Quantum groups, christian kassel

Graduate Texts in Mathematics I.H Ewing 155 Editorial Board F.W Gehring P.R Halmos Graduate Texts in Mathematics 10 11 12 I3 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 TAKEUTIIZARING Introduction to Axiomatic Set Theory 2nd ed OXTOBY Measure and Category 2nd ed SCHAEFFER Topological Vector Spaces HILTON/STAMMBACH A Course in Homological Algebra MAC LANE Categories for the Working Mathematician HUGHES/PIPER Projective Planes SERRE A Course in Arithmetic TAKEUTI/ZARING Axiomatic Set Theory HUMPHREYS Introduction to Lie Algebra~ and Representation Theory COHEN A Course in Simple Homotopy Theory CONWAY Functions of One Complex Variable 2nd ed BEALS Advanced Mathematical Analysis ANDERSON/FULLER Rings and Categories of Modules 2nd ed GOLUBITSKy/GUILEMIN Stable Mappings and Their Singularities BERBERIAN Lectures in Functional Analysis and Operator Theory WINTER The Structure of Fields ROSENBLATT Random Processes 2nd ed HALMos Measure Theory HALMos A Hilbert Space Problem Book 2nd ed HUSEMOLLER Fibre Bundles 3rd ed HUMPHREYS Linear Algebraic Groups BARNES/MACK An Algebraic Introduction to Mathematical Logic GREUB Linear Algebra 4th ed HOLMES Geometric Functional Analysis and Its Applications HEWITT/STROMBERG Real and Abstract Analysis MANES Algebraic Theories KELLEY General Topology ZARISKIISAMUEL Commutative Algebra Vol.l ZARISKIISAMUEL Commutative Algebra Vol.lI JACOBSON Lectures in Abstract Algebra I Basic Concepts JACOBSON Lectures in Abstract Algebra II Linear Algebra JACOBSON Lectures in Abstract Algebra III Theory of Fields and Galois Theory 33 HIRSCH Differential Topology 34 SPITZER Principles of Random Walk 2nd ed 35 WERMER Banach Algebras and Several Complex Variables 2nd ed 36 KELLEy/NAMIOKA et al Linear Topological Spaces 37 MONK Mathematical Logic 38 GRAUERT/FRITZSCHE Several Complex Variables 39 ARVESON An Invitation to C*-Algebras 40 KEMENy/SNELL/KNAPP Denumerable Markov Chains 2nd ed 41 ApOSTOL Modular Functions and Dirichlet Series in Number Theory 2nd ed 42 SERRE Linear Representations of Finite Groups 43 GILLMAN/JERISON Rings of Continuous Functions 44 KENDIG Elementary Algebraic Geometry 45 LoEVE Probability Theory I 4th ed 46 LoEvE Probability Theory II 4th ed 47 MOISE Geometric Topology in Dimensions and 48 SACHSlWu General Relativity for Mathematicians 49 GRUENBERGIWEIR Linear Geometry 2nd ed 50 EDWARDS Fermat's Last Theorem 51 KLINGENBERG A Course in Differential Geometry 52 HARTSHORNE Algebraic Geometry 53 MANIN A Course in Mathematical Logic 54 GRAVERIWATKINS Combinatorics with Emphasis on the Theory of Graphs 55 BROWN/PEARCY Introduction to Operator Theory I: Elements of Functional Analysis 56 MASSEY Algebraic Topology: An Introduction 57 CROWELL/Fox Introduction to Knot Theory 58 KOBUTZ p-adic Numbers, p-adic Analysis, and zeta-Functions 2nd ed 59 LANG Cyclotomic Fields 60 ARNOLD Mathematical Methods in Classical Mechanics 2nd ed 61 WHITEHEAD Elements of Homotopy Theory 62 KARGAPOLOVIMERLZJAKOv Fundamentals of the Theory of Groups 63 BOLLOBAS Graph Theory 64 EDWARDS Fourier Series Vol I 2nd ed continued after index Christian Kassel Quantum Groups With 88 Illustrations Springer-Science+Business Media, LLC Christian Kassel Institut de Recherche Mathematique Avancee Universite Louis Pasteur-C.N.R.S 67084 Strasbourg France Editorial Board J.H Ewing Department of Mathematics Indiana University Bloomington, IN 47405 USA F W Gehring Department of Mathematics University of Michigan Ann Arbor, MI 48109 USA P.R Halmos Department of Mathematics Santa Clara University Santa Clara, CA 95053 USA Mathematics Subject Classification (1991): Primary-17B37, 18DlO, 57M25, 81R50; Secondary-16W30, 17B20, 17B35, 18D99, 20F36 Library of Congress Cataloging-in-Publication Data Kassel, Christian Quantum groups/Christian Kassel p cm - (Graduate texts in mathematics; voI 155) Includes bibliographical references and index ISBN 978-1-4612-6900-7 ISBN 978-1-4612-0783-2 (eBook) DOI 10.1007/978-1-4612-0783-2 Quantum groups Hopf algebras Topology Mathematical physics Title II Series: Graduate texts in mathematics; 155 QC20.7.G76K37 1995 512'.55-dc20 94-31760 Printed on acid-free paper © 1995 Springer Science+Business Media New York Originally published by Springer-Verlag New York, Inc in 1995 Softcover reprint of the hardcover 1st edition 1995 All rights reserved This work may not be translated or copied in whole or in part without the written permission ofthe publisher (Springer Science+Business Media, LLC), except for brief excerpts in connection with reviews or scholarly anaJysis Use in connection with any form of information storage and retrievaJ, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely byanyone Production managed by Francine McNeill; manufacturing supervised by Genieve Shaw Photocomposed pages prepared using Patrick D.F Ion's TeX files 987654321 ISBN 978-1-4612-6900-7 Preface {( Eh bien, Monsieur, que pensez-vous des x et des y ?» Je lui repondu : {( C'est bas de plafond » V Hugo [Hug51] The term "quantum groups" was popularized by Drinfeld in his address to the International Congress of Mathematicians in Berkeley (1986) It stands for certain special Hopf algebras which are nontrivial deformations of the enveloping Hopf algebras of semisimple Lie algebras or of the algebras of regular functions on the corresponding algebraic groups As was soon observed, quantum groups have close connections with varied, a priori remote, areas of mathematics and physics The aim of this book is to provide an introduction to the algebra behind the words "quantum groups" with emphasis on the fascinating and spectacular connections with low-dimensional topology Despite the complexity of the subject, we have tried to make this exposition accessible to a large audience We assume a standard knowledge of linear algebra and some rudiments of topology (and of the theory of linear differential equations as far as Chapter XIX is concerned) We divided the book into four parts we now briefly describe In Part I we introduce the language of Hopf algebras and we illustrate it with the Hopf algebras SLq(2) and Uq(.s((2)) associated with the classical group 8L These are the simplest examples of quantum groups, and actually the only ones we treat in detail Part II focuses on two classes of Hopf algebras that provide solutions of the Yang-Baxter equation in a systematic way We review a method due to Faddeev, Reshetikhin, and Takhtadjian as well as Drinfeld's quantum double construction, both designed to produce quantum groups Parts I and II may form the core of a one-year introductory course on the subject Parts III and IV are devoted to some of the spectacular connections alluded to before The avowed objective of Part III is the construction of isotopy invariants of knots and links in R , including the Jones polynomial, VI Preface from certain solutions of the Yang-Baxter equation To this end, we introduce various classes of tensor categories that are responsible for the close relationship between quantum groups and knot theory Part IV presents more advanced material: it is an account of Drinfeld's elegant treatment of the monodromy of the Knizhnik-Zamolodchikov equations Our aim is to highlight Drinfeld's deep result expressing the braided tensor category of modules over a quantum enveloping algebra in terms of the corresponding semisimple Lie algebra We conclude the book with the construction of a "universal knot invariant" This is a nice, far-reaching application of the algebraic techniques developed in the preceding chapters I wish to acknowledge the inspiration I drew during the composition of this text from [Dri87] [Dri89a] [Dri89b] [Dri90] by Drinfeld, [JS93] by Joyal and Street, [Tur89] [RT90] by Reshetikhin and Turaev After having become acquainted with quantum groups, the reader is encouraged to return to these original sources Further references are given in the notes at the end of each chapter Lusztig's and Turaev's monographs [Lus93] [Tur94] may complement our exposition advantageously This book grew out of two graduate courses I taught at the Department of Mathematics of the Universite Louis Pasteur in Strasbourg during the years 1990-92 Part I is the expanded English translation of [Kas92] It is a pleasure to express my thanks to C Bennis, R Berger, C Mitschi, P Nuss, C Reutenauer, M Rosso, V Turaev, M Wambst for valuable discussions and comments, and to Raymond Seroul who coded the figures lowe special thanks to Patrick Ion for his marvellous job in preparing the book for printing, with his attention to mathematical, English, typographical, and computer details Christian Kassel March 1994, Strasbourg Notation - Throughout the text, k is a field and the words "vector space", "linear map" mean respectively "k-vector space" and "k-linear map" The boldface letters N, Z, Q, R, and C stand successively for the nonnegative integers, all integers, the field of rational, real, and complex numbers The Kronecker symbol l5ij is defined by l5 ij = if i = j and is zero otherwise We denote the symmetric group on n letters by Sri' The sign of a permutation u is indicated by c(u) The symbol indicates the end of a proof Roman figures refer to the numbering of the chapters Contents Preface Part One v Quantum 8L(2) I Preliminaries Algebras and Modules Free Algebras The Affine Line and Plane Matrix Multiplication Determinants and Invertible Matrices Graded and Filtered Algebras Ore Extensions Noetherian Rings Exercises 10 Notes 3 10 10 12 14 18 20 22 II Tensor Products Tensor Products of Vector Spaces Tensor Products of Linear Maps Duality and Traces Tensor Products of Algebras Tensor and Symmetric Algebras Exercises Notes 23 23 26 29 32 34 36 38 viii Contents III The Language of Hopf Algebras Coalgebras Bialgebras Hopf Algebras Relationship with Chapter I The Hopf Algebras GL(2) and SL(2) Modules over a Hopf Algebra Comodules Comodule-Algebras Coaction of SL(2) on the Affine Plane Exercises Notes 39 39 45 49 57 57 61 64 66 70 IV The Quantum Plane and Its Symmetries The Quantum Plane Gauss Polynomials and the q- Binomial Formula The Algebra Mq(2) Ring-Theoretical Properties of Mq(2) Bialgebra Structure on Mq(2) The Hopf Algebras GLq(2) and SLq(2) Coaction on the Quantum Plane Hopf *-Algebras Exercises 10 Notes 72 72 74 77 81 82 83 85 86 88 90 V The Lie Algebra of SL(2) Lie Algebras Enveloping Algebras The Lie Algebra 5[(2) Representations of 5[(2) The Clebsch-Gordan Formula Module-Algebra over a Bialgebra Action of 5[(2) on the Affine Plane Duality between the Hopf Algebras U(.5[(2)) and SL(2) Exercises Notes 93 93 94 99 101 105 The Quantum Enveloping Algebra of 5[(2) The Algebra Uq(.5[(2)) Relationship with the Enveloping Algebra of 5[(2) Representations of Uq The Harish-Chandra Homomorphism and the Centre of Uq 121 121 125 127 130 107 109 11 119 VI Contents Case when q is a Root of Unity Exercises Notes VII A Hopf Algebra Structure on Uis[(2)) Comultiplication Semi simplicity Action of Uq(.s[(2)) on the Quantum Plane Duality between the Hopf Algebras Uq(.s[(2)) and SLq(2) Duality between Uq (.s[(2))-Modules and SL q(2)-Comodules Scalar Products on Uq (.s[(2))-Modules Quantum Clebsch-Gordan Exercises Notes Part Two Universal R-Matrices VIII The Yang-Baxter Equation and (Co)Braided Bialgebras The Yang-Baxter Equation Braided Bialgebras How a Braided Bialgebra Generates R- Matrices The Square of the Antipode in a Braided Hopf Algebra A Dual Concept: Cobraided Bialgebras The FRT Construction Application to GLq(2) and SLq(2) Exercises Notes ix 134 138 138 140 140 143 146 150 154 155 157 162 163 165 167 167 172 178 179 184 188 194 196 198 IX Drinfeld's Quantum Double Bicrossed Products of Groups Bicrossed Products of Bialgebras Variations on the Adjoint Representation Drinfeld's Quantum Double Representation-Theoretic Interpretation of the Quantum Double Application to Uq(.s[(2)) R- Matrices for U q Exercises Notes 199 199 202 207 213 220 223 230 236 238 518 References [Ros88] M Rosso Finite-dimensional representations of the quantum analog of the enveloping algebra of a complex simple Lie algebra Comm Math Phys 117 (1988), 581-593 MR 90c:17019 [Ros89] M Rosso An analogue of P.B.W theorem and the universal R-matrix for Uhs[(N + 1) Comm Math Phys 124 (1989), 307-318 MR 90h:17019 [Ros90] M Rosso Analogues de la formule de Killing et du theoreme d'Harish-Chandra pour les groupes quantiques Ann Sci Ecole Norm Sup {4} 23 (1990), 445-467 MR 93e:17026 [Ros92] M Rosso Representations des groupes quantiques In Seminaire Bourbaki, volume 201-203 of Asterisque, pages 443483 S.M.F., Paris, 1992 MR 93b:17050 [RSTS88] N.Yu Reshetikhin and M.A Semenov-Tian-Shansky Quantum R-matrices and factorization problems J Geom Phys (1988), 533-550 MR 92g:17019 [RT90] N.Yu Reshetikhin and V.G Turaev Ribbon graphs and their invariants derived from quantum groups Comm Math Phys 127 (1990), 1-26 MR 91c:57016 [RT91] N.Yu Reshetikhin and V.G Turaev Invariants of 3-manifolds via link polynomials and quantum groups Invent Math 103 (1991), 547-597 MR 92b:57024 [RTF89] N.Yu Reshetikhin, L.A Takhtadjian, and L.D Faddeev Quantization of Lie groups and Lie algebras Algebra i Analiz (1989),178-206 English transl.: Leningrad Math J (1990), 193-225 MR 90j:17039 [Sa190] H Saleur Representations of Uq (s[(2)) for q a root of unity In Number theory and physics {Les Houches, 1989}, volume 47 of Springer Proc Phys., pages 68-76 Springer, Berlin, 1990 MR 91i:17023 [Sch92] P Schauenburg On Coquasitriangular Hop! Algebras and the Quantum Yang-Baxter Equation, volume 67 of Algebra Berichte Verlag Reinhard Fischer, Miinchen, 1992 [SC56] Seminaire Cartan Algebres d'Eilenberg-Mac Lane et homotopie {1954-55} Ecole Normale Superieure, Paris, 1956 (also published by W.A Benjamin, Inc., New York, Amsterdam, 1967) [Ser65] J.-P Serre Lie Algebras and Lie Groups W.A Benjamin, Inc., New York, Amsterdam, 1965 References 519 [Ser93] J.-P Serre Gebres Enseign Math (2) 39 (1993), 33-85 [Shu90] M.C Shum Tortile tensor categories Ph.D Thesis, Macquarie University, Nov 1989, Macquarie Math Report No 900047, 1990 J Pure Appl Algebra, to appear in 1994 [Sk185] E.K Sklyanin ~n an algebra generated by quadratic relations Uspekhi Mat Nauk 40:2 (242) (1985), 214 (in Russian) [Sla66] L.J Slater Generalized Hypergeometric Functions bridge University Press, Cambridge, 1966 [SR72] N Saavedra Rivano Categories Tannakiennes, volume 265 of Lecture Notes in Math Springer-Verlag, Berlin, New York, 1972 [Sta92] T Stanford Finite-type invariants of knots, links, and graphs Columbia University, Preprint, 1992 [Sta93] T Stanford Braid commutators and Vassiliev invariants Preprint, 1993 [Sud90] A Sudbery Consistent multiparametric quantisation of GL(n) J Phys A 23 (1990), L697-L704 MR 91m:17022 [Swe69] M Sweedler Hopf Algebras W.A Benjamin, Inc., New York, 1969 [Taf71] E.J Taft The order of the antipode of finite-dimensional Hopf algebras Proc Nat Acad Sci U.S.A 68 (1971), 2631-2633 [Tak81] M Takeuchi Matched pairs of groups and bismash products of Hopf algebras Comm Algebra (1981), 841-882 MR 83f:16013 [Tak85] M