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Anatoli Klimyk Konrad Schmiidgen

Quantum Groups and Their Representations

, Springer

Professor Dr. Anatoli Klimyk Ukrainian Academy of Sciences Institute for Theoretical Physics Kiev 252143, Ukraine

Editors

Roger Balian

Professor Dr. Konrad Schmiidgen Universitat Leipzig Fakultat fUr Mathematik und Informatik 0-04109 Leipzig, Germany

Nicolai Reshetikhin CEA Department of Mathematics Service de Physique TMorique de Saclay University of California F-9119l Gif-sur-Yvette, France Berkeley, CA 94720-3840, USA

Wolf Beiglbock Institut fUr Angewandte Mathematik Universitat Heidelberg 1m Neuenheimer Feld 294 0-69120 Heidelberg, Germany

Harald Grosse Institut fiir Theoretische Physik Universitat Wien BoItzmanngasse 5 A-I090 Wien, Austria

Elliott H. Lieb Jadwin Hall Princeton University, P. O. Box 708 Princeton, NJ 08544-0708, USA

Herbert Spohn Theoretische Physik Ludwig-Maximilians-Universitat Milnchen Theresienstra8e 37 0-80333 Milnchen, Germany

Walter Thirring Institut filr Theoretische Physik Universitlit Wien Boltzmanngasse 5 A-I090 Wien, Austria

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Die Deutsche Bibliothek - CIP-Einheitsaufnahme

Klimyk,Anatolij V.: Quantum groups and their representations 1 Anatoli Klimyk ; Konrad SchmUdgen. -Berlin; Heidelberg; New York; Barcelona; Budapest; Hong Kong; London; Milan; Paris; Santa Clara; Singapore; Tokyo: Springer, 1997 (Texts and monographs in physics)

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Preface

The invention of quantum groups is one of the outstanding achievements of mathematical physics and mathematics in the late twentieth century. The birth of the new theory and its rapid development are results of a strong interrelation between mathematics and physics.

Quantu~ groups arose in the work of L.D. Faddeev and the Leningrad school on the inverse scattering method in order to solve integrable models. The algebra Uq(sh) appeared first in 1981 in a paper by P.P. Kulish and N.Yu. Reshetikhin on the study of integrable XYZ models with highest spin. Its Hopf algebra structure was discovered later by E.K. Sklyanin. A major event was the discovery by V.G. Drinfeld and M. Jimbo around 1985 of a class of Hopf algebras which can be considered as one-parameter deforma­tions of universal enveloping algebras of semisimple complex Lie algebras. These Hopf algebras will be called Drinfeld-Jimbo algebras in this book. Al­most simultaneously, S.L. Woronowicz invented the quantum group SUq (2) and developed his theory of compact quantum matrix groups. An algebraic approach to quantized coordinate algebras was given about this time by Yu.I. Manin.

A striking feature of quantum group theory is the surprising connec­tions with many, sometimes at first glance unrelated, branches of mathe­matics and physics. There are links with mathematical fields such as Lie groups, Lie algebras and their representations, special functions, knot theory, low-dimensional topology, operator algebras, noncommutative geometry, and combinatorics. On the physical side there are interrelations with the quantum inverse scattering method, the theory of integrable models, elementary parti­cle physics, conformal and quantum field theories, and others. It is expected that quantum groups will lead to a deeper understanding of the concept of symmetry in physics.

Currently there is no satisfactory general definition of a quantum group. It is commonly accepted that quantum groups are certain "nice" Hopf alge­bras and that the standard deformations of the enveloping Hopf algebras of semisimple Lie algebras and of coordinate Hopf algebras of the corresponding Lie groups are guiding examples. Instead of searching for a rigorous definition of a quantum group it seems to be more fruitful to look for classes of Hopf algebras that give rise to a rich theory with important applications and con-

VI Preface

tain enough interesting examples. In this book at least three such classes are extensively studied: quasitriangular Hopf algebras, coquasitriangular Hopf algebras, and compact quantum group algebras.

The aim of this book is to provide a treatment of the theory of quantum algebras (quantized universal enveloping algebras), quantum groups (quan­tized algebras of functions), their representations and corepresentations, and the noncommutative differential calculus on quantum groups. The exposition is organized such that different parts of the text can be read and used (al­most) independently of others. Sections 1.2 and 1.3 contain the main general definitions and notions on Hopf algebras needed in the text. This book is divided into four parts.

Part I serves (among others) as an introduction to the theory of Hopf algebras, to the quantum algebra Uq(sb), the quantum group SLq(2), the q-oscillator algebra, and to their representations. The reader can use the corresponding chapters as first steps in order to learn the theory of quantum groups. A beginner might try to become aquainted with the language of Hopf algebras by reading Sect. 1.1 and portions of Sects. 1.2 and 1.3 and then passing immediately to Chaps. 3 or 4 (or start with Chaps. 3 or 4 and read parallel to them the relevant parts of Chap. 1). The main parts of the material of Chaps. 1,3, and 4 can also be taken as a basis for an introductory course on quantum groups.

Parts II-IV cover some of the more advanced topics of the theory. In Part II (quantized universal enveloping algebras)- and Part III (quantized algebras of functions) both fundamental approaches to quantum groups are developed in detail and as independently as possible, so readers interested in only one of these parts can restrict themselves to the corresponding chapters. Nevertheless the connections between both approaches appear to be very fruitful and instructive (see Sects. 4.4, 4.5.5, 9.4, 11.2.3, 11.5, and 11.6.6). A reader who is interested in only noncommutative differential calculus should pass directly to Part IV of the book and begin with Chaps. 12 or 14 (of course, some knowledge about the corresponding quantum groups from Chap. 9 and the L-functionals from Subsect. 10.1.3 is still required there). Together with Sects. 1.2 and 1.3, Parts II-IV form an advanced text on quantum groups. Selected material from these parts can also be used for graduate courses or seminars on quantum groups. Moreover, a large number of explicit formulas and new material (for instance, in Sects. 8.5, 10.1.3, 10.3.1, 13.2, and 14.3-5) are provided throughout the text, so we hope the book may be useful for experts as well.

Let us say a few words about the selected topics and the presentation in the book. Our objective in choosing the material was to cover important and useful tools and methods for (possible) applications in theoretical and mathematical physics (especially in representation theory and in noncommu­tative differential calculus). Of course, this depends on our personal view of the matter. We have tried to give a comprehensive treatment of the chosen

Preface VII

topics at the price of not including some concepts (for instance, the quantum Weyl group). Although we develop a number of general concepts too, the em­phasis in the book is always placed on the study of concrete quantum groups and quantum algebras and their representations. Most of the results are pre­sented with complete (but sometimes concise) proofs. Often, missing proofs or gaps in the existing literature have been filled. For some rather technical proofs (in particular of advanced algebraic results) readers are referred to the original papers. In many cases we have omitted proofs that are similar to the classical case. Having the potential reader in mind, we have avoided abstract mathematical theories whenever it was possible. For instance, we do not use cohomology theory, category theory (apart from Subsect. 10.3.4), Poisson-Lie groups, deformation theory, and knot theory in the book. We assume, however, that the reader has some standard knowledge of Lie groups and Lie algebras and their representation theory.

The book is organized as follows. Formulas, results, definitions, examples, and remarks are numbered and quoted consecutively within the chapters. When a reference to an item in another chapter is made, the number of the chapter is added. For instance, (30) means formula (30) in the same chapter and Propostion 9.7 refers to Proposition 7 in Chap. 9. The end of a proof is marked by D and of an example or a remark by 6. The reader should also notice that often assumptions are fixed and kept in force throughout the whole chapter, section, or subsection. Bibliographical comments are usually gathered at the end of each chapter. There the sources of some results or notions are cited (as far as the authors are aware) and some related references are listed, but no attempt has been made to report the origins of all items.

We want to express our gratitude to A. Schuler and 1. Heckenberger for their indispensible help and valuable suggestions in writing this book and to Mrs. K. Schmidt for typing parts of the manuscript. We also thank Yu. Bespalov, A. Gavrilik, and L. Vainerman for reading parts of the book.

Kiev and Leipzig, March 1997 A. U. Klimyk, K. Schmudgen

Table of Contents

Part I. An Introduction to Quantum Groups

1. Hopf Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1 Prolog: Examples of Hopf Algebras of Functions on Groups. . 3 1.2 Coalgebras, Bialgebras and Hopf Algebras . . . . . . . . . . . . . . . . . 6

1.2.1 Algebras........................................ 6 1.2.2 Coalgebras...................................... 8 1.2.3 Bialgebras....................................... 11 1.2.4 Hopf Algebras ................................... 13 1.2.5* Dual Pairings of Hopf Algebras .................... 16 1.2.6 Examples of Hopf Algebras . . . . . . . . . . . . . . . . . . . . . . .. 18 1.2.7 *-Structures..................................... 20 1.2.8* The Dual Hopf Algebra AO . . . . . . . . . . . . . . . . . . . . . . .. 22 1.2.9* Super Hopf Algebras. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 23 1.2.10* h-Adic Hopf Algebras. . . . . . . . . . . . . .. . . . . . . . . . . . . .. 25

1.3 Modules and Comodules of Hopf Algebras . . . . . . . . . . . . . . . .. 27 1.3.1 Modules and Representations. . . . . . . . . . .. . . . . . . . . .. 27 1.3.2 Comodules and Corepresentations . . . . . . . . . . . . . . . . .. 29 1.3.3 Comodule Algebras and Related Concepts. . . . . . . . . .. 32 1.3.4* Adjoint Actions and Coactions of Hopf Algebras ..... 34 1.3.5* Corepresentations and Representations

of Dually Paired Coalgebras and Algebras . . . . . . . . . .. 35 1.4 Notes................................................. 36

2. q-Calculus............. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 37 2.1 Main Notions on q-Calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 37

2.1.1 q-Numbers and q-Factorials . . . . . . . . . . . . . . