whenever kr = kr  .

2 Quantum Toroidal gl1 , its Representations and Geometry

46

First, we note that the wheel conditions (2.21) for F guarantee that φπ (F), which is a Laurent polynomial in {y1 , . . . , yt }, vanishes under the following specializations: 

(i) q2x yr = q1 · q2x yr  for any 1 ≤ r < r  ≤ t, 1 ≤ x < kr , 1 ≤ x  ≤ kr  ,  (ii) q2x yr = q3 · q2x yr  for any 1 ≤ r < r  ≤ t, 1 ≤ x < kr , 1 ≤ x  ≤ kr  . Second, the condition φ π > π also implies that φπ (F) vanishes π (F) = 0 for any  under the following specializations: 

(iii) yr = q2x yr  for any 1 ≤ r < r  ≤ t, 1 ≤ x  ≤ kr  ,  (iv) q2kr +1 yr = q2x yr  for any 1 ≤ r < r  ≤ t, 1 ≤ x  ≤ kr  . μ,π

Combining the above vanishing conditions (i)–(iv) for φπ (F) with F ∈ Sk,d , we conclude that φπ (F) is divisible by Q π ∈ C[y1 , . . . , yt ], defined as a product of the linear terms in yr ’s arising from (i)–(iv), counted with the correct multiplicities. The total degree of Q π as well as the degree with respect to each variable yr are: 

tot.deg(Q π ) = k 2 −

kr2 and deg yr (Q π ) = 2kkr − 2kr2 .

(2.31)

1≤r ≤t

On the other hand, the specialization φπ (F) satisfies: tot.deg(φπ (F)) = d + k 2 − k and deg yr (φπ (F)) ≤ μkr + 2kkr − kr2 − kr . (2.32) To see the above inequality one should multiply by ξ only those x∗ -variables that got specialized to q2? -multiples of yr and use the condition that the slope of F is ≤ μ. Define rπ := φπ (F)/Q π ∈ C[y1±1 , . . . , yt±1 ]. Combining (2.31) and (2.32), we get: tot.deg(rπ ) = d +

t 

kr (kr − 1) and deg yr (rπ ) ≤ kr2 − kr + μkr .

(2.33)

r =1

The space of Laurent polynomials satisfying (2.33) is spanned by the monomials k 2 −k1 +d1

y1 1

k 2 −kt +dt

· · · yt t

with d1 + · · · + dt = d and dr ≤ μkr ∀ r .

Finally, we note that both φπ (F) and Q π are symmetric in yr and yr  whenever  kr = kr  , hence, so is rπ . This implies (2.30) and thus completes the proof. Now we are ready to complete the proof of Theorem 2.3. To this end, for any μ  k ≥ 1, d ∈ Z, μ ∈ R, we consider the following subspace Ek,d of E: μ Ek,d = Span u k1 ,d1 · · · u kt ,dt |k1 + · · · + kt = k, d1 + · · · + dt = d, dr ≤ μkr , kr ≥ 1

!

2.1 Quantum Toroidal Algebras of gl1

47

b∈Z According to [1, Lemma 5.6], the “positive” subalgebra E> generated by {u a,b }a>0 μ + has a basis consisting of convex paths in Conv (cf. [1]), while Ek,d has a basis consisting of such convex paths of slope ≤ μ starting from (0, 0) and ending at (k, d). The latter are in bijection with the collections (2.27). Combining this observation with the injectivity of ϒ from (2.24) and Proposition 2.4, it thus remains to prove the following result:

Proposition 2.5 ([17]) ϒ(Ek,d ) ⊆ Sk,d . μ

μ

Proof Due to Lemma 2.2, it suffices to verify that elements Fk,d := ϒ(u k,d ) have slope ≤ dk . This is proved by an induction on k, the base case k = 1 being trivial. To prove the induction step, for a fixed (k, d) pick an empty triangle with vertices {(0, 0), (k2 , d2 ), (k, d)} such that dk22 < dk < dk11 and gcd(k1 , d1 ) = 1 = gcd(k2 , d2 ) with (k1 , d1 ) = (k − k2 , d − d2 ) and k1 , k2 > 0 (for example, pick a triangle with the minimal area). By the induction hypothesis and the relation (2.10), it remains to show that the commutator [Fk1 ,d1 , Fk2 ,d2 ] has a slope ≤ dk . By the induction hypothesis, a slope of Fk1 ,d1 is ≤ dk11 , that is, as we multiply 1 ≤

d1

 

variables by ξ the result grows at the speed ξ k1 1 , and similarly for Fk2 ,d2 . Let us now consider any summand from the symmetrization appearing in Fk1 ,d1  Fk2 ,d2 and Fk2 ,d2  Fk1 ,d1 . As we multiply  variables by ξ , let 1 and 2 denote the number of those x∗ -variables that got placed into Fk1 ,d1 and Fk2 ,d2 , respectively, so that ≤

d1

 +

d2

 

 = 1 + 2 . Then, the corresponding summand grows at the speed ξ k1 1 k2 2 . However, the choice of the triangle guarantees that  dk11 1  +  dk22 2  ≤ dk (1 + 2 ) for any 0 ≤ 1 ≤ k1 and 0 ≤ 2 ≤ k2 , with the only exception when 1 = k1 , 2 = 0. But in the latter case, the corresponding summands in both Fk1 ,d1  Fk2 ,d2 and Fk2 ,d2  Fk1 ,d1 with  variables multiplied by ξ have the form ξ d1 · Fk1 ,d1 (x1 , . . . , xk1 ) · Fk2 ,d2 (xk1 +1 , . . . , xk ) + O(ξ

k1 d k

),

and hence the commutator [Fk1 ,d1 , Fk2 ,d2 ] indeed has a slope ≤ dk , as claimed.



μ μ Remark 2.5 (a) The result above establishes a vector space isomorphism Ek,d Sk,d which implies that the inequality of Proposition 2.4 is actually an equality.

(b) The proof of Proposition 2.4 implies that the subspace of all F ∈ Sk,d satisfying lim

ξ →∞

F(ξ · x1 , . . . , ξ · x , x+1 , . . . , xk ) ξ kμ d

=0

∀0 <  < k

(2.34)

is at most one-dimensional. It is proved in [17, Sect. 5] that ϒ(u k,d ) satisfies this property, while an explicit formula for ϒ(u k,d ) ∈ Sk,d is derived in [17, Theorem 1.1, Sect. 6].

2 Quantum Toroidal gl1 , its Representations and Geometry

48

2.1.6 Commutative Subalgebra A Following [6], let us consider an important N-graded subspace A =

where

k≥0

Ak of S:

    Ak = F ∈ Sk,0  ∂ (∞,) F exists ∀ 1 ≤  ≤ k ,

(2.35)

∂ (∞,) F := lim F(ξ · x1 , . . . , ξ · x , x+1 , . . . , xk ) .

(2.36)

ξ →∞

0 , see (2.26). By Lemma 2.2, A is a subalgebra of S. Equivalently, we have Ak = Sk,0 This subalgebra obviously contains the following family of elements:

K 1 (x1 ) = x10 ,

K k (x1 , . . . , xk ) =

 1≤r =r  ≤k

x r − q2 x r  ∀k ≥ 2. xr − xr 

(2.37)

The following description of A was established in [6, Proposition 2.20]: Theorem 2.4 ([6]) The subalgebra A ⊂ S is shuffle-generated by {K k }k≥1 and is a polynomial algebra in them. In particular, A is a commutative subalgebra of S. Remark 2.6 The same result obviously holds if q2 in (2.37) is replaced with q1 or q3 . Proof According to Remark 2.5(a), we have: dim Ak = p(k) = # {partitions of k} .

(2.38)

We shall identify partitions π with Young diagrams λ = {λ1 ≥ λ2 ≥ · · · ≥ λt > 0} and use λ for the transposed Young diagram. We define K λ := K λ1  K λ2  · · ·  K λt , so that K λ ∈ A|λ| by Lemma 2.2. Analogously to Lemmas 1.5–1.6, we have. Lemma 2.3 φλ (K λ ) = 0 and φμ (K λ ) = 0 for any μ > λ with |μ| = |λ|. Combining this lemma with the dimension formula (2.38), we immediately see that {K λ }|λ|=k is a basis of Ak . Thus, it remains only to prove the commutativity: Lemma 2.4 [K m 1 , K m 2 ] = 0 for any m 1 , m 2 ≥ 1. Proof It is sufficient to prove that φ(2,1m1 +m2 −2 ) ([K m 1 , K m 2 ]) = 0. Indeed, this would imply that the numerator f from (2.20) in [K m 1 , K m 2 ] is divisible by  r =r  (xr − q2 xr  ), and hence for degree reasons [K m 1 , K m 2 ] = νm 1 ,m 2 K m 1 +m 2 with νm 1 ,m 2 ∈ C. Let us multiply both sides of this equality by r =r  (xr − xr  ) and consider a specialization xr → y for all 1 ≤ r ≤ m 1 + m 2 . The left-hand side will clearly specialize to 0, while the right-hand side - to ((1 − q2 )y)(m 1 +m 2 )(m 1 +m 2 −1) · νm 1 ,m 2 . Thus, νm 1 ,m 2 = 0 and so [K m 1 , K m 2 ] = 0.

2.2 Representations of Quantum Toroidal gl1

49

Let us now prove the stated equality φ(2,1m1 +m2 −2 ) ([K m 1 , K m 2 ]) = 0 for m 1 , m 2 ≥ 0, where we set K 0 := 1 ∈ S0 . The proof is by induction on min{m 1 , m 2 }, with the base of induction (m 1 = 0 or m 2 = 0) being obvious. By a direct computation, we get φ(2,1m1 +m2 −2 ) (K m 1  K m 2 ) =

1 Sym(A1 (m 1 −1)!(m 2 −1)!

· B1 )

1 +m 2 −1 with the symmetrization Sym taken over all permutations of {yr }rm=2 , where A1 m 1 +m 2 −1 is an (m 1 , m 2 )-independent rational function in {yr }r =1 which is symmetric in 1 +m 2 −1 {yr }rm=2 and B1 is explicitly given by:



B1 =

2≤r =r  ≤m 1

yr − q2 yr  yr − yr 

 m 1 0 , λk = ∅ for k  0 , (2.66) that depend on an extra parameter K ∈ C× . The resulting modules M(u, K ) are called (vacuum) Macmahon modules. As we will not presently need explicit formulas for the action, we rather refer the reader to [7, Sect. 3.1] for more details. We shall only need: Proposition 2.10 ([7, Sect. 3.1]) Assuming that q1 , q3 are generic (2.39) and / q1Z q3Z , the module M(u, K ) can be invariantly described as the lowest K2 ∈  weight U¨ q1 ,q2 ,q3 (gl1 )-module in the category O− with the lowest weight φ K (z/u) := −1 K z/u−K . In the above realization, the empty plane partition ∅¯ = (∅, ∅, . . .) is its z/u−1 lowest weight vector.

2.2.3 Vertex Representations and Their Relation to Fock Modules In this section, we recall an important family of vertex-type U¨ q1 ,q2 ,q3 (gl1 )-modules. To this end, we consider a Heisenberg algebra A generated by {Hk }k∈Z subject to: 

|k|

[H0 , Hk ] = 0 ,

[Hk , Hl ] = k

1 − q1

δ H −|k| k,−l 0

1 − q3

(k, l = 0) .

(2.67)

Let A≥ be the subalgebra generated by {Hk }k≥0 , and Cv0 be the A≥ -representation with Hk v0 = δk0 v0 . The induced representation F := IndA A≥ (Cv0 ) is called the Fock representation of A. We also define the degree operator d acting on F via d(H−k1 H−k2 · · · H−k N v0 ) = (k1 + k2 + · · · + k N ) · H−k1 H−k2 · · · H−k N v0 . The following result provides a natural structure of a U¨ q1 ,q2 ,q3 (gl1 )-module on F: 

Proposition 2.11 ([6, Proposition A.6]) For c, u ∈ C× , the following formulas  define an action of U¨ q1 ,q2 ,q3 (gl1 ) on F: ρu,c (e(z)) =

     z k  z −k  1 − qk  1 − q −k −q1 q3 c 3 3 · exp H−k Hk exp − , (1 − q1 )2 (1 − q3 )2 k u k u k>0

k>0

2.2 Representations of Quantum Toroidal gl1

ρu,c ( f (z)) =  c

−1

· exp −

 1 − qk 3

k>0



±

ρu,c (ψ (z)) = exp ∓

k

 1 − q ∓k 3

k>0 ±1/4

ρu,c (γ ±1/2 ) = q2

,

k/2 q2 H−k

k

59

 z k



u

(1 −

exp

  1 − q −k 3

k

k>0 −k/4 q2k )q2 H±k

 z ∓k u

k/2 q2 Hk



 z −k



u

,

,

ρu,c (q ±d1 ) = q ±d .

 Remark 2.11 This module is irreducible since so is its restriction to U¨ q01 ,q2 ,q3 . To this end, we note the following compatibility between the relations (t1 ) and (2.67):

k −k −k/2 −βk2 −k(γ − γ )|γ →q21/2 −(1 − q3−k )(1 − q3k )(1 − q2k )2 q2 k(1 − q1k ) · = · . k2 βk k2 1 − q3−k

Let us now relate the above vertex-type U¨ q1 ,q2 ,q3 (gl1 )-modules ρv,c to the Fock ¨ Uq1 ,q2 ,q3 (gl1 )-modules τu . To this end, we recall that τu is the lowest weight ϑ denote the ϑ-twisted module in O− with the lowest weight φ(z/u). Let τ1/q 2u  ϑ action of U¨ q1 ,q2 ,q3 (gl1 ) on F(1/q2 u) given by τ1/q2 u (x) = τ1/q2 u (ϑ(x)), with the  ∼  automorphism ϑ : U¨ q1 ,q2 ,q3 (gl1 )−→ U¨ q1 ,q2 ,q3 (gl1 ) of (2.40). As pointed out in the ϑ end of Sect. 2.2.1, τ1/q2 u is a highest weight module in the category O+ and its  highest weight is φ(q 2 u/z) = φ(z/u)−1 . We also consider ρv,c , the  -twisted   ¨ action of Uq1 ,q2 ,q3 (gl1 ) on F given by ρv,c (x) = ρv,c ( (x)), with the isomorphism  ∼   : U¨ q1 ,q2 ,q3 (gl1 ) −→ U¨ q1 ,q2 ,q3 (gl1 ) of (2.18). Then, we have the following result: 



 Theorem 2.5 For any v, c ∈ C× , we have a U¨ q1 ,q2 ,q3 (gl1 )-module isomorphism:

 ϑ

τ1/q , ρv,c 2u

(2.68)

where u = −(1 − q1 )(1 − q3 )/c.  Proof Since both ρv,c and τ1/q2 u are irreducible modules, so are U¨ q1 ,q2 ,q3 (gl1 )ϑ ϑ   and τ1/q . Thus, it suffices to verify that v0 ∈ ρv,c and |∅ ∈ τ1/q modules ρv,c 2u 2u are the highest weight vectors of the same highest weight. ϑ is a highest weight module with the As noted right before the theorem, τ1/q 2u −1 highest weight φ(z/u) and the highest weight vector |∅. Therefore:

ϑ ϑ (ek )|∅ = 0 , τ1/q (ψ±r )|∅ = ±(q − q −1 )(q 2 u)±r |∅ ∀ k ∈ Z, r > 0 . τ1/q 2u 2u  (ek )v0 = 0 for any k ∈ Z, which follows from On the other hand, we also have ρv,c Theorem 2.2(b) as all eigenvalues of the degree operator d on F are nonnegative. Thus, it just remains to prove:  (ψ±r )(v0 ) = ±(q − q −1 )(q 2 u)±r v0 ∀ r > 0 . ρv,c

(2.69)

2 Quantum Toroidal gl1 , its Representations and Geometry

60

According to (t4), we have:      (ψ±r )(v0 ) = ±β1 · [ρv,c (e±r ), ρv,c ( f 0 )](v0 ) = ±β1 · ρv,c (e±r )(ρv,c ( f 0 )(v0 )) ρv,c (2.70) for all r > 0. Since  ( f 0 ) = h −1 γ 1/2 = β1−1 ψ−1 ψ0 γ 1/2 by (2.14), we get:  ( f 0 )(v0 ) = ρv,c

1 H−1 v0 . v(1 − q1 )

(2.71)

On the other hand, we have  (e0 ) = −h 1 γ −1/2 = β1−1 ψ1 ψ0−1 γ −1/2 by (2.14), hence:  (e0 ) (H−1 v0 ) = − ρv,c

v v0 . q(1 − q3 )

(2.72)

 We can now evaluate all ρv,c (e±r )(H−1 v0 ) for r > 0 by applying (t5 ):    (e±(r +1) )(H−1 v0 ) = [ρv,c (h ±1 ), ρv,c (e±r )](H−1 v0 ) . ρv,c

(2.73)

To this end, we note that  (h 1 ) = − f 0 and  (h −1 ) = e0 by (2.14), hence we have:  ρv,c (h 1 )(v0 ) = −c−1 v0 ,

q1 q3 c v0 , (1 − q1 )2 (1 − q3 )2 (2.74)  (h 1 )(H−1 v0 ) = −c−1 (1 − q2 (1 − q1 )(1 − q3 )) H−1 v0 , ρv,c q1 q3 c  ρv,c (h −1 )(H−1 v0 ) = − (1 − (1 − q1 )(1 − q3 )) H−1 v0 . (1 − q1 )2 (1 − q3 )2

 (h −1 )(v0 ) = − ρv,c

Combining (2.72)–(2.74), we thus get by induction on r :  ±r v · − q2 (1 − q1 )(1 − q3 )/c v0 ∀ r > 0 . q(1 − q3 ) (2.75) Combining (2.70, 2.71, 2.75), we obtain (2.69) with u = −(1 − q1 )(1 − q3 )/c.   ρv,c (e±r )(H−1 v0 ) = −

Following [10], we note that the vertex representations ρv,c allow to interpret the elements K k of (2.37) generating the commutative subalgebra A and their “elliptic” generalizations via the transfer matrices as explained in Sect. 3.5, see Remark 3.17. Remark 2.12 The above observation together with the shuffle realization of the Fock modules (as discussed in the next subsection) played the crucial role in [8]. Remark 2.13 Analogously to Theorem 2.5, one can show that τ−(1−q1 )(1−q3 )/c

 ∗, ∗, ∗ . Here, ρv,c is the  -twist of the U¨ q1 ,q2 ,q3 (gl1 )-module ρv,c , dual to ρv,c , ρv,c ∗ (e(z))= which can be naturally realized on F by similar vertex-type formulas: ρv,c k/2 k/2 −k " # " #k k $ $ (1−q3 )q2 (1−q3 )q2 q1 q3 c z −k · exp (− k>0 H−k v ) exp ( k>0 Hk vz ), (1−q1 )2 (1−q3 )2 k k

2.2 Representations of Quantum Toroidal gl1

61

" #−k " #k $ $ 1−q −k 1−q k ∗ ρv,c ( f (z)) = −c−1 · exp ( k>0 k 3 H−k vz ) exp (− k>0 k 3 Hk vz ), ∓1/4 ∗ ∗ ∗ (γ ±1/2 ) = q2 , as well as ρv,c (q ±d1 ) = q ∓d , ρv,c (ψ ± (z)) = ρv,c ∓k " # k −k/4 $ ∓k (1−q3 )(1−q2 )q2 H∓k vz ). exp(± k>0 k

2.2.4 Shuffle Bimodules and Their Relation to Fock Modules Following [8], we recall the shuffle realization of the Fock modules and their tensor  products by constructing natural S-bimodules that become U¨ q1 ,q2 ,q3 (gl1 )-modules. × For u ∈ C , consider an N-graded C-vector space S1 (u) = k∈N S1 (u)k , where the degree k component S1 (u)k consists of k -symmetric rational functions F in the variables {xr }rk=1 satisfying the following three conditions: (i) Pole conditions on F:  k  . f ∈ C x1±1 , . . . , xk±1

f (x1 , . . . , xk ) , k r =r  (xr − xr  ) · r =1 (xr − u)

F=

(2.76)

(ii) First kind wheel conditions on F:  F(x1 , . . . , xk ) = 0 once

xr1 xr2 xr3 , , xr2 xr3 xr1

 = {q1 , q2 , q3 } for some r1 , r2 , r3 .

(iii) Second kind wheel conditions on F (with f from (2.76)): f (x1 , . . . , xk ) = 0 once xr1 = u and xr2 = q2 u for some 1 ≤ r1 = r2 ≤ k . Given F ∈ Sk and G ∈ S1 (u) , we define F  G, G  F ∈ S1 (u)k+ via: 1 (F  G)(x1 , . . . , xk+ ) := × k! · !  Sym F(x1 , . . . , xk )G(xk+1 , . . . , xk+ )

 r >k

r ≤k

 ζ

xr xr 

  k x  r φ , u r =1

1 × (G  F)(x1 , . . . , xk+ ) := k! · !  Sym G(x1 , . . . , x )F(x+1 , . . . , xk+ )

 r >

r ≤

 ζ

xr xr 

(2.77)

 .

(2.78)

2 Quantum Toroidal gl1 , its Representations and Geometry

62

These formulas clearly endow S1 (u) with an S-bimodule structure. Identifying S   with U¨ q>1 ,q2 ,q3 via (2.25), we thus get two commuting U¨ q>1 ,q2 ,q3 -actions on S1 (u).  In fact, the left action can be extended to an action of the entire U¨ q1 ,q2 ,q3 (gl1 ): Proposition 2.12 ([8, Proposition 3.2]) The following formulas define an action of U¨ q1 ,q2 ,q3 (gl1 ) on S1 (u):



πu (ψ0 )G = q −1 · G , πu (h l )G =

πu (q d2 )G = q n · G ,  n  r =1

n πu ( f k )G = β1

xrl

πu (ek )G = x k  G ,

ul − l (1 − q1 )(1 − q3l )

 ·G,

&  % k z G(x1 , . . . , xn−1 , z) dz Res + Res , n−1 xr z=∞ z=0 z r =1 ζ ( z )

for k ∈ Z, l = 0, and G ∈ S1 (u)n . To relate this to the Fock modules F(u), let us consider the following subspace: J0 = spanC {G  F | G ∈ S1 (u), F ∈ Sk with k ≥ 1} ⊂ S1 (u) .  Proposition 2.13 ([8, Proposition 3.3]) (a) J0 is a U¨ q1 ,q2 ,q3 (gl1 )-submodule of πu .  (b) We have a U¨ q1 ,q2 ,q3 (gl1 )-module isomorphism S1 (u)/J0 F(u).

The above admits a higher rank counterpart. For any m ≥ 1 and generic u = (2.64), so that F(u 1 ) ⊗ · · · ⊗ F(u m ) is irreducible (u 1 , . . . , u m ) ∈ (C× )m satisfying by Lemma 2.8, define S(u) = k∈N S(u)k similarly to S1 (u) with the following changes: (i’) Pole conditions for F from a degree k component should read as follows: f (x1 , . . . , xk ) , m k s=1 r =1 (xr − u s ) r =r  (xr − xr  ) ·

F=

 k  . f ∈ C x1±1 , . . . , xk±1

(iii’) Second kind wheel conditions for such F should read as follows: f (x1 , . . . , xk ) = 0 once xr1 = u s and xr2 = q2 u s for some s, 1 ≤ r1 = r2 ≤ k .

We endow S(u) with an S-bimodule structure by the formulas (2.77, 2.78) with 

φ(xr /u) 

1≤r ≤k





φ(xr /u s ) .

1≤s≤m 1≤r ≤k

The resulting left action of U¨ q>1 ,q2 ,q3 S on S(u) can be extended to the action  of the entire U¨ q1 ,q2 ,q3 (gl1 ), denoted by πu , cf. Sect. 3.3.3. Consider the subspace 

2.3 Geometric Realizations

63

J0 = spanC {G  F | G ∈ S(u), F ∈ Sk with k ≥ 1} ⊂ S(u) . Proposition 2.14 ([8, Proposition 6.1]) J0 is a U¨ q1 ,q2 ,q3 (gl1 )-submodule of πu . More over, we have a U¨ q1 ,q2 ,q3 (gl1 )-module isomorphism S(u)/J0 F(u 1 ) ⊗ · · · ⊗ F(u m ). 

Yet another natural generalization of S1 (u) is provided by the S-bimodules S1K (u). As a vector space, S1K (u) is defined similarly to S1 (u) but without imposing the second kind wheel conditions. The S-bimodule structure on S1K (u) is defined by the formulas (2.77, 2.78) with the following modification in (2.77): φ(t)  φ K (t) := (K −1 · t − K )/(t − 1) .

(2.79)

 The resulting left action of U¨ q>1 ,q2 ,q3 S on S1K (u) can be extended to the action of  the entire U¨ q1 ,q2 ,q3 (gl1 ), denoted by πuK . Consider the subspace

J0 = spanC {G  F | G ∈ S1K (u), F ∈ Sk with k ≥ 1} ⊂ S1K (u) .  Proposition 2.15 ([22]) J0 is a U¨ q1 ,q2 ,q3 (gl1 )-submodule of πuK . Moreover, we have  / q1Z q3Z . a U¨ q1 ,q2 ,q3 (gl1 )-module isomorphism S1K (u)/J0 M(u, K ) if K 2 ∈

One can naturally define the higher rank counterparts S K (u) of S1K (u) similarly to S(u). Then, the analogue of the Proposition above provides a shuffle realization of the tensor product M(u 1 , K 1 ) ⊗ · · · ⊗ M(u N , K N ) of vacuum Macmahon modules (for general parameters u = (u 1 , . . . , u N ) and K = (K 1 , . . . , K N )).

2.3 Geometric Realizations In this section, we provide some flavour of the geometric applications of the quantum toroidal and shuffle algebras by recalling the realization of the Fock modules F(u) and their tensor products through the geometry of the Hilbert schemes of points on the plane (A2 )[n] and more generally the Gieseker varieties M(r, n). We also discuss how the commutative subalgebra A ⊂ S from Sect. 2.1.6 and its “negative” version Aop ⊂ S op give rise to an action of the Heisenberg algebra on the localized equivariant K -theory of (A2 )[n] , providing a K -theoretic counterpart of the famous Nakajima’s construction from [15]. Our exposition closely follows that of [9, 21].

2.3.1 Correspondences and Fixed Points for (A2 )[n] Throughout this section X = A2 . Let X [n] be the Hilbert scheme of' n points on X . Its C-points are the codimension n ideals J ⊂ C[x, y]. Let P[k] ⊂ n X [n] × X [n+k]

2 Quantum Toroidal gl1 , its Representations and Geometry

64

be the Nakajima-Grojnowski correspondence. For k > 0, P[k] =

( n

P[k]n with

    P[k]n = (J1 , J2 ) ∈ X [n] × X [n+k]  J2 ⊂ J1 , supp(J1 /J2 ) is a single point . (2.80) It is known that P[1] is a smooth variety. It is equipped with two natural projections: p

q

X [n] ←−P[1]n −→ X [n+1] , J1 → (J1 , J2 ) → J2 .

(2.81)

Let L be the tautological line bundle on P[1] whose fiber at (J1 , J2 ) equals J1 /J2 . Consider a natural action of a torus T = C× × C× on each X [n] , induced from the action on X given by the formula (t1 , t2 ) · (x, y) = (t1 x, t2 y). The set (X [n] )T of T-fixed points in X [n] is finite and is in bijection with size n Young diagrams. For such a Young diagram λ = (λ1 , . . . , λl ), the corresponding ideal Jλ ∈ (X [n] )T is given by: " # Jλ = C[x, y] · Cx λ1 y 0 ⊕ · · · ⊕ Cx λl y l−1 ⊕ Cy l . For a Young diagram λ, let λ denote its transpose and |λ| = λ1 + · · · + λl , as before.

2.3.2 Geometric Action I

Let  M be a direct sum of equivariant (complexified) K -groups:  M = n K T (X [n] ). ±1 ±1 It is a module over K T (pt) = C[t1 , t2 ], the equivariant (complexified) K -group of a point. Set t3 := t1−1 t2−1 ∈ C(t1 , t2 ) and consider F = C(t1 , t2 )[t]/(t 2 − t3 ). We define (2.82) M :=  M ⊗ K T (pt) F . It has a natural N-grading: M=



Mn ,

Mn = K T (X [n] ) ⊗ K T (pt) F .

n∈N

According to the Thomason localization theorem, restriction to the T-fixed point set induces an isomorphism ∼

K T (X [n] ) ⊗ K T (pt) F −→ K T ((X [n] )T ) ⊗ K T (pt) F .

The structure sheaves of T-fixed points Jλ form a basis in n K T ((X [n] )T ) ⊗ K T (pt) F, [|λ|] is a proper morphism, the denoted by {λ}. Since embedding of a point Jλ into X direct image in the equivariant K -theory is well-defined, and we denote by [λ] ∈ M|λ| the direct image of the structure sheaf {λ}. Thus, the set {[λ]} is an F-basis of M.

2.3 Geometric Realizations

65

Let F be the tautological vector bundle on X [n] whose fiber at a point J ∈ X [n] equals the quotient C[x, y]/J . Define the generating series a(z), c(z) ∈ M(z) via: a(z) := •−1/z (F) =



[k (F)](−1/z)k ,

(2.83)

k≥0

c(z) :=

a(zt1 )a(zt2 )a(zt3 ) a(zt1−1 )a(zt2−1 )a(zt3−1 )

.

(2.84)

We also define the linear operators ek , f k , ψk , ψ0−1 , d2 (k ∈ Z) acting on M via:

±

ψ (z) =

ek = q∗ (L ⊗k ⊗ p∗ ) : Mn −→ Mn+1 ,

(2.85)

f k = −t −1 · p∗ (L ⊗(k−1) ⊗ q∗ ) : Mn −→ Mn−1 ,

(2.86)

ψ0±1

+

∞  k=1

ψ±k z

∓k

 :=

t −1 z − t c(z) z−1

± ∈



End(Mn )[[z ∓1 ]] ,

n

d2 = n · Id : Mn −→ Mn ,

(2.87) (2.88)

where γ (z)± denotes the expansion of a rational function γ (z) in z ∓1 , respectively. The following is the key result of this section:  Theorem 2.6 The operators (2.85)–(2.88) give rise to an action of U¨ q1 ,q2 ,q3 (gl1 ) on M from (2.82) with the parameters q1 = t1 , q2 = t3 , q3 = t2 .

This theorem was proved independently and simultaneously in [9, 19]. We refer the interested reader to [9, Sect. 4] for a straightforward verification of the defining relations that crucially uses the properties (2.42, 2.50) of the δ-functions. In fact, this representation recovers the previously constructed Fock module:  Proposition 2.16 We have a U¨ q1 ,q2 ,q3 (gl1 ) -module isomorphism M F(1).

Proof Such an isomorphism is provided by the assignment [λ] → cλ |λ for a unique  choice of nonzero scalars cλ with c∅ = 1, see [4, Corollary 4.5].

2.3.3 Heisenberg Algebra Action on the Equivariant K -Theory Following [9], let us now construct an action of a Heisenberg algebra on M of (2.82), whose cohomological version [11, 15] utilizes higher order correspondences P[k]. To this end, we first recall the notion of Macdonald polynomials following [12]. The algebra  of symmetric polynomials in the variables {xk }∞ k=1 over F = C(q, t) is

2 Quantum Toroidal gl1 , its Representations and Geometry

66

$ freely generated by the power-sum symmetric polynomials pr = k xkr , that is,  = F[ p1 , p2 , . . .]. For a Young diagram λ = (λ1 , . . . , λl ) = (1m 1 2m 2 · · · ), we define: pλ := pλ1 · · · pλl ∈  ,

z λ :=



r mr m r ! ∈ N .

r ≥1

Consider the Macdonald inner product (·, ·)q,t on  such that: ( pλ , pμ )q,t = δλμ z λ

 1 − q λr . 1 − t λr 1≤r ≤l

Definition 2.5 The Macdonald polynomials {Pλ }λ∈P + ⊂  are defined as the unique family of polynomials such that: $ • Pλ = m λ + μ 0 , U¨ q,d , and U¨ q,d be the C-subalgebras of U¨ q,d (sln ) generated by { f i,r }ri∈[n] , Let U¨ q,d −1 r ∈Z ±1/2 ±d1 ±d2 r ∈Z , q , q }i∈[n] , respectively. The following result {ei,r }i∈[n] , and {ψi,r , ψi,0 , γ is completely analogous to Propositions 1.1 and 2.2: Proposition 3.1 (a) (Triangular decomposition of U¨ q,d (sln )) The multiplication map < 0 > ⊗ U¨ q,d ⊗ U¨ q,d −→ U¨ q,d (sln ) m : U¨ q,d is an isomorphism of C-vector spaces. > 0 < (resp. U¨ q,d and U¨ q,d ) is isomorphic to the associative C-algebra (b) The algebra U¨ q,d −1 r ∈Z ∈Z ∈Z and { f i,r }ri∈[n] ) with generated by {ei,r }i∈[n] (resp. {ψi,r , ψi,0 , γ ±1/2 , q ±d1 , q ±d2 }ri∈[n] the defining relations (T2, T7) (resp. (T0.1, T0.2, T1) and (T3, T8).

We equip U¨ q,d (sln ) with a Z[n] × Z-grading via the following assignment: deg(ei,r ) := (1i , r ) , deg( f i,r ) := (−1i , r ) , deg(ψi,r ) := (0, r ) , −1 , γ ±1/2 , q ±d1 , q ±d2 and any i ∈ [n], r ∈ Z , deg(x) := (0, 0) for x = ψi,0

(3.5)

where 1 j ∈ Z[n] has the j-th coordinate equal to 1 and all other coordinates being zero. We shall also need the following modifications of the above algebra U¨ q,d (sln ):  • Let U¨ q,d (sln ) be obtained from U¨ q,d (sln ) by “ignoring” the generators q ±d2 and   taking a quotient by the ideal (c ), that is, setting i∈[n] ψi,0 = 1.

3 Quantum Toroidal sln , Its Representations and Bethe subalgebras

76

 • Let U¨ q,d (sln ) be obtained from U¨ q,d (sln ) by “ignoring” the generators q ±d1 and taking a quotient by the ideal (c), that is, setting γ 1/2 = 1.

tor • Let Uq,d (sln ) be obtained from U¨ q,d (sln ) by “ignoring” q ±d1 and q ±d2 .   tor The above algebras U¨ q,d (sln ), U¨ q,d (sln ), Uq,d (sln ) are also Z[n] × Z-graded via (3.5). Furthermore, they also satisfy the analogues of Proposition 3.1 with the correspond < ¨ > ¨ 0  ¨ <  ¨ >  ¨ 0 tor < tor > tor 0 , Uq,d , Uq,d , Uq,d , Uq,d , Uq,d , Uq,d , Uq,d , Uq,d , ing subalgebras denoted by U¨ q,d respectively. In particular, their “positive” subalgebras are naturally isomorphic:   > > > tor > U¨ q,d U¨ q,d Uq,d . U¨ q,d

(3.6)

3.1.2 Hopf Pairing, Drinfeld Double, and a Universal R-Matrix Let us recall the general notion of a Hopf pairing, following [17, Chap. 3]. Given two Hopf C-algebras A and B with invertible antipodes S A and S B , a bilinear map ϕ : A × B −→ C is called a Hopf pairing if it satisfies the following properties: ϕ(a, bb ) = ϕ(a1 , b)ϕ(a2 , b ) ϕ(aa  , b) = ϕ(a, b2 )ϕ(a  , b1 )

∀ a ∈ A , b, b ∈ B , ∀ a, a  ∈ A , b ∈ B ,

(3.7)

as well as ϕ(a, 1 B ) =  A (a) , ϕ(1 A , b) =  B (b) , ϕ(S A (a), b) = ϕ(a, S B−1 (b)) ∀ a ∈ A , b ∈ B ,

where we use the Sweedler notation (suppressing the summation) for the coproduct: (x) = x1 ⊗ x2 .

(3.8)

For such a data, one defines the generalized Drinfeld double Dϕ (A, B) as follows (Dϕ (A, B) is called the Drinfeld double when B = A,cop and ϕ is the natural pairing): Theorem 3.1 ([17, Theorem 3.2]) There is a unique Hopf algebra Dϕ (A, B) such that: (i) Dϕ (A, B) A ⊗ B as coalgebras. (ii) Under the natural inclusions

3.1 Quantum Toroidal Algebras of sln

77

A → Dϕ (A, B) given by a → a ⊗ 1 B , B → Dϕ (A, B) given by b → 1 A ⊗ b , A and B are Hopf subalgebras of Dϕ (A, B). (iii) For any a ∈ A and b ∈ B, we have: (a ⊗ 1 B ) · (1 A ⊗ b) = a ⊗ b , (1 A ⊗ b) · (a ⊗ 1 B ) = ϕ(S −1 A (a1 ), b1 )ϕ(a3 , b3 ) a2 ⊗ b2 , where we use the Sweedler notation again (suppressing the summation): (Id ⊗ )((x)) = ( ⊗ Id)((x)) = x1 ⊗ x2 ⊗ x3 . A Hopf algebra A is called quasitriangular if there is an invertible element  A) satisfying the following propR ∈ A ⊗ A (or in a proper completion R ∈ A⊗ erties: ( ⊗ Id)(R) = R 13 R 23 , (Id ⊗ )(R) = R 13 R 12 , R(x) = op (x)R for all x ∈ A ,

(3.9)

where for decomposable tensors (and extended to all tensors by linearity) we define (a ⊗ b)12 = a ⊗ b ⊗ 1 , (a ⊗ b)13 = a ⊗ 1 ⊗ b , (a ⊗ b)23 = 1 ⊗ a ⊗ b ∈ A ⊗ A ⊗ A

as well as (a ⊗ b)op = b ⊗ a. Such an element R is called a universal R-matrix of A. The basic property of the generalized Drinfeld doubles is their quasitriangularity: Theorem 3.2 ([17, Theorem 3.2]) For a nondegenerate Hopf pairing ϕ : A × B → C, the generalized Drinfeld double Dϕ (A, B) of Theorem 3.1 is quasitriangular with a universal R-matrix given by: R=



ei ⊗ ei∗ ,

i∈I

where {ei }i∈I and {ei∗ }i∈I are bases of A and B, respectively, dual with respect to ϕ. To apply the above constructions to the quantum toroidal algebra U¨ q,d (sln ) and its subalgebras, we first need to endow the former with a Hopf algebra structure. This was carried out in [5, Theorem 2.1] in a more general setup: Theorem 3.3 U¨ q,d (sln ) is endowed with a (topological) Hopf algebra structure via:

3 Quantum Toroidal sln , Its Representations and Bethe subalgebras

78

coproduct  : ei (z) → ei (z) ⊗ 1 + ψi− (γ(1) z) ⊗ ei (γ(1) z) , 1/2

f i (z) → 1 ⊗ f i (z) + f i (γ(2) z) ⊗ ψi+ (γ(2) z) , 1/2

ψi± (z)

→

±1/2 ψi± (γ(2) z)

∓1/2 ⊗ ψi± (γ(1) z) , ±1/2 ±d1 ±d2

,q

x → x ⊗ x for x = γ

,q

(H1)

,

counit  : ei (z) → 0 , f i (z) → 0 , ψi± (z) → 1 , x → 1 for x = γ ±1/2 , q ±d1 , q ±d2 ,

(H2)

antipode S : ei (z) → −ψi− (γ −1/2 z)−1 ei (γ −1 z) , f i (z) → − f i (γ −1 z)ψi+ (γ −1/2 z)−1 , x → x

−1

1/2

for x = γ

±1/2

,q

±d1

,q

±d2

(H3) , ψi± (z) ,

1/2

where we set γ(1) := γ 1/2 ⊗ 1 and γ(2) := 1 ⊗ γ 1/2 . 



 ≥ ≤ ≥ ¨ ≤ ≥  ¨≤ , U¨ q,d (resp. U¨ q,d , Uq,d or U¨ q,d , Uq,d ) be the subalgebras of U¨ q,d (sln ) Let U¨ q,d   ∈Z,l∈−N ±1 (resp. U¨ q,d (sln ) or U¨ q,d (sln )) generated by {ei,r , ψi,l , ψi,0 , γ ±1/2 , q ±d1 , q ±d2 }ri∈[n] ∈Z,l∈N ±1 and { f i,r , ψi,l , ψi,0 , γ ±1/2 , q ±d1 , q ±d2 }ri∈[n] . Then, we have the following result:

Theorem 3.4 (a) There exists a unique Hopf algebra pairing ≥ ≤ ϕ : U¨ q,d × U¨ q,d −→ C

(3.10)

satisfying ϕ(ei (z), f j (w)) =

z

δi j mi j , ϕ(ψi− (z), ψ + · δ z/w) , (P1) j (w)) = gai j (d q − q −1 w

1/2 , q d1 , q d2 , ϕ(ei (z), x − ) = ϕ(x + , f i (z)) = 0 for x ± = ψ ∓ j (w), γ

(P2)

ϕ(γ 1/2 , q d1 ) = ϕ(q d1 , γ 1/2 ) = q −1/2 , ϕ(ψi− (z), q d2 ) = q −1 , ϕ(q d2 , ψi+ (z)) = q , (P3) (P4) ϕ(ψi− (z), x) = ϕ(x, ψi+ (z)) = 1 for x = γ 1/2 , q d1 , ϕ(γ 1/2 , q d2 ) = ϕ(q d2 , γ 1/2 ) = ϕ(γ 1/2 , γ 1/2 ) = 1 , ϕ(q d1 , q d1 ) = ϕ(q d1 , q d2 ) = ϕ(q d2 , q d1 ) = 1 , ϕ(q d2 , q d2 ) = q

(P5) n 3 −n 12

.

(P6)

≥ ≤ (b) The natural Hopf algebra homomorphism Dϕ (U¨ q,d , U¨ q,d ) → U¨ q,d (sln ) gives rise to the isomorphism

3.1 Quantum Toroidal Algebras of sln

79

∼

≥ ≤ ±1 Dϕ (U¨ q,d , U¨ q,d )/J −→ U¨ q,d (sln ) , J := (x ⊗ 1 − 1 ⊗ x | x = ψi,0 , γ ±1/2 , q ±d1 , q ±d2 ) .   (c) Likewise, the algebras U¨ q,d (sln ), U¨ q,d (sln ) admit the Drinfeld double realizations     ≥ ≤ ≥ ≤ via D ϕ ( U¨ q,d , U¨ q,d ), Dϕ  (U¨ q,d , U¨ q,d ) with similarly defined pairings  ϕ, ϕ  .

(d) The pairings ϕ, ϕ  ,  ϕ are nondegenerate iff q, qd, qd −1 are not roots of unity.   (e) If q, qd, qd −1 are not roots of unity, then U¨ q,d (sln ), U¨ q,d (sln ), U¨ q,d (sln ) admit   universal R-matrices R, R , R associated to the Hopf pairings ϕ, ϕ  ,  ϕ, respectively.

Remark 3.1 The existence of such Hopf pairings and the identification of the algebras given by generators and relations with the Drinfeld doubles are always verified by direct computations. The hardest part (often used as “folklore”) is establishing the nondegeneracy of such pairings. For general quantum affine algebras this was proved in [NT, Proposition 6.7], while in the present toroidal setup it follows from [N]. mi j Remark 3.2 (a) We note that ϕ(ψi− (z), ψ + z/w) is equivalent to: j (w)) = gai j (d − −ai j , ψ+ , ϕ(h i,−k , h j,l ) = δkl ϕ(ψi,0 j,0 ) = q

d km i j [kai j ]q (k, l > 0) . k(q − q −1 )

(3.11)

(b) The specific choice of values in (P6) is only to simplify the formulas of [23, Sect. 4].

3.1.3 Horizontal and Vertical Copies of Quantum Affine sln Following [7], we define the quantum affine algebra of sln (in the new Drinsln ), to be the associative C-algebra generated by feld realization), denoted by Uq ( −1 ±1 }r ∈Z × with the defining relations similar to , C ±1 , D {ei,r , f i,r , ψi,r , ψi,0 i∈[n] (T0.1)–(T8): ±1 − central , [ψi± (z), ψ ± j (w)] = 0 , C ±1 ∓1 ±1 · D ∓1 = 1 , · ψi,0 = C ±1 · C ∓1 = D ψi,0

−1 = qei (q −n z) , D  f i (z) D −1 = q −1 f i (q −n z) , Dψ  ± (z) D −1 = ψ ± (q −n z) ,  i (z) D De i i − + gai j (C −1 z/w)ψi+ (z)ψ − j (w) = gai j (C z/w)ψ j (w)ψi (z) ,

ei (z)e j (w) = gai j (z/w)e j (w)ei (z) , f i (z) f j (w) = gai j (z/w)−1 f j (w) f i (z) ,   (q − q −1 )[ei (z), f j (w)] = δi j δ(Cw/z)ψi+ (Cw) − δ(C z/w)ψi− (C z) ,

3 Quantum Toroidal sln , Its Representations and Bethe subalgebras

80

ψi+ (z)e j (w) = gai j (z/w)e j (w)ψi+ (z) , ψi− (z)e j (w) = gai j (C −1 z/w)e j (w)ψi− (z) , ψi+ (z) f j (w) = gai j (C −1 z/w)−1 f j (w)ψi+ (z) , ψi− (z) f j (w) = gai j (z/w)−1 f j (w)ψi− (z) ,

ei (z)e j (w) = e j (w)ei (z) if ai j = 0 , [ei (z 1 ), [ei (z 2 ), e j (w)]q −1 ]q + [ei (z 2 ), [ei (z 1 ), e j (w)]q −1 ]q = 0 if ai j = −1 , f i (z) f j (w) = f j (w) f i (z) if ai j = 0 , [ f i (z 1 ), [ f i (z 2 ), f j (w)]q −1 ]q + [ f i (z 2 ), [ f i (z 1 ), f j (w)]q −1 ]q = 0 if ai j = −1 , where the generating series {ei (z), f i (z), ψi± (z)}i∈[n]× are defined as before. Let ±1 . sln ) obtained by “ignoring” the generators D Uqaff (sln ) be the subalgebra of Uq (  The above algebra Uq (sln ) is known to admit the original Drinfeld-Jimbo realizasln ) be the associative tion of [6, 15]. To present this identification explicitly, let UqDJ ( C-algebra generated by {E i , Fi , K i±1 , D ±1 }i∈[n] with the following defining relations: D ±1 D ∓1 = 1 , K i±1 K i∓1 = 1 ,

D K i D −1 = K i ,

Ki K j = K j Ki ,

D E i D −1 = q E i ,

K i E j K i−1 = q ai j E j ,

D Fi D −1 = q −1 Fi , K i F j K i−1 = q −ai j F j ,

(q − q −1 )[E i , F j ] = δi j (K i − K i−1 ) , E i E j = E j E i if ai j = 0 ,

E i2 E j − (q + q −1 )E i E j E i + E j E i2 = 0 if ai j = −1 ,

Fi F j = F j Fi if ai j = 0 ,

Fi2 F j − (q + q −1 )Fi F j Fi + F j Fi2 = 0 if ai j = −1 .

According to [7], there is a C-algebra isomorphism ∼

: UqDJ ( sln ) −→ Uq ( sln )

(3.12)

given by: ±1 K i±1 → ψi,0 for 1 ≤ i ≤ n − 1 , −1  K 0 → C · (ψ1,0 · · · ψn−1,0 ) , D → D ,

E i → ei,0 ,

Fi → f i,0 ,

E 0 → C(ψ1,0 · · · ψn−1,0 )−1 · [· · · [ f 1,1 , f 2,0 ]q , . . . , f n−1,0 ]q , F0 → [en−1,0 , . . . , [e2,0 , e1,−1 ]q −1 · · · ]q −1 · (ψ1,0 · · · ψn−1,0 )C −1 .

(3.13)

3.1 Quantum Toroidal Algebras of sln

81

Remark 3.3 (a) The isomorphism (3.12, 3.13) was presented in [7] without a proof. The inverse −1 was constructed in [1] by using Lusztig’s braid group action, while the direct verification of (3.13) giving rise to a C-algebra homomorphism sln ) → Uq ( sln ) appeared in [16]. However, proofs of injectivity of −1 ,

: UqDJ ( in [1, 16] had gaps that were fixed more recently in [3].  satisfy slightly different (b) In some standard literature, the grading elements D and D relations. However, the present conventions are better adapted to fit into the toroidal story (in particular, our discussion in the next section) and follow that of [19]. (c) The quantum loop algebra Uq (Lsln ) from Chap. 1 is the quotient of Uqaff (sln ) by the ideal (C − 1), thus corresponding to a trivial central charge. (d) Likewise, the algebra UqDJ (Lsln ) from Remark 1.4 is obtained from UqDJ ( sln ) by “ignoring” the generators D ±1 and taking a quotient by the ideal (K 0 K 1 · · · K n−1 − 1). Following [24], let us introduce the vertical and horizontal copies of the quantum affine Uq ( sln ) inside U¨ q,d (sln ). To this end, we consider two algebra homomorphisms sln ) → U¨ q,d (sln ) ← Uq ( sln ) , Uq ( v

h

(3.14)

defined by:

◦ h : E i → ei,0 ,

Fi → f i,0 ,

K i → ψi,0 ,

D → q d2 for i ∈ [n] (3.15)

with of (3.12, 3.13), and v : ei,r → d ir ei,r , C → γ ,

f i,r → d ir f i,r , ψi,r → d ir γ r/2 ψi,r ,

 → q −nd1 · q D

n−1 j=1

j (n− j) h j,0 2

for 0 ≤ i < n , r ∈ Z ,

(3.16)

h

j,0 where we follow the conventions of Sect. 3.1.1 and add elements q 2 to U¨ q,d (sln ). According to [24], these algebra homomorphisms are injective. The corresponding sln ). images of h and v are called the horizontal and the vertical copies of Uq (   tor (sln ) Following Remark 3.3(c, d), we also note that U¨ q,d (sln ), U¨ q,d (sln ), Uq,d admit similar constructions, that is, we have the following algebra embeddings: 



v h  sln ) → U¨ q,d (sln ) ← Uq (Lsln ) , Uq ( 



v  h Uq (Lsln ) → U¨ q,d (sln ) ← Uq ( sln ) , v

(3.17)

h

tor Uqaff (sln ) → Uq,d (sln ) ← Uqaff (sln ) .

(3.18) (3.19)

3 Quantum Toroidal sln , Its Representations and Bethe subalgebras

82

By abuse of notations, we shall use U˙ qh (sln ) and U˙ qv (sln ) to denote the corresponding horizontal and vertical copies of Uq ( sln ), Uqaff (sln ), Uq (Lsln ) in one of the algebras   tor U¨ q,d (sln ), U¨ q,d (sln ), U¨ q,d (sln ), Uq,d (sln ) from the embeddings (3.14, 3.17–3.19).

3.1.4 Miki’s Isomorphism In this section, we recall the beautiful result of K. Miki that provides an algebra iso  ∼ morphism U¨ q,d (sln ) −→ U¨ q,d (sln ) intertwining the vertical and horizontal embeddings of the quantum affine/loop algebras of sln in (3.17, 3.18), similar to (2.18). To state this result explicitly, we shall need some more notations: sln ) UqDJ ( sln ) determined by: • Let σ be the algebra anti-automorphism of Uq ( σ : Ei → Ei ,

Fi → Fi ,

K i → K i−1 ,

D → D −1 for i ∈ [n] .

sln ) determined by: • Let η be the algebra anti-automorphism of Uq ( −1 f i,r → f i,−r , h i,l → −C l h i,−l , ψi,0 → ψi,0 ,

−i(n−i)  → D  · ψi,0 for 1 ≤ i < n , r ∈ Z , l = 0 . D

η : ei,r → ei,−r , C → C ,

1≤i≤n−1

• We shall also use the corresponding anti-automorphisms σ, η of Uq (Lsln ).   • Let Q be the algebra automorphism of U¨ q,d (sln ) determined by: 

Q : ei,r → (−d)r ei+1,r ,

f i,r → (−d)r f i+1,r , h i,l → (−d)l h i+1,l ,

ψi,0 → ψi+1,0 , q d2 → q d2 for i ∈ [n] , r ∈ Z , l = 0 . • Let Q be the algebra automorphism of U¨ q,d (sln ) such that it maps the generators  other than γ ±1/2 , q ±d1 as Q, while 





Q : γ 1/2 → γ 1/2 , q d1 → q d1 · γ −1 .   • Let Y j (1 ≤ j ≤ n) be the algebra automorphism of U¨ q,d (sln ) determined by: 

Y j : h i,l → h i,l , ψi,0 → ψi,0 , q d2 → q d2 , ¯

ei,r → (−d)−nδi0 δ jn −i δi j +iδi, j−1 ei,r −δ¯i j +δi, j−1 for i ∈ [n], r ∈ Z, l = 0 ,  1 if j ≡ i (mod n) nδi0 δ jn +i δ¯i j −iδi, j−1 f i,r +δ¯i j −δi, j−1 , δ¯i j := f i,r → (−d) 0 otherwise

3.1 Quantum Toroidal Algebras of sln

83

• Let Y j (1 ≤ j ≤ n) be the algebra automorphism of U¨ q,d (sln ) such that it maps  ±1 , γ ±1/2 , q ±d1 as Y j , while the generators other than ψi,0 



¯



Y j : γ 1/2 → γ 1/2 , ψi,0 → γ −δi j +δi, j−1 ψi,0 , q d1 → q d1 · γ − j = where K

 j−1

l

q n hl,0 ·

n−1

j , ·K

l−n n h l,0

and we follow the conventions of  to U¨ q,d (sln ). Sect. 3.1.1 in that we add elements γ , q l=1

l= j

q

n+1 2n

h j,0 2n

1 2n

Theorem 3.5 ([19, Proposition 1]) There exists an algebra isomorphism   ∼  : U¨ q,d (sln ) −→ U¨ q,d (sln )

(3.20)

satisfying the following properties: 







 ◦  h = v ,  ◦  v ◦ η ◦ σ = h , Q ◦ Yn ◦  =  ◦ Y1−1 ◦ Q .

(3.21)

Remark 3.4 (a) The construction of  in [19] was based on the previous work [18], tor (sln ) was constructed. where an analogous algebra automorphism  of Uq,d (b) In [19], the parameters q, ξ are related to our q, d via q ↔ q and ξ ↔ d −n , while ± ±1 } of [19, Sect. B.1] are related to ours , h i,l , ki±1 , C ±1 , D ±1 , D the generators {xi,r via: + − ↔ d ir ei,r , xi,r ↔ d ir f i,r , h i,l ↔ d il γ l/2 h i,l , xi,r ±1 ki±1 ↔ ψi,0 , C ±1 ↔ γ ±1 ,

D ±1 ↔ q ±d2 ,

±1 ↔ q ∓nd1 · q ± D h j,0

n−1 j=1

c

d1

j (n− j) h j,0 2

.

d2

(c) While the generators of [19] do not require one to add q N , q N , q N , q N , the present choice is more symmetric and is better suited for the rest of our exposition. We conclude this section by computing explicitly  -images of some generators: Proposition 3.2 (a) We have  : ei,0 → ei,0 ,

±1 ±1 f i,0 → f i,0 , ψi,0 → ψi,0 for i ∈ [n]× ,

±1 ±1 → γ ±1 · ψ0,0 , q ±d2 → q ∓nd1 · q ± ψ0,0

n−1 j=1

j (n− j) h j,0 2

,

e0,0 → d · γ ψ0,0 · [· · · [ f 1,1 , f 2,0 ]q , . . . , f n−1,0 ]q , −1 −1 γ . f 0,0 → d −1 · [en−1,0 , . . . , [e2,0 , e1,−1 ]q −1 · · · ]q −1 · ψ0,0

(b) For 1 ≤ i < n, we have

3 Quantum Toroidal sln , Its Representations and Bethe subalgebras

84

 : h i,1 → (−1)i+1 d −i × [[· · · [[· · · [ f 0,0 , f n−1,0 ]q , . . . , f i+1,0 ]q , f 1,0 ]q , . . . , f i−1,0 ]q , f i,0 ]q 2 , h i,−1 → (−1)i+1 d i × [ei,0 , [ei−1,0 , . . . , [e1,0 , [ei+1,0 , . . . , [en−1,0 , e0,0 ]q −1 · · · ]q −1 ]q −1 · · · ]q −1 ]q −2 . (c) For i = 0, we have  : h 0,1 → (−1)n d 1−n · [[· · · [ f 1,1 , f 2,0 ]q , . . . , f n−1,0 ]q , f 0,−1 ]q 2 , h 0,−1 → (−1)n d n−1 · [e0,1 , [en−1,0 , . . . , [e2,0 , e1,−1 ]q −1 · · · ]q −1 ]q −2 . (d) We have  : e0,−1 → (−d)n e0,1 ,

f 0,1 → (−d)−n f 0,−1 .

Proof Part (a) follows directly by applying the equality  ◦  h = v to the explicit formulas for (E i ), (Fi ), (K i ), (D) with being the isomorphism of (3.12, 3.13). sln ) in the Drinfeld-Jimbo presentation: (b) First, let us express h i,±1 ∈ Uq (

−1 (h i,1 ) = (−1)i [[· · · [[· · · [E 0 , E n−1 ]q −1 , . . . , E i+1 ]q −1 , E 1 ]q −1 , . . . , E i−1 ]q −1 , E i ]q −2 ,

−1 (h i,−1 ) = (−1)i [Fi , [Fi−1 , . . . , [F1 , [Fi+1 , . . . , [Fn−1 , F0 ]q · · · ]q ]q · · · ]q ]q 2 .

(3.22)

(3.23)

These formulas are proved by iteratively applying two useful identities on q-brackets (we leave details to the interested reader): [a, [b, c]u ]v = [[a, b]x , c]uv/x + x · [b, [a, c]v/x ]u/x , [[a, b]u , c]v = [a, [b, c]x ]uv/x + x · [[a, c]v/x , b]u/x . Applying the equality  ◦  v ◦ η = h ◦ σ −1 to the formulas (3.22) and (3.23), we obtain the stated formulas for the images  (h i,1 ) and  (h i,−1 ), respectively.     (c) Applying the equality Q ◦ Yn ◦  =  ◦ Y1−1 ◦ Q to h n−1,±1 and using the formulas for  (h n−1,±1 ) from part (b), we obtain the stated formulas for  (h 0,±1 ). (d) The explicit formulas for  (e0,−1 ) and  ( f 0,1 ) follow by applying the equality      Q ◦ Yn ◦  =  ◦ Y1−1 ◦ Q to en−1,0 and f n−1,0 , respectively.

3.1 Quantum Toroidal Algebras of sln

85

3.1.5 Horizontal and Vertical Extra Heisenberg Subalgebras Following [9], let us now extend the vertical and horizontal U˙ qv (sln ), U˙ qh (sln ) to U˙ qv (gln ), U˙ qh (gln ) by adjoining vertical and horizontal Heisenberg algebras hv1 and hh1 . For r = 0, pick a nontrivial solution { pi,r }i∈[n] of the following linear system: 

pi,r d −r m i j [rai j ]q = 0 ∀ j ∈ [n]× ,

(3.24)

i∈[n] tor and let hv1 be the subspace of Uq,d (sln ) spanned by

 h rv

=

γ

i∈[n] 1/2

pi,r h i,r if r = 0 . if r = 0

(3.25)

We note that hv1 is well-defined (as (3.24) has a one-dimensional space of solutions) and commutes with U˙ qv (sln ) due to (T1 , T5 , T6 ). Since [h vk , h lv ] = 0 for l = −k, hv1 tor is isomorphic to a Heisenberg algebra. Let U˙ qv (gln ) be the subalgebra of Uq,d (sln ) v v v aff ˙ ˙ generated by Uq (sln ) and h1 , so that Uq (gln ) Uq (gln )–the quantum affine gln . Let r =0 U˙ qv (h1 ), U˙ qv (hn ) be the subalgebras generated by {h rv , γ ±1/2 }r =0 and {h i,r , γ ±1/2 }i∈[n] . tor (sln ) from Remark 3.4(a) Let us recall the algebra automorphism  of Uq,d satisfying    : U˙ qv (sln ) → U˙ qh (sln ) , U˙ qh (sln ) → U˙ qv (sln ) , q c → q 2c , q c → q −c /2 .

tor (sln ) generated by U˙ qh (sln ) Then, we define U˙ qh (gln ) as the subalgebra of Uq,d and the horizontal extra Heisenberg hh1 :=  (hv1 ). Thus U˙ qh (gln ) =  (U˙ qv (gln )) Uqaff (gln ). We define U˙ qh (h1 ) :=  (U˙ qv (h1 )), U˙ qh (hn ) :=  (U˙ qv (hn )), and consider their “positive” subalgebras U˙ qh> (h1 ) and U˙ qh> (hn ) generated by { (h v−r )}r >0 and >0 { (h i,−r )}ri∈[n] .

Remark 3.5 (a) The other three versions of the quantum toroidal algebra of sln admit natural counterparts of these constructions, but with Uqaff (gln ) being replaced either n ) (obtained by adjoining by Uq (Lgln ) (the trivial central charge version) or Uq (gl n ), Uqv (gl n ) ⊂ U¨ q,d (sln ), the degree generator). To be more precise, we have: Uqh (gl   h v  h  v ¨ ¨ Uq (Lgln ), Uq (gln ) ⊂ Uq,d (sln ), and Uq (gln ), Uq (Lgln ) ⊂ Uq,d (sln ). (b) The explicit formulas for the aforementioned images  (h v−r ),  (h i,−r ) expressing them via the original algebra generators are complicated and unknown. However, their shuffle realization is provided in the end of this chapter, see Theorem 3.19.

3 Quantum Toroidal sln , Its Representations and Bethe subalgebras

86

(c) We note that [20] uses an apparently different horizontal quantum affine gln arising through the RTT realization [4] of Uqaff (gln ). However, it actually coincides with U˙ qh (gln ) as follows from Sect. 3.2.5, see Corollary 3.1.

3.2 Shuffle Algebra and Its Commutative Subalgebras > In this section, we recall the shuffle algebra realization of U¨ q,d established in [20]. Following [12], we also provide a shuffle realization of a family of its commutative subalgebras, reminiscent to the commutative subalgebra A of the quantum toroidal gl1 from Sect. 2.1.6. These commutative subalgebras will be identified with the Bethe subalgebras in Sect. 3.5.4, while their particular limit recovers U˙ qh > (hn ), the >0 subalgebra generated by { (h i,−r )}ri∈[n] (that is, half of the horizontal Heisenberg).

3.2.1 Shuffle Algebra Realization Consider an N[n] -graded C-vector space 

S =

Sk ,

k=(k0 ,...,kn−1 )∈N[n] 1≤r ≤ki . where Sk consists of k -symmetric rational functions in the variables {xi,r }i∈[n] We fix a matrix of rational functions (ζi, j (z))i, j∈[n] ∈ Mat [n]×[n] (C(z)) via:

ζi,i (z) =

z − q −2 d −1 z − q z − qd −1 (z) = 1 . , ζi,i+1 (z) = , ζi,i−1 (z) = , ζi, j ∈{i,i±1} / z−1 z−1 z−1

Similarly to (1.32), let us now introduce the bilinear shuffle product  on S: given F ∈ Sk and G ∈ S , define F  G ∈ Sk+ via 1 (F  G)(x0,1 , . . . , x0,k0 +0 ; . . . ; xn−1,1 , . . . , xn−1,kn−1 +n−1 ) := × k! · ! ⎛ ⎞     (3.26) >ki  ∈[n] r



i  x i,r k

 : U¨ q,d −→ S

(3.30)

r for any i ∈ [n], r ∈ Z. such that ei,r → xi,1

The following result was established in [20]: ∼ > Theorem 3.6 ([20, Theorem 1.1])  : U¨ q,d −→ S of (3.30) is an isomorphism of N[n] × Z-graded C-algebras.

Remark 3.6 We note that while the shuffle product of [20, Sect. 3.1] as well as the pole and wheel conditions of [20, Sect. 3.2] differ from (3.26) and (3.27, 3.28), the shuffle algebra A+ of loc.cit. is easily seen to be isomorphic to our S, see [20, Remark 3.4].

3 Quantum Toroidal sln , Its Representations and Bethe subalgebras

88

The proof of Theorem 3.6 is similar in spirit to its gl1 -counterpart (2.25). In analogy to Definition 2.2, the key tool of loc.cit. is the study of slope ≤ μ subalgebras: Sμ =



μ

μ

Sk,d with Sk,d consisting of all F ∈ Sk,d for which

(k,d)∈N[n] ×Z

F(. . . , ξ · xi,1 , . . . , ξ · xi,i , xi,i +1 , . . . , xi,ki , . . .) ∀0 ≤  ≤ k . ∃ lim ξ →∞ ξ ||μ

(3.31)

Similarly to Lemma 2.2, the subspace S μ is easily seen to be a subalgebra of S. The μ key point is that each graded component Sk,d is finite dimensional and we have: μ

Proposition 3.4 ([20, Lemma 3.17]) dim(Sk,d ) ≤ number of unordered collections L = {(i 1 , j1 , d1 ), . . . , (i t , jt , dt )} such that

(3.32)

[i 1 , j1 − 1] + · · · + [i t , jt − 1] = k , d1 + · · · + dt = d , dr ≤ μ( jr − ir ) , with t ≥ 1, ir < jr ∈ Z, dr ∈ Z, and the notation [a, b] ∈ N[n] defined as in (3.35). Furthermore, we identify (ir , jr , dr ) ∼ (ir − n, jr − n, dr ) in the above counting. This result is proved similarly to Proposition 2.4 and uses the specialization maps φ L of (3.38) utilized in our study of A(s) below (we disregard dr ’s in (3.38) as (3.32) forces all dr = 0). A more refined object from [20] is the following subalgebra of S μ : μ|k|∈Z μ

B =



k∈N[n]

μ Bk

μ|k|=d∈Z

=



μ

Sk,d .

(3.33)

k∈N[n]

n ) be the “positive” subalgebra of the Drinfeld-Jimbo quantum affine gln , Let UqDJ> (gl obtained from UqDJ> ( sln ) by adding central elements {Pk }k≥1 (Heisenberg’s half). For b μ = a ∈ Q with gcd(a, b) = 1, the beautiful result [20, Lemma 3.15] identifies:  n )⊗g B μ UqDJ> (gl g

where

g = gcd(n, a) .

(3.34)

3.2.2 Commutative Subalgebras A(s0 , . . . , sn−1 ) In this section, we introduce a natural counterpart of the commutative subalgebra A from Sect. 2.1.6, which now depends on n − 1 additional parameters.  For any 0 ≤  ≤ k ∈ N[n] , ξ ∈ C× , F ∈ Sk , define Fξ ∈ C(x0,1 , . . . , xn−1,kn−1 ) via:

3.2 Shuffle Algebra and Its Commutative Subalgebras

89



Fξ := F(ξ · x0,1 , . . . , ξ · x0,0 , x0,0 +1 , . . . , x0,k0 ; . . . ; ξ · xn−1,1 , . . . , ξ · xn−1,n−1 , . . . , xn−1,kn−1 )

For any pair of integers a ≤ b, we define the degree vector  := [a; b] ∈ N[n] via:      = (0 , . . . , n−1 ) with i = # c ∈ Z  a ≤ c ≤ b and c ≡ i (mod n) . (3.35) 

For such a choice (3.35) of , we will denote Fξ simply by Fξ(a,b) . We also define: ∂ (∞;a,b) F := lim Fξ(a,b) and ∂ (0;a,b) F := lim Fξ(a,b) if they exist . ξ →∞

ξ →0

(3.36)

Definition 3.1 For any s = (s0 , . . . , sn−1 ) ∈ (C× )[n] , consider an N[n] -graded subspace A(s) ⊂ S whose degree k = (k0 , . . . , kn−1 ) component is defined by:    A(s)k := F ∈ Sk,0  ∂ (∞;a,b) F = (sa sa+1 · · · sb ) · ∂ (0;a,b) F

 for any a ≤ b such that [a; b] ≤ k ,

where we again use the cyclic (that is, modulo n) notations si := si

mod n .

A certain class of such elements is provided by the following result: Lemma 3.2 For any k ∈ N, μ ∈ C, and s ∈ (C× )[n] , define Fkμ (s) ∈ S(k,...,k) by μ

Fk (s) :=   i∈[n]

1≤r =r  ≤k (x i,r

− q −2 xi,r  ) ·   i∈[n]



i∈[n] (s0

1≤r,r  ≤k (x i,r

· · · si

k

r =1

xi,r − μ

k

r =1

xi+1,r )

− xi+1,r  )

,

μ

where we set xn,r := x0,r as before. If s0 · · · sn−1 = 1, then Fk (s) ∈ A(s). Proof It suffices to verify the claim for μ = 0, a = 0, b = nr + c with 0 ≤ r < k and 0 ≤ c ≤ n − 1. Then 0 = . . . = c = r + 1, c+1 = . . . = n−1 = r . As μ grows at the speed ξ i∈[n] i (i+1 −i −1)+ i∈[n] max{i ,i+1 } , ξ → ∞, the function Fk (s)(a,b) ξ and as ξ → 0, the function Fkμ (s)(a,b) grows at the speed ξ i∈[n] i (−i+1 +i −1)+ i∈[n] min{i ,i+1 } . ξ For the above values of i , both powers of ξ are zero and hence both limits μ μ Fk (s) do exist. Moreover, for  being 0 or ∞, we have ∂ (∞;a,b) Fk (s) and ∂ (0;a,b)  μ  ( ∂ (;a,b) Fk (s) = (−1) i∈[n] i i −i−1 ) q −2 i∈[n] i (k−i ) · G · i∈[n] G ,i , where  G=



−2 xi,r  ) 1≤r =r  ≤i (x i,r − q  1≤r  ≤i+1 i∈[n] 1≤r ≤i (x i,r − x i+1,r  ) i∈[n]

· ·

 i∈[n]



i∈[n]



i L if there exists s, such that bs − as > bs − as and bt − at = bt − at for 1 ≤ t < s. Similarly to (1.40), any partition L  k gives rise to a specialization map φ L : A(s)k −→ C[y1±1 , . . . , yr±1 ] .

(3.38)

To this end, we split the x∗,∗ -variables into r groups, corresponding to the above intervals in L, and specialize those corresponding to the interval [at ; bt ] (1 ≤ t ≤ r ) in the natural order to: (qd)−at · yt , . . . , (qd)−bt · yt For F =

f (x0,1 ,...,xn−1,kn−1 ) 1≤r  ≤ki+1 (xi,r −xi+1,r  ) i∈[n] 1≤r ≤k



∈ A(s)k , we define φ L (F) as the corresponding

i

specialization of f (which is independent of our splitting of the x∗,∗ -variables since f is symmetric). This gives rise to the filtration on A(s)k via: A(s)kL :=



Ker(φ L  ) .

(3.39)

L  >L

We shall now consider the images φ L (F) for F ∈ A(s)kL . First, we note that the wheel conditions (3.28) for F guarantee that φ L (F), which is a Laurent polynomial in {y1 , . . . , yr }, vanishes under the following specializations: 

(i) (qd)−x yt  = (q/d)(qd)−x yt for any 1 ≤ t < t  ≤ r, at ≤ x < bt , at  ≤ x  ≤ bt  with x  ≡ x + 1 mod n, 

(ii) (qd)−x yt  = (d/q)(qd)−x yt for any 1 ≤ t < t  ≤ r, at < x ≤ bt , at  ≤ x  ≤ bt  with x  ≡ x − 1 mod n. Second, the condition φ L  (F) = 0 for any L  > L also implies that φ L (F) vanishes under the following specializations: 

(iii) (qd)−x yt  = (qd)−bt −1 yt for any 1 ≤ t < t  ≤ r, at  ≤ x  ≤ bt  with x  ≡ bt + 1 mod n, 

(iv) (qd)−x yt  = (qd)−at +1 · yt for any 1 ≤ t < t  ≤ r, at  ≤ x  ≤ bt  with x  ≡ at − 1 mod n. Combining the vanishing conditions (i)–(iv) for φ L (F) with F ∈ A(s)kL , we conclude that the specialization φ L (F) is divisible by a polynomial Q L ∈ C[y1 , . . . , yr ], defined as a product of the linear terms in yt ’s arising from (i)–(iv), counted with the correct multiplicities.

3 Quantum Toroidal sln , Its Representations and Bethe subalgebras

92

For 1 ≤ t ≤ r , we define t := [at ; bt ] ∈ N[n] . Then, the total degree of Q L equals 



1≤t 0}. It suffices to show that dim A (s)kδ ≥ dim Rk .

(3.45)

To prove (3.45), for any Young diagram λ = (λ1 , . . . , λl ), we introduce the vertical specialization map

1≤t≤l (3.46) ϕλ : S|λ|·δ −→ C {yi,t }i∈[n] by specializing the x∗,∗ -variables as follows: xi,λ1 +···+λt−1 +a → q 2a yi,t for any 1 ≤ t ≤ l , 1 ≤ a ≤ λt , i ∈ [n] . It is clear that for any π = {k1 ≥ · · · ≥ k N > 0} with |π | = k1 + · · · + k N = k = |λ| and λ > π  (where > denotes the dominance order on size k Young diagrams and π  denotes the transposed Young diagram, as before), we have (cf. Lemmas 1.5, 2.3): μ

ϕλ (Fπ (s)) = 0 for any μ ∈ C N .

(3.47)

Therefore, it suffices to prove (cf. Lemma 1.6): 

  μ ,...,μ 

 i iN dim ϕπ  span Fπ 1 (s)  1 ≤ i 1 , . . . , i N ≤ n ≥ dim Rk .

(3.48)

πk

To this end, we first consider the case k1 = . . . = k N ⇒ k = N k1 = |π |. Then ϕπ 



μ Fπ (s)



=Z·

N

r =1 i∈[n]

 s0 · · · si

k1

t=1

yi,t − μir

k1

 yi+1,t

(3.49)

t=1

for a certain nonzero common factor Z . Define Yi := yi,1 · · · yi,k1 for i ∈ [n]. Since fr (Y1 , . . . , Yn ) :=

i∈[n]

(s0 · · · si Yi − μr Yi+1 ) , 1 ≤ r ≤ n ,

3.2 Shuffle Algebra and Its Commutative Subalgebras

95

N are algebraically independent, we see that dim Span{ t=1 f it (Y1 , . . . , Yn )} is ≥ the n . Generalizing this argument to a general number of degree N monomials in {Ti,k1 }i=1 π implies the stated inequality (3.48).  μ

μ∈C

By Lemmas 3.3–3.5, the subspace A(s) is shuffle-generated by {Fk (s)}k≥1 and thus it is a subalgebra of S. Moreover, it has the prescribed dimensions of each graded component, with the equality taking place in Lemma 3.3(a). The following result will μ be derived in Sect. 3.5.4 from the transfer matrix realization of Fk (s): μ

μ∈C

Lemma 3.6 The elements {Fk (s)}k≥1 pairwise commute. The above Lemmas 3.3–3.6 complete our proof of Theorem 3.7. Remark 3.7 We note that the proof of that A(s) = C for any Lemma 3.3 Zimplies / q · d Z unless all integers ai = 0. s = (s0 , . . . , sn−1 ) ∈ (C× )[n] such that i∈[n] siai ∈

3.2.4 Special Limit Ah Let us consider a special limit Ah of A(s), corresponding to the case s1 , . . . , sn−1 → 0 and s0 = 1/(s1 · · · sn−1 ). Explicitly, we define Ah as follows: Definition 3.2 For k ∈ N[n] , let Ahk ⊂ Sk be the subspace of all F ∈ Sk,0 such that: • ∂ (0;a,b) F exists for any a ≤ b satisfying  := [a; b] ≤ k, and furthermore vanishes whenever 0 > min{i }i∈[n] . • ∂ (∞;a,b) F exists for any a ≤ b satisfying  := [a; b] ≤ k, and furthermore vanishes whenever 0 < max{i }i∈[n] . • ∂ (0;a,b) F = ∂ (∞;a,b) F for any a ≤ b satisfying δ = [a; b] ≤ k. " We set Ah := k∈N[n] Ahk . For any p ∈ [n] and k ∈ N, consider the elements  0p;k ∈ Skδ,0 defined via:   0p;k

:=

i∈[n]



1≤r =r  ≤k (x i,r



i∈[n]



− q −2 xi,r  ) ·

1≤r,r  ≤k (x i,r

 i∈[n]

− xi+1,r  )

k

r =1

k xi,r x0,r · . x r =1 p,r

(3.50)

Remark 3.8 The elements (3.50) arise as the leading terms of the μn− p -coefficients μ in Fk (s) from Lemma 3.2 in the aforementioned limit s1 , s2 , . . . , sn−1 → 0. It is straightforward to see that  0p;k ∈ Ah , cf. Lemma 3.2. In fact, we have the following description of Ah (proved completely analogously to Theorem 3.7): Theorem 3.8 The subspace Ah is a subalgebra of S. Moreover, Ah is a polynomial h algebra in the generators { 0p;k }k≥1 p∈[n] of (3.50). In particular, A is a commutative subalgebra of S.

3 Quantum Toroidal sln , Its Representations and Bethe subalgebras

96

Combining this result with Theorem 3.19 from the end of this chapter, we obtain: ∼ > Theorem 3.9 The restriction of the algebra isomorphism  : U¨ q,d −→ S from The∼ orem 3.6 gives rise to an algebra isomorphism  : U˙ qh> (hn ) −→ Ah .

" Remark 3.9 We note that Ah = k∈N Ahkδ and Ahkδ ⊂ Skδ,0 can be described only by the first condition of Definition 3.2 (the other two being implied).

3.2.5 Identification of Two Extra Horizontal Heisenbergs # Recall the slope filtration S = μ∈R S μ of (3.31). For a slope μ = 0, consider the " subalgebra B 0 := k∈N[n] Bk0 with Bk0 := S 0 ∩ Sk,0 , see (3.33). Explicitly, we have: 

F ∈ Bk0 ⇐⇒ F ∈ Sk,0 and ∃ lim Fξ ∀ 0 ≤  ≤ k .

(3.51)

ξ →∞

Let U˙ qh> (sln ) be the subalgebra of U˙ qh (sln ) generated by {ei,0 }i∈[n] . In [20] n ) it was enlarged to U˙ qh> (gln ), isomorphic to the “positive” subalgebra UqDJ> (gl DJ  of the quantum group Uq (gln ), by adjoining a family of commuting elements {X k }k≥1 (the subalgebra generated by X k coincides with the subalgebra generated by h hk =  (h v−k ) from Sect. 3.1.5, due to Corollary 3.1). As the key ingredient in the proof of Theorem 3.6, the following result was established in [20], cf. (3.34): > Proposition 3.5 ([20]) (a) The restriction of the algebra embedding  : U¨ q,d → S ∼ h> 0 of Proposition 3.3 gives rise to the algebra isomorphism  : U˙ q (gln ) −→ B .

(b) The images Pk := (X k ) are uniquely (up to a constant) characterized by: 

lim (Pk )ξ = 0 ∀ 0 <  < kδ

ξ →∞

Pk is Z/nZ − invariant ,

and

(3.52)

where the cyclic group Z/nZ acts on Skδ,0 by permuting xi,r → xi+1,r for all i, r . As an immediate corollary, we get the following result: Theorem 3.10  −1 (A(s)) ⊂ U˙ qh> (gln ) s0 · · · sn−1 = 1.

for

generic

{si }i∈[n]

such

that

μ

Proof By Theorem 3.7 and Proposition 3.5(a), it suffices to show that Fk (s) ∈ B 0 .  μ The latter is equivalent to the existence of limits lim (Fk (s))ξ for all 0 ≤  ≤ kδ. As 

ξ →∞





i∈[n] i (i+1 −i +1)− i∈[n] min{i ,i+1 } ξ → ∞, the function (Fkμ (s))ξ grows at the speed ξ (see the proof of Lemma 3.2). Since i∈[n] i (i+1 − i + 1) − i∈[n] min{i , i+1 }

3.3 Representations of Quantum Toroidal sln

=

i∈[n]

i i+1 −

2 i2 +i+1 −i −i+1 2

97

− min{i , i+1 } and each summand is nonposiμ



tive by (3.44), so is the aforementioned power of ξ . Hence, the limits lim (Fk (s))ξ ξ →∞

exist and are finite for any 0 ≤  ≤ kδ. This completes the proof.



We conclude this section by providing explicit formulas for Pk . To this end, define: 

 Fk :=

i∈[n]

1≤r =r  ≤k (q



i∈[n]

−1



xi,r − q xi,r  ) ·

1≤r,r  ≤k (x i+1,r 

 i∈[n]

 1≤r ≤k

xi,r

− xi,r )

∈ Skδ,0

(3.53)

for k > 0,  and F0 := 1 ∈ S(0,...,0),0 . We note that Fk is a multiple of Fk0 (s) ∈ A(s) whenever i∈[n] si = 1. Furthermore, Fk is clearly Z/nZ-invariant. We also define L k ∈ Skδ,0

via

exp



 L k z −k = Fk z −k .

k≥1

(3.54)

k≥0

The relevant properties of these elements are stated in the next theorem: 

Theorem 3.11 (a) For  ∈ / {0, δ, 2δ, . . . , kδ}, we have lim (Fk )ξ = 0. (b) For any 0 ≤  ≤ k, we have lim (Fk )δ ξ = ξ →∞

ξ →∞ ≤ F ({xi,r }ri∈[n] )·

> Fk− ({xi,r }ri∈[n] ).



(c) For any 0 <  < kδ, we have lim (L k )ξ = 0. ξ →∞





Proof For any 0 ≤  ≤ kδ, the function (Fk )ξ grows at the speed ξ i∈[n] i (i+1 −i ) as ξ → ∞. Note that i∈[n] i (i+1 − i ) = − 21 i∈[n] (i − i+1 )2 ≤ 0 and the equality takes place iff 0 = . . . = n−1 ⇔  ∈ {0, δ, 2δ, . . . , kδ}. This implies part (a). Part (b) is straightforward. Finally, part (c) follows directly from parts (a, b), due to  the logarithmical definition (3.54) of L k . The shuffle elements L k ’s are clearly Z/nZ-invariant as so are Fk ’s. Combining this with Theorem 3.11(c), the characterization from Proposition 3.5(b), and Theorem 3.19 from the end of this chapter, we obtain: Corollary 3.1 Pk is a nonzero multiple of L k from (3.54). The subalgebra generated by {X k } coincides with the subalgebra generated by h hk =  (h v−k ) from Sect. 3.1.5. In particular, the horizontal quantum affine gln of [9] and [20] actually coincide.

3.3 Representations of Quantum Toroidal sln In this section, following [9, 22, 23], we provide three different constructions of the modules over quantum toroidal algebras of sln that are reminiscent of the corresponding constructions for the quantum toroidal algebras of gl1 from Chap. 2. In

3 Quantum Toroidal sln , Its Representations and Bethe subalgebras

98

  fact, these will be actually modules of either U¨ q,d (sln ) or U¨ q,d (sln ), as no U¨ q,d (sln )modules are presently known (in the presence of the degree operators q d1 , q d2 ) and tor (sln )-modules one of the two central elements c, c acts trivially. in all known Uq,d

3.3.1 Vector, Fock, and Macmahon Modules  The general theory of the category O− and the lowest weight modules for U¨ q,d (sln )  is similar to that for U¨ q1 ,q2 ,q3 (gl1 ) from Sect. 2.2.1. The category O− is defined as  in Definition 2.3 with the algebra U¨ q,d (sln ) being Z-graded via the adjoint action x → q d2 xq −d2 , cf. (T0.4). Likewise, the lowest weight modules are defined as in Definition 2.4, but with the lowest weight being now encoded by a nonzero constant ν ∈ C× and a tuple of series φ = {φi± (z)}i∈[n] , φi± (z) ∈ C[[z ∓1 ]], so that on the lowest weight vector v0 we have:

q d2 v0 = ν · v0 , ψi± (z)v0 = φi± (z) · v0 ,

f i (z)v0 = 0 for i ∈ [n] .

Let Vφ,ν be the corresponding irreducible lowest weight U¨ q,d (sln )-module. The following result is completely analogous to Proposition 2.7: 

Proposition 3.6 Vφ,ν is in O− if and only if there are rational functions φi (z) ∈ C(z) such that φi (0)φi (∞) = 1 and φi± (z) = φi (z)± –the expansions of φi (z) in z ∓1 . Similarly to the observation from the end of Sect. 2.2.1, all the above admits natural opposite counterparts: the category O+ and the notion of the highest weight modules. Then, the analogue of Proposition 3.6 still holds. Moreover, we have natural  equivalences O± → O∓ given by twists with the automorphisms ϑk of U¨ q,d (sln ): ϑk : ei (z) → f i  (1/z) , f i (z) → ei  (1/z) , ψi± (z) → ψi∓ (1/z) , q d2 → q −d2 (3.55) for k ∈ [n], where i  := 2k − i mod n ∈ [n]. We note that ϑk are just cyclic rotations of ϑ0 , that is, ϑk = k ◦ ϑ0 ◦ −k , where  is the following automorphism of  U¨ q,d (sln ): ± (z) , q d2 → q d2 . (3.56)  : ei (z) → ei+1 (z) , f i (z) → f i+1 (z) , ψi± (z) → ψi+1

We shall now describe some of such modules combinatorially, following [9]. All of them depend on two parameters: p ∈ [n], u ∈ C× . Following (2.59, 2.79), define: φ(t) :=

q −1 t − q t −1

and

φ K (t) :=

K −1 t − K . t −1

(3.57)

We shall assume that q and d are generic (equivalently, q1 and q3 are generic (2.39)):

3.3 Representations of Quantum Toroidal sln

99

q a d b = 1 (a, b ∈ Z) ⇐⇒ a = b = 0 .

(G)

For integers a and b, we shall write a ≡ b if a − b is divisible by n, use a ∈ [n] to denote the residue a mod n, and finally set δ¯ab := δab (vanishing unless a ≡ b). • Vector representations and their tensor products The “building block” of all the constructions below is the family of vector represenp∈[n] tations {V ( p) (u)}u∈C× with a basis parametrized by Z, see [9, Lemma 3.1]: Proposition 3.7 (Vector representations) Let V ( p) (u) be a C-vector space with a  basis {[u]r }r ∈Z . The following formulas define a U¨ q,d (sln )-action on it: 

δ(q1r u/z) · [u]r +1 if i + r + 1 ≡ p , 0 otherwise  δ(q1r −1 u/z) · [u]r −1 if i + r ≡ p f i (z)[u]r = , 0 otherwise

ei (z)[u]r =

q ±d2 [u]r = q ±r · [u]r , ⎧ r −1 ± ⎪ ⎨φ(q1 u/z) · [u]r if i + r ≡ p ψi± (z)[u]r = φ(z/q1r u)± · [u]r if i + r + 1 ≡ p . ⎪ ⎩ otherwise [u]r

(3.58)

Proof The proof is completely analogous to its gl1 -counterpart from Sect. 2.2.2.  Furthermore, for any p1 , . . . , p N ∈ [n] and u 1 , . . . , u N ∈ C× satisfying uu ij ∈ / q1Z  for i = j, the analogue of (2.45)–(2.48) gives rise to a natural U¨ q,d (sln )-action on V ( p1 ) (u 1 ) ⊗ · · · ⊗ V ( p N ) (u N ). In particular, if q and d are generic (G), then we get a  U¨ q,d (sln )-action on the tensor products (cf. (2.52)) V ( p)N (u) := V ( p) (u) ⊗ V ( p) (q2−1 u) ⊗ V ( p) (q2−2 u) ⊗ · · · ⊗ V ( p) (q2−N +1 u) ,  and W ( p)N (u) ⊂ V ( p)N (u), spanned by {|λu }λ∈P N , is a U¨ q,d (sln )-submodule.

• Fock representations and their tensor products  Similarly to the gl1 -counterpart, the above action of U¨ q,d (sln ) on W ( p)N (u) can be used to construct an action on the inductive limit of the subspaces W ( p)N ,+ (u) spanned p∈[n] by {|λu }λ∈P N ,+ , see [9, Proposition 3.3]. The resulting Fock modules {F ( p) (u)}u∈C× have bases {|λ}λ∈P+ , see (2.58), and are explicitly described via:  Proposition 3.8 Assume that q and d are generic (G). Then, the U¨ q,d (sln )-action on F ( p) (u) is given by the following explicit matrix coefficients:

3 Quantum Toroidal sln , Its Representations and Bethe subalgebras

100 λ + 1l |ei (z)|λ = δ¯ p+l−λl ,i+1

p+k−λ

k ≡i

k ≡i+1

p+k−λ



λ −λ −1 λ −λ λ φ q1 k l q3k−l φ q1 l k q3l−k δ q1 l q3l−1 u/z ,

1≤kl

λ −1 φ q1 k q3k−1 u/z

k≥1

k ≡i+1

p+k−λ

⎞±

−1 λk −1 k−2 ⎠ , φ q1 q3 u/z

k≥1

λ|q ±d2 |λ = q ±|λ| ,

and all other matrix coefficients vanishing. It is an irreducible lowest weight module in the category O− with the lowest weight (1; φ(z/u)δi p )i∈[n] , and |∅ is its lowest weight vector. Proof The proof is completely analogous to its gl1 -counterpart from Sect. 2.2.2.  Definition 3.3 For c ∈ (C× )[n] , let τu,c denote the twist of this representation by the  algebra automorphism χ p,c of U¨ q,d (sln ) defined via (cf. Remark 3.13): p

ei,r → ci ei,r , f i,r → ci−1 f i,r , ψi,r → ψi,r , q d2 → q −

p(n− p) 2

· q d2 ∀ i ∈ [n], r ∈ Z .

N ⊂ [n] × C× × (C× )[n] generic if We call a collection {( pk , u k , ck )}k=1

∀ 1 ≤ s  < s ≤ N  a, b ∈ Z s.t. b − a ≡ ps  − ps and u s = u s  q1−a q3−b . (3.59) We have the following simple result (see [9, Lemma 4.1]): N , an analogue of (2.45)–(2.48) Lemma 3.7 For a generic collection {( pk , u k , ck )}k=1  p1 pN ¨ defines a Uq,d (sln )-action on τu 1 ,c1 ⊗ · · · ⊗ τu N ,c N . Moreover, it is an irreducible low est weight U¨ q,d (sln )-module in O− , and |∅ ⊗ · · · ⊗ |∅ is its lowest weight vector.

Proof The proof is completely analogous to its gl1 -counterpart from Sect. 2.2.2. In ∈Z act particular, the irreducibility of this module follows from the fact that {ψi,r }ri∈[n] 1 N  diagonally in the basis |λ  ⊗ · · · ⊗ |λ  with a simple joint spectrum. • Macmahon representations  Finally, as in the gl1 -case, one can further construct a U¨ q,d (sln )-action on the subspace of F ( p) (u) ⊗ F ( p) (q2 u) ⊗ · · · ⊗ F ( p) (q2N −1 u) spanned by |λ1  ⊗ · · · ⊗ |λ N  satisfying λlk ≥ λlk+1 for 1 ≤ k < N , l ≥ 1. Considering the N → ∞ limit again, one  further obtains a one-parametric family of U¨ q,d (sln )-actions on the vector space with a basis labeled by plane partitions λ¯ of (2.66), depending on a parameter K ∈ C× . The resulting modules M( p) (u, K ) are called (vacuum) Macmahon modules. As we will not presently need explicit formulas for the action, we refer the interested reader to [9, Sect. 4.2] for more details. We shall only need the following result, cf. (3.57):

3.3 Representations of Quantum Toroidal sln

101

Proposition 3.9 ([9, Theorem 4.3]) Let p ∈ [n], u ∈ C× , and assume that q and / q Z d Z . Then, the Macmahon module M( p) (u, K ) can d are generic (G) and K 2 ∈  be invariantly described as the lowest weight U¨ q,d (sln )-module in the category O− K δi p with the lowest weight (1; φ (z/u) )i∈[n] . In the above realization, the empty plane partition ∅¯ = (∅, ∅, . . .) is its lowest weight vector.

3.3.2 Vertex Representations In this section, we recall an important family of vertex-type U¨ q,d (sln )-modules from [22], generalizing the construction of Frenkel-Jing [13] for quantum affine algebras. To this end, we consider a Heisenberg algebra An generated by a central element k=0 H0 and {Hi,k }i∈[n] with the following defining relations, cf. (T1 ): 

[Hi,k , H j,l ] = d −km i j

[k]q · [kai j ]q δk,−l · H0 . k

(3.60)

k>0 ≥ Let A≥ n be the subalgebra generated by {Hi,k }i∈[n] ∪ {H0 }, and Cv0 be the An -module An with Hi,k v0 = 0 and H0 v0 = v0 . The induced representation Fn := IndA≥ (Cv0 ) is n called the Fock representation of An . n−1 , the fundamental weights of sln by We denote the simple roots of sln by {α¯ i }i=1 "n−1 n−1 n−1 ¯ i }i=1 , the simple coroots of sln by {h¯ i }i=1 . Let Q¯ := i=1 { Zα¯ i be the root lattice "n−1 "n−1 ¯ i = i=2 ¯ n−1 be the weight lattice of sln . We set Z Zα¯ i ⊕ Z of sln , P¯ := i=1

α¯ 0 := −



α¯ i ∈ Q¯ ,

¯ 0 := 0 ∈ P¯ , 

h¯ 0 := −

1≤i≤n−1



h¯ i .

1≤i≤n−1 ¯

¯ be the C-algebra generated by eα¯ 2 , . . . , eα¯ n−1 , en−1 subject to the relations: Let C{ P} ¯

eα¯ i · eα¯ j = (−1)h i ,α¯ j  eα¯ j · eα¯ i , For α =

n−1 i=2

¯

¯

eα¯ i · en−1 = (−1)δi,n−1 en−1 · eα¯ i .

¯ via: ¯ n−1 , we define eα¯ ∈ C{ P} m i α¯ i + m n  ¯

eα¯ := (eα¯ 2 )m 2 · · · (eα¯ n−1 )m n−1 (en−1 )m n . n−1 ¯ be the subalgebra of C{ P} ¯ generated by {eα¯ i }i=1 . Let C{ Q} For every 0 ≤ p ≤ n − 1, consider the subspace ¯

¯  p ⊂ Fn ⊗ C{ P} ¯ . W ( p)n := Fn ⊗ C{ Q}e

3 Quantum Toroidal sln , Its Representations and Bethe subalgebras

102

We define the operators eα¯ , ∂α¯ i , Hi,l , z Hi,0 , d acting on W ( p)n , which assign to every n−1   ¯ ¯ v ⊗ eβ = Hi1 ,−k1 · · · Hi N ,−k N v0 ⊗ e j=1 m j α¯ j + p ∈ W ( p)n

(3.61)

the following values: ¯

¯

eα¯ (v ⊗ eβ ) := v ⊗ eα¯ eβ , ¯

¯

¯ · v ⊗ eβ , ∂α¯ i (v ⊗ eβ ) := h¯ i , β ¯

¯

Hi,l (v ⊗ eβ ) := (Hi,l v) ⊗ eβ , ¯ h¯ i ,β

β¯

1 2

(3.62)

n−1

¯ ¯ j m i j j=1 h i ,m j α

β¯

(v ⊗ e ) := z d ·v⊗e ,

 ¯ β)−( ¯ ¯ p , ¯ p) ¯ (β,  β¯ d(v ⊗ e ) := · v ⊗ eβ . ki + 2 z

Hi,0

The following result provides a natural structure of a U¨ q,d (sln )-module on W ( p)n : 

Proposition 3.10 ([22, Proposition 3.2.2]) For c = (c0 , . . . , cn−1 ) ∈ (C× )[n] ,  u ∈ C× , 0 ≤ p ≤ n − 1, the following formulas define an action of U¨ q,d (sln ) on W ( p)n : p (ei (z)) = ρu,c     z k z −k z 1+Hi,0  q −k/2  q −k/2 Hi,−k Hi,k , ci · exp exp − · eα¯ i [k]q u [k]q u u k>0 k>0

p

ρu,c ( f i (z)) =

    z k z −k z 1−Hi,0  q k/2  q k/2 (−1)nδi0 · exp − Hi,−k Hi,k , exp · e−α¯ i ci [k]q u [k]q u u k>0

k>0

 p ρu,c (ψi± (z)) p ρu,c (γ ±1/2 )

= exp ±(q − q

=q

±1/2

,

−1

)



Hi,±k

k>0 p ±d1 ρu,c (q ) = q ±d

z ∓k

 · q ±∂α¯ i ,

u

.

Remark 3.10 For q and d generic (G), the U¨ q,d (sln )-module ρu,c is irreducible since so its restriction to U˙ qv (sln ), recovering the representation of [13], as proved in [2]. 

p

3.3 Representations of Quantum Toroidal sln

103

3.3.3 Shuffle Bimodules Following [23] (and generalizing the gl1 -counterparts from Sect. 2.2.4), let us intro duce three families of S-bimodules which also become U¨ q,d (sln )-modules. • Shuffle bimodules S1, p (u) For u ∈ C× and 0 ≤ p ≤ n − 1, consider an N[n] -graded C-vector space 

S1, p (u) =

k=(k0 ,...,kn−1

S1, p (u)k , )∈N[n]

where the degree k component S1, p (u)k consists of k -symmetric rational functions 1≤r ≤ki F in the variables {xi,r }i∈[n] satisfying the following three conditions: (i) Pole conditions on F:   f (x0,1 , . . . , xn−1,kn−1 ) ±1 1≤r ≤ki k } . , f ∈ C {x k p r  ≤ki+1 i,r i∈[n] (xi,r − xi+1,r  ) · r =1 (x p,r − u) i∈[n] r ≤k

F= 

i

(ii) First kind wheel conditions on F:   F {xi,r } = 0 once xi,r1 = qd  xi+,s = q 2 xi,r2 for some , i, r1 , r2 , s . where  ∈ {±1}, i ∈ [n], 1 ≤ r1 = r2 ≤ ki , and 1 ≤ s ≤ ki+ . (iii) Second kind wheel conditions on F (with f from (i)):   f {xi,r } = 0 once x p,r1 = u and x p,r2 = q 2 u for some 1 ≤ r1 = r2 ≤ k p . Fix c = (c0 , c1 , . . . , cn−1 ) ∈ (C× )[n] . Given F ∈ Sk and G ∈ S1, p (u) , we define F  G, G  F ∈ S1, p (u)k+ via (cf. (2.77), (2.78) and (3.26)):

k 1 · ci× k! · ! i∈[n] i ⎛ ⎞     kp >ki  ∈[n] r



i x

 x p,r ⎠ i,r r >k  ≤ki Sym ⎝ F {xi,r }ri∈[n] G {xi  ,r  }i  ∈[n]i ζi,i  φ  ,r  x u i r =1 i∈[n] r ≤k

(F  G)(x0,1 , . . . , x0,k0 +0 ; . . . ; xn−1,1 , . . . , xn−1,kn−1 +n−1 ) :=

i

(3.63) and

3 Quantum Toroidal sln , Its Representations and Bethe subalgebras

104

1 (G  F)(x0,1 , . . . , x0,k0 +0 ; . . . ; xn−1,1 , . . . , xn−1,kn−1 +n−1 ) := × k! · ! ⎛ ⎞     >i  ∈[n] r



i  x i,r r >  ≤i ⎠. F {xi  ,r  }i  ∈[n]i ζi,i  Sym ⎝G {xi,r }ri∈[n]  ,r  x i i∈[n] r ≤

(3.64)

i

These formulas clearly endow S1, p (u) with an S-bimodule structure. Identifying S   > > via Theorem 3.6, we thus get two commuting U¨ q,d -actions on S1, p (u). with U¨ q,d In fact, similarly to Proposition 2.12, the left action can be extended to an action  of the entire quantum toroidal algebra U¨ q,d (sln ):  Proposition 3.11 The following formulas define a U¨ q,d (sln )-action on S1, p (u):

p πu,c (q d2 )G = q − p πu,c (ei,k )G p πu,c (h i,0 )G

=

p(n− p) +|k| 2

k xi,1

·G,

G,

= (2ki − ki−1 − ki+1 − δi p ) · G , ⎛ ⎞ ki    [l]q l l ⎠ 1 p πu,c (h i,l )G = ⎝ [laii  ]q d −lm ii  xil ,r  − δi p q u · G for l = 0 , l i  ∈[n] r  =1 l $  k −1   ,r  }|x G({x ) z c k dz i  → z i,k i i p Res + Res πu,c ( f i,k )G = −1 i .  ki  −δii  xi  ,r  q − q z=0 z=∞  ,i ( ζ ) z   i i r =1 z Here, k ∈ Z, c = (c0 , . . . , cn−1 ) ∈ (C× )[n] , G ∈ S1, p (u)k , and |k| = The factor q −

p(n− p) 2



i∈[n] ki .

p

in πu,c (q d2 ) is solely needed for Sect. 3.4, see Remark 3.13.

Remark 3.11 The formulas of Proposition 3.11 can be equivalently written as follows: p πu,c (q d2 )G = q −

p(n− p) +|k| 2

·G,

p (ei (z))G = δ(xi,1 /z)  G , πu,c

p πu,c (ψi± (z))G = ± k ki+1 ki−1 i z δi p

q 2 z − xi,r z − qd xi+1,r dz − q xi−1,r · · ·φ · G, z − q 2 xi,r r =1 qz − d xi+1,r r =1 qdz − xi−1,r u r =1

(3.65)

(3.66)

3.3 Representations of Quantum Toroidal sln

105

ki c−1 p πu,c ( f i (z))G = −1 i × q −q  + − $  G({xi  ,r  }|xi,ki →z ) G({xi  ,r  }|xi,ki →z ) −  k  −δ  ,  ki  −δii  xi  ,r  xi  ,r  i ii i i r  =1 ζi  ,i ( z ) r  =1 ζi  ,i ( z )

(3.67)

where γ (z)± denotes the expansion of a rational function γ (z) in z ∓1 , respectively. p

Proof We need to verify the compatibility of the above assignment πu,c with the defining relations (T0.1)–(T8). The only nontrivial of those are (T3, T4, T6, T8). To check (T3, T6), we use formulas (3.66, 3.67) together with an obvious identity ζi, j (z/w) = gai j (d m i j z/w) for any i, j ∈ [n]. The verification of (T8) boils down to: ζ j,i (w/z)  Sym z 1 ,z 2

ζi,i (z 1 /z 2 )−1 (q + q −1 )ζi,i (z 1 /z 2 )−1 − + ζi,i±1 (z 1 /w)ζi,i±1 (z 2 /w) ζi,i±1 (z 1 /w)ζi±1,i (w/z 2 ) ζi,i (z 1 /z 2 )−1 ζi±1,i (w/z 1 )ζi±1,i (w/z 2 )

 = 0.

Finally, to verify (T4) we note that the ki + 1 − δi j different summands from the p p symmetrization appearing in πu,c (ei (z))πu,c ( f j (w))G cancel the ki + 1 − δi j terms p p (out of ki + 1) from the symmetrization appearing in πu,c ( f j (w))πu,c (ei (z))G.  • Shuffle bimodules S(u) The above construction admits a higher rank counterpart. For m ∈ N[n] , consider u = (u 0,1 , . . . , u 0,m 0 ; . . . ; u n−1,1 , . . . , u n−1,m n−1 ) with u i,s ∈ C× . Define S(u) =

" k∈N[n]

S(u)k similarly to S1, p (u) with the following modifications:

(i’) Pole conditions for F from a degree k component should read as follows: F= i∈[n]

r  ≤ki+1 r ≤ki

(xi,r

f (x0,1 , . . . , xn−1,kn−1 )  i ki  − xi+1,r  ) · i∈[n] m s=1 r =1 (x i,r − u i,s )

k  ±1 1≤r ≤ki }i∈[n] . with f ∈ C {xi,r (iii’) Second kind wheel conditions for such F should read as follows:   f {xi,r } = 0 once xi,r1 = u i,s and xi,r2 = q 2 u i,s for some i , s , r1 = r2 . We endow S(u) with an S-bimodule structure by the formulas (3.63, 3.64) with

1≤r ≤k p

φ(x p,r /u) 



i∈[n] 1≤s≤m i 1≤r ≤ki

φ(xi,r /u i,s ) .

3 Quantum Toroidal sln , Its Representations and Bethe subalgebras

106

 > The resulting left action of U¨ q,d S on S(u) can be extended to the action of the  ¨ entire quantum toroidal Uq,d (sln ), denoted by πu,c . Explicitly, the latter is defined by the formulas (3.65)–(3.67) with the following two modifications:

φ(z/u)δi p 

φ(z/u i,s ) ,

q−

p(n− p) 2

 q−

n−1 p=0

p) m p · p(n− 2

.

1≤s≤m i

• Shuffle bimodules S1,K p (u) and S K (u) Another natural generalization of the above S1, p (u) is provided by the S-bimodules S1,K p (u). As a vector space, S1,K p (u) is defined similarly to S1, p (u) but without imposing the second kind wheel conditions. The S-bimodule structure on S1,K p (u) is defined by the formulas (3.63, 3.64) with the following modification in (3.63): φ(t)  φ K (t) = (K −1 · t − K )/(t − 1) .   > S on S1,K p (u) can be extended to the U¨ q,d (sln )The resulting left action of U¨ q,d

p,K

action, denoted by πu,c , defined by (3.65)–(3.67) but with φ being replaced by φ K . Inspired by the above construction of S(u), it is clear how to define the higher rank counterparts S K (u) of S1,K p (u), equip them with the S-bimodule structure, and   K > S to the action πu,c of U¨ q,d (sln ) on S K (u). extend the resulting left action of U¨ q,d

3.4 Identification of Representations In this section, we present explicit relation between three different families of modules from Sect. 3.3 (assuming that q and d are generic in the sense of (G)).

3.4.1 Shuffle Realization of Fock Modules  p For 0 ≤ p ≤ n − 1, u ∈ C× , c ∈ (C× )[n] , recall the action πu,c of U¨ q,d (sln ) on S1, p (u) from Proposition 3.11. We define

S  :=



Sk ⊂ S

k=(0,...,0)

and consider the following subspace ! J0 := S1, p (u)  S  = spanC G  F | G ∈ S1, p (u), F ∈ S  ⊂ S1, p (u) . The following result is straightforward, cf. Proposition 2.13(a):

(3.68)

3.4 Identification of Representations

107

 p Lemma 3.8 J0 of (3.68) is a submodule of the U¨ q,d (sln )-action πu,c on S1, p (u).  > Proof The invariance of J0 with respect to the left action of U¨ q,d S is obvious as it coincides with (3.63) and hence commutes with the right action of S from (3.64).  0 -action follows from the equality The invariance of J0 with respect to the left U¨ q,d

⎛ ⎞ kj

−lm i j  [la ] d ij q p p πu,c (h i,l )(G  F) = πu,c (h i,l )G  F + G  ⎝ x lj,s · F ⎠ l j∈[n] s=1 for any i ∈ [n], l = 0, and G ∈ S1, p (u) , F ∈ Sk with k,  ∈ N[n] . Finally, for i ∈ [n], l ∈ Z, and F, G as above, using the formula (3.67) one can easily see that p ( f i,l )(G  F) = G   F + G  F  πu,c

for some G  ∈ S1, p (u)−1i and F  ∈ Sk−1i ∩ S  , with the two terms corresponding to cases when the x∗,∗ -variable specialized to z in (3.67) is placed either in G or F.   p Let π¯ u,c denote the corresponding quotient representation of U¨ q,d (sln ) on

S¯1, p (u) := S1, p (u)/J0 .

(3.69)

The following result is analogous to Proposition 2.13(b):  Theorem 3.12 For any p, u, c as above, we have a U¨ q,d (sln )-module isomorphism

p p τu,c . π¯ u,c

(3.70)

 p Proof By Proposition 3.8, τu,c is an irreducible U¨ q,d (sln )-module. Moreover, both 1¯ u ∈ S¯1, p (u) (the image of 1 ∈ S1, p (u)(0,...,0) ) and |∅ ∈ F ( p) (u) are the lowest weight vectors of the same lowest weight. Hence, it suffices to compare dimensions of the N[n] -graded components of the quotient space S¯1, p (u) from (3.69):



  dim S¯1, p (u)k = p(m) := # size m Young diagrams ∀ m ∈ N .

(†)

k∈N[n] : |k|=m

Descending filtration " m,λ m To prove (†) we equip S1, S1, p (u)k with a filtration {S1, p (u) := p (u)}λ labeled |k|=m

by Young diagrams λ of size ≤ m, with specialization maps ρλ defined below: m,λ S1, p (u) :=

 μ#λ

m Ker(ρμ ) ⊂ S1, p (u) ,

(3.71)

3 Quantum Toroidal sln , Its Representations and Bethe subalgebras

108

cf. (2.29), (3.39), where # is the lexicographical order on size ≤ m Young diagrams. Consider an [n]-coloring of the boxes of a Young diagram λ by assigning color c() := p − a + b ∈ [n] to a box  = (a, b) ∈ λ located at the b-th row and a-th column (with 1 ≤ b ≤ λ1 and 1 ≤ a ≤ λb ). We define the N[n] -degree of λ via: λ ) ∈ N[n] k λ := (k0λ , . . . , kn−1

with

kiλ = #{ ∈ λ | c() = i} .

 p p Remark 3.12 Let τu denote τu,(1,...,1) . Then, the assignment |λ → ∈λ cc() · |λ  p ∼ p gives rise to an isomorphism of U¨ q,d (sln )-modules τu −→ τu,c for any × [n] c ∈ (C ) . Let us fill each box  = (a, b) ∈ λ with q1a−1 q3b−1 u. For F ∈ S1, p (u)k , the specialization ρλ (F) shall be obtained by specializing k λ variables of F to the corresponding entries of λ, while taking care of the zeroes in the numerator and denominator. Specialization maps ρλ For k ∈ N[n] and a Young diagram λ, define  := k − k λ ∈ Z[n] . If  ∈ / N[n] , we set ρλ (F) = 0 for any F ∈ S1, p (u)k . Otherwise, we define ρλ (F) via (3.72) as follows: • First, we pick the corner box  = (1, 1) ∈ λ of color p and specialize x p,k p → u. Since F has the first order pole at x p,k p = u, the following is well-defined: % & ρλ(1) (F) := (x p,k p − u) · F |x p,k

p  →u

.

• Next, we specialize more variables to the entries of the remaining boxes from the first row and the first column. For every box (a + 1, 1) ∈ λ (0 < a < λ1 ) of color p − a, we choose an unspecified yet x p−a,∗ -variable and set it to q1a u. Likewise, for every box (1, b + 1) ∈ λ (0 < b < λ1 ) of color p + b, we choose an unspecified yet x p+b,∗ -variable and set it to q3b u. We perform this procedure step-by-step moving (λ +λ −1)

from (1, 1) to the right and then from (1, 1) upwards. Let ρλ 1 1 (F) denote the resulting overall specialization of F. (λ +λ −1) • If (2, 2) ∈ / λ, then we set ρλ (F) := ρλ 1 1 (F). Otherwise, we pick another variable of the p-th family, say x p,k p −1 , and specialize it to q1 q3 u, but first cancel the (λ1 +λ1 −1)

zero of ρλ

(λ1 +λ1 )

ρλ

(F) due to the first kind wheel conditions: '

(F) :=

1 (λ +λ −1) · ρλ 1 1 (F) x p,k p −1 − q1 q3 u

( . |x p,k p −1 →q1 q3 u

• Next, we move step-by-step from (2, 2) to the right and then from (2, 2) upwards, on each step specializing the corresponding x∗,∗ -variable to the prescribed entry of  ∈ λ located in the second row or column, but first eliminating order 1 zeroes as above (due to the first kind wheel conditions). (|λ|) 1≤r ≤i ). • Having performed this procedure |λ| times, we get ρλ (F) ∈ C({xi,r }i∈[n] Then, we finally set (|λ|) (3.72) ρλ (F) := ρλ (F) .

3.4 Identification of Representations

109

Key properties of ρλ Let us now identify the key properties of the specializations maps ρλ similar to those of φd from Lemmas 1.5–1.6 and of φ L from Lemma 3.3. To do so, we define  p−λ1

Q ,λ :=

(x p−λ1 ,r − q1λ1 u) ·

r =1

λb+1 0 q[k]q Hi,k vz ) · exp( k>0 q[k]q Hi,−k vz ) ·  ∓k p,∗ " ) · q ∓∂α¯ i , e−α¯ i ( qvz ) Hi,0 −1 q ∂α¯ i , ρv,c (ψi± (z)) = exp(∓(q − q −1 ) k>0 Hi,±k vz

3.5 Bethe Algebra Realization of A(s0 , . . . , sn−1 ) p,∗

ρv,c ( f i (z)) = − ci1q 2 · exp( e

α¯ i

( qvz )−Hi,0 −1 q −∂α¯ i ,



p,∗ ρv,c (γ

" q −k/2 k>0 [k]q Hi,k ±1/2 ∓1/2

)=q

117

 z −k v

, and

) · exp(−

p,∗ ρv,c (q ±d1 )



q −k/2 k>0 [k]q ∓d

=q

"

Hi,−k

 z k v

)·

.

3.5 Bethe Algebra Realization of A(s0 , . . . , sn−1 ) In this section, we finally prove the commutativity of the subalgebras A(s) from Sect. 3.2.2(and their one-parameter deformations) for generic s = (s0 , . . . , sn−1 ) satisfying i∈[n] si = 1 by providing a transfer matrix realization of their generators  μ ≤ Fk (s). To this end, we study certain functionals on U¨ q,d (arising as linear combinations of matrix coefficients), and realize them as pairings with explicit elements of  ≥ via ϕ  from Theorem 3.4(c). This can be interpreted via the universal R-matrix U¨ q,d of Theorem 3.2, thus relating A(s) to the standard Bethe subalgebras of Uq (Lgln ).

3.5.1 Trace Functionals 

≤ In this section, we introduce and evaluate three functionals on U¨ q,d . The corresponding computations use the standard technique of the normal ordering (which is also how Proposition 3.10 is proved, see [22]). Explicitly, it is a procedure that for any given unordered product P of terms exp(y Hi,−k ), exp(x Hi,k ), eα¯ i , z Hi,0 , q ∂α¯ i (with i ∈ [n], k > 0) reorders them to : P : by placing specifically in the order indicated above (the relative order of {exp(y Hi,−k )} as well as {exp(x Hi,k )} is not important). When doing so, we note that most of the terms pairwise commute except for:

exp(x Hi,k ) exp(y H j,−l ) =   x yd −km i j [k]q [kai j ]q · exp(y H j,−l ) exp(x Hi,k ) exp δkl k

(3.83)

(due to the defining relations (3.60)) as well as ¯

eα¯ i eα¯ j = (−1)h i ,α¯ j  · eα¯ j eα¯ i , ¯

q ∂α¯ i eα¯ j = q h i ,α¯ j  · eα¯ j q ∂α¯ i , z Hi,0 eα¯ j = z

h¯ i ,α¯ j  m i j h¯ i ,α¯ j /2

d

(3.84) · eα¯ j z Hi,0 .

• Top matrix coefficient Consider the functional   , +  ¯  p ¯  ≤ −→ C defined by φ 0p,c (y) := v0 ⊗ e p ρ1,c (y)v0 ⊗ e p φ 0p,c : U¨ q,d

(3.85)

3 Quantum Toroidal sln , Its Representations and Bethe subalgebras

118



p ≤ evaluating the top-to-top matrix coefficient in the vertex U¨ q,d -representation ρ1,c , ¯

cf. Corollary 3.3. Since ρ1,c (h i,k )v0 ⊗ e p = 0 for any i ∈ [n], k > 0, the functional (3.85) is determined by its values at p

r

r0 n−1 · · · ψn−1,0 · (γ 1/2 )a (q d1 )b y = f i1 ,k1 f i2 ,k2 · · · f im ,km · ψ0,0 [n] with a, b ∈ Z, r := (r and 0m, . . . , rn−1 ) ∈ Z , m ∈ N, k1 , . . . , km ∈ Z, ¯ The latter is equivalent to {i 1 , . . . , i m } i 1 , . . . , i m ∈ [n] satisfying s=1 α¯ is = 0 ∈ Q. containing an equal number of each of the indices {0, . . . , n − 1}. Evoking the quadratic relation (T3), we thus conclude that φ 0p,c is determined by the following generating series:

φ 0p,c;N ,r ,a,b (z 0,1 , . . . , z n−1,N ) := ⎛ ⎞ N

r

  f 0 (z 0,k ) · · · f n−1 (z n−1,k ) · φ 0p,c ⎝ ψi,0i · γ a/2 q bd1 ⎠ k=1

(3.86)

i∈[n]

for any N ∈ N, a, b ∈ Z, r ∈ Z[n] , with the above z ∗,∗ -variables ordered as follows: z 0,1 , . . . , z n−1,1 , z 0,2 , . . . , z n−1,2 , . . . , z 0,N , . . . , z n−1,N .

(3.87)

Proposition 3.12 Set c = c0 · · · cn−1 and ε = (−1)(n−1)(n+2)/2 . We have: φ 0p,c;N ,r ,a,b (z 0,1 , . . . , z n−1,N ) = (εc)−N q a/2+r p −r0 d 

  i∈[n]

i∈[n]



1≤k0



  q2 y y

1− , = 1− x x

(3.88)

(3.89)

and the commutation rules (3.83, 3.84). We note that ε appears due to the equality

3.5 Bethe Algebra Realization of A(s0 , . . . , sn−1 )

119

(−1)n e−α¯ 0 e−α¯ 1 · · · e−α¯ n−1 = εe0 .

(3.90) 

We leave the remaining details to the interested reader.

• Top level graded trace Recall the degree operator d of (3.62) acting diagonally in the natural basis (3.61) of W ( p)n with all eigenvalues in N. Let M( p)n := Ker(d) ⊂ W ( p)n denote  its kernel. Finally, let Uq (sln ) denote the subalgebra of U¨ q,d (sln ) generated by ±1 }i∈[n]× , which is isomorphic to UqDJ (sln )–the Drinfeld-Jimbo quantum {ei,0 , f i,0 , ψi,0 group of sln . We leave the following result as a simple exercise to the reader: Lemma 3.13 (a) The subspace M( p)n is Uq (sln )-invariant and is isomorphic to ¯ p ), the irreducible highest weight UqDJ (sln )-module of the highest weight  ¯ p. L q ( ¯ σp¯ be the sln -weight having entries (b) For any σ¯ = {1 ≤ σ1 < · · · < σ p ≤ n}, let  1−

p n

¯ σ¯

at the places {σi }i=1 and − np elsewhere. Then {v0 ⊗ e p }σ¯ is a basis of M( p)n . p

Furthermore, we also define the degree operators d1 , . . . , dn−1 acting on W ( p)n via:

n−1 n−1 ¯ ¯ dr v ⊗ e j=1 m j α¯ j + p = −m r · v ⊗ e j=1 m j α¯ j + p

∀ v ∈ Fn .

For u = (u 1 , . . . , u n−1 ) ∈ (C× )n−1 , consider the functional φ up,c :



≤ U¨ q,d −→ C ,

y →

  , ¯ σ¯  p ¯ σ¯ dn−1  v0 ⊗ e p ρ1,c (y)u d11 · · · u n−1 v0 ⊗ e p

+ σ¯ 

≤ ¯ evaluating the Q-graded trace of the U¨ q,d -action on the subspace (not a submodule!) ¯ σ¯

M( p)n . We note that ρ1,c (h i,k )v0 ⊗ e p = 0 for i ∈ [n], k > 0. Therefore, similarly u to the previous computation, the functional φ p,c is determined by the generating series analogous to (3.86): p

u

φ p,c;N ,r ,a,b (z 0,1 , . . . , z n−1,N ) := ⎛ ⎞ N

r   φ up,c ⎝ f 0 (z 0,k ) · · · f n−1 (z n−1,k ) · ψi,0i · γ a/2 q bd1 ⎠ . k=1

(3.91)

i∈[n]

This series  Ncan be evaluated similarly to Proposition 3.12 by normally ordering the product k=1 ( f 0 (z 0,k ) · · · f n−1 (z n−1,k )) and using (3.83, 3.84) as well as (3.88, 3.89): Proposition 3.13 Set c = c0 · · · cn−1 and ε = (−1)(n−1)(n+2)/2 . We have:

3 Quantum Toroidal sln , Its Representations and Bethe subalgebras

120

N (n−2)

φ p,c;N ,r ,a,b (z 0,1 , . . . , z n−1,N ) = (εc)−N q a/2 d 2 ×   2 i∈[n] 1≤k 0 ,

(3.95)

i,−k , H0 }k>0 generate a as in Remark 3.14. In particular, the elements {Hi,k , H Heisenberg algebra ai for any i ∈ [n], and moreover ai commutes with a j for any i = j ∈ [n]. According to (3.60), we have  i∈[n] r >0

u i,r Hi,−r =

 i∈[n] r >0

i,−r with   u i,r H u i,r =



d −r m i j

[r ]q [rai j ]q r

· u j,r .

j∈[n]

Using the pairwise commutativity of ai and a j for i = j one can rewrite Tr 2 as a product of the corresponding traces over the Fock modules of each ai . Applying Lemma 3.14 to each of these, it is now straightforward to see that Tr 2 recovers exactly the product of factors from the last two lines in the right-hand side of (♦). This completes the proof of Theorem 3.16. 

3.5 Bethe Algebra Realization of A(s0 , . . . , sn−1 )

123

3.5.2 Functionals via Pairing 



≥ ≤ Recall the Hopf pairing ϕ  : U¨ q,d × U¨ q,d → C from Theorem 3.4. As ϕ  is non  u u,t ≥,∧ ≥,∧ and X p,c ∈ U¨ q,d [[t]] degenerate, there exist unique elements X 0p,c , X p,c ∈ U¨ q,d ∧ (with denoting an appropriate completion) such that  u,t φ 0p,c (y) = ϕ  (X 0p,c , y) , φ up,c (y) = ϕ  (X up,c , y) , φ u,t p,c (y) = ϕ (X p,c , y) (3.96) 

≤ . We shall now obtain explicit shuffle formulas for these elements. for any y ∈ U¨ q,d To this end, we extend the isomorphism  of Theorem 3.6 to the isomorphism ∼ ≥  ≥ : U¨ q,d −→ S ≥ . 

(3.97)

Here, the extended shuffle algebra S ≥ is obtained from S by adjoining generators  ≥ −1 s∈N , γ ±1/2 , q ±d1 }i∈[n] with defining relations compatible with those for U¨ q,d . {ψi,−s , ψi,0 In particular, we have: (3.98) q d1 Fq −d1 = q −d · F , ⎡

⎤ ≤k j

r ζ (z/x ) 1≤r ≤k i, j j,r ⎦ ψi− (z)  F = ⎣ F {x j,r } j∈[n] j ·  ψi− (z) ζ (x /z) j,i j,r j∈[n]

(3.99)

− z s ,  denotes the multiplication for any F ∈ Sk,d , where we set ψi− (z) := s≥0 ψi,−s ≥ in S , and the ζ -factors in (3.99) are all expanded in the nonnegative powers of z. u u,t u u,t Let  0p,c ,  p,c ,  p,c denote the images of X 0p,c , X p,c , X p,c under  ≥ of (3.97): ≥ u,t  0p,c :=  ≥ (X 0p,c ) ,  up,c :=  ≥ (X up,c ) ,  u,t p,c :=  (X p,c ) .

(3.100)

The following is the main result of this section (δ = (1, 1, . . . , 1) ∈ N[n] as before): Theorem 3.17 (a) We have  0p,c =



¯

(εc)−N (1 − q −2 )n N (−q n d −n/2 ) N ·  0p;N · q  p q −d1 2

N ≥0

with  0p;N ∈ S N δ defined in (3.50), and c = c0 · · · cn−1 , ε = (−1)(n−1)(n+2)/2 as before. (b) We have  up,c =



(εc)−N (1 − q −2 )n N (−q n d −n/2 ) N ·  p;N · q −d1

N ≥0

with  p;N ∈ S N≥δ given by: u

2

u

3 Quantum Toroidal sln , Its Representations and Bethe subalgebras

124 ¯  up;N

  −2 xi,r  ) 1 i∈[n] 1≤r =r  ≤N (x i,r − q  × = ·  u · · · u j−1 i∈[n] 1≤r,r  ≤N (x i,r − x i+1,r  ) j=1 1 ⎧ N ⎫ N ⎨ ⎬

¯ ¯ (−1) p [μ p ] xi+1,r − μ · u 1 · · · u i xi,r · q i+1 −i , ⎩ ⎭ p

i∈[n]

r =1

r =1

¯

¯

where in the last product we take all xi,r ’s to the left and all q i+1 −i to the right. (c) We have  u,t p,c =



¯

(εc)−N (1 − q −2 )n N (−q n d −n/2 ) N ·  p;N · q  p q −d1 u,t

2

N ≥0

with  p;N ∈ (S N≥δ )∧ [[t]] given by: u,t

 u,t

 p;N =

i∈[n]



1≤r  =r  ≤N (x i,r





i∈[n]

− q −2 xi,r  ) ·

1≤r,r  ≤N (x i,r

x N (t¯ xi,A ; t¯)∞

1 i,B · (t¯; t¯)n∞ ¯qd xi+1,A ¯ ( t xi,B ; t )∞ i∈[n] A,B=1

= where t¯ = tγ −1 , %

1 √ 2π −1



N

r =1 x i,r

i∈[n]

− xi+1,r  )

N

x0,r ) × · θ(x; % x p,r

r =1

x

· (t¯q 2 xi,A ; t¯)∞ i,B

· (t¯qd −1

·

xi−1,A ¯ xi,B ; t )∞

·

N



ψ¯ i− (t¯k q 1/2 xi,r ) ,

k>0 i∈[n] r =1

 n−1 · ai j ln(t¯) i, j=1 , and 

x = (x1 , . . . , xn−1 ) with xi =

1 √ 2π −1

ln

u i−1 t¯δ pi ψi,0

N

xi−1,r xi+1,r 2 xi,r r =1

 .

In the above products, we take all xi,r ’s to the left and all ψi,k ’s to the right. This theorem follows directly from Propositions 3.12, 3.13 and Theorem 3.16 by applying the following technical lemma: Lemma 3.15 (a) We have ϕ  (aa  , bb ) = ϕ  (a, b) · ϕ  (a  , b ) 



≥ ≤ > 0 < 0 , a  ∈ U¨ q,d ∩ U¨ q,d , b ∈ U¨ q,d , b ∈ U¨ q,d ∩ U¨ q,d . for any elements a ∈ U¨ q,d 







(b) For any ki , ki ∈ N, ai , ai ∈ Z (with i ∈ [n]), and A, B, A , B  ∈ Z, we have:

3.5 Bethe Algebra Realization of A(s0 , . . . , sn−1 )

⎛ ϕ ⎝

r ≤ki

i∈[n]

ψ¯ i− (z i,r )

r  ≤k j ai A/2 Bd1 ψi,0 γ q ,

i∈[n]

q

125

ψ¯ + j (w j,r  )

j∈[n]

− 21 A B− 21 AB  + i, j∈[n] ai a j ai j

⎞

a   ψ j,0j γ A /2 q B d1 ⎠

=

j∈[n]   j∈[n] 1≤r ≤k j

·



i∈[n] 1≤r ≤ki

w j,r  − q ai j d m i j z i,r . w j,r  − q −ai j d m i j z i,r

> , we have: (c) For any k ∈ N, i 1 , . . . , i k ∈ I, d1 , . . . , dk ∈ Z, X ∈ U¨ q,d 

ϕ  (X, f i1 ,−d1 · · · f ik ,−dk ) = 4 k (3.101) (X )(z 1 , . . . , z k )z 1−d1 . . . z k−dk dzr  (q − q −1 )−k , √ 2π −1zr |z 1 |'···'|z k | 1≤a