Takeuchi Topological coalgebras J Algebra 97 (1985), 505-539 [Tak89] M Takeuchi Quantum orthogonal and symplectic groups and their embedding into quantum GL Proc Japan Acad Ser A Math Sci 65 (1989), 55-58 MR 90j:16024 [Tak92a] M Takeuchi Finite-dimensional representations of the quantum Lorentz group Comm Math Phys 144 (1992), 557-580 MR 93c:17030 [Tak92b] M Takeuchi Hopf algebra techniques applied to the quantum group U/5[(2» In Deformation Theory and Quantum Groups with Applications to Mathematical Physics (Amherst, 1990), Cam- 520 References volume 134 of Contemp Math., pages 309-323 Amer Math Soc., Providence, RI, 1992 [Tak92c] M Takeuchi Some topics on GLq(n) J Algebra 147 (1992), 379-410 MR 93b:17055 [Tan90] T Tanisaki Harish-Chandra isomorphisms for quantum algebras Comm Math Phys 127 (1990), 555-571 [Trj33] W.J Trjitzinsky Analytic theory of linear q-difference equations Acta Math 61 (1933), 1-38 [Tur88] V.G Turaev The Yang-Baxter equation and invariants of links Invent Math 92 (1988), 527-553 MR 8ge:57003 [Tur89] V.G Turaev Operator invariants of tangles and R-matrices Izv Akad Nauk SSSR Ser Math 53:5 (1989), 1073-1107 English transl.: Math USSR-Izv 35 (1990), 411-444 MR 91e:17011 [Tur92] V.G Turaev Modular categories and 3-manifold invariants Intern J Modern Phys B (1992), 1807-1824 [Tur94] V.G Turaev Quantum invariants of knots and 3-manifolds W de Gruyter, Berlin, 1994 [TW80] E.J Taft and R.L Wilson There exist finite-dimensional Hopf algebras with antipodes of arbitrary even order J Algebra 62 (1980), 283-291 MR 81d:16008 [Vak89] L.L Vaksman q-analogues of Clebsch-Gordan coefficients, and the algebra of functions on the quantum SU(2) group Dokl Akad Nauk SSSR 306:2 (1989), 269-271 English transl.: Soviet Math Dokl 39 (1989), 467-470 MR 90k:33018 [Var74] V.S Varadarajan Lie groups, Lie algebras, and their representations Prentice Hall, Inc., Englewood Cliffs, NJ, 1974 [Vas90] V.A Vassiliev Cohomology of knot spaces In V.I Arnold, editor, Theory of singularities and applications, volume of Adv Soviet Math., pages 23-69 Amer Math Soc., Providence, RI, 1990 MR 92a:57016 [Vas92] V.A Vassiliev Complements of discriminants of smooth maps, volume 98 of Transl Math Monographs Amer Math Soc., Providence, RI, 1992 [Vog93] P Vogel Invariants de Vassiliev des namds (d'apres D BarNatan, M Kontsevich et V.A Vassiliev) In Seminaire Bourbaki, volume 216 of Asterisque, pages 213-232 S.M.F., Paris, 1993 References 521 [Was87] W Wasow Asymptotic Expansions for Ordinary Differential Equations Dover Pub!., New York, 1987 [Wor87a] S.L Woronowicz Compact matrix pseudo-groups Math Phys 111 (1987), 613-665 MR 88m:46079 [Wor87b] S.L Woronowicz Twisted SU(2) group An example of noncommutative differential calculus Publ Res Inst Math Sci 23 (1987), 117-181 MR 88h:46130 [Wor88] S.L Woronowicz Tannaka-Krein duality for compact matrix pseudo-groups twisted SU(N) groups Invent Math 93 (1988), 35-76 MR 90e:22033 [Yam89] H Yamane A Poincare-Birkhoff-Witt theorem for quantized enveloping algebras of type AN' Publ Res Inst Math Sci 25 (1989), 503-520 MR 91a:17016 [Yan67] C.N Yang Some exact results for the many-body problem in one dimension with repulsive delta-function interaction Phys Rev Lett 19 (1967), 1312-1315 [Yet88] D.N Yetter Markov algebras In Braids (Santa Cruz, 1986), volume 78 of Contemp Math., pages 705-730 Amer Math Soc., Providence, RI, 1988 [Yet90] D.N Yetter Quantum groups and representations of monoidal categories Math Proc Cambridge Philos Soc 108 (1990), 261-290 MR 91k:16028 [Zag93] D Zagier Values of zeta functions and their applications M.P.I., Bonn, Preprint, 1993 Proc E.C.M., to appear Comm Index 0,390,410 ~-operation, 243, 259 p~, 378 p~, 377 p~, 307 p~z, 456, 457, 459 p~h, 459 T(Pl,ql, ,Pk,qk)' 466, 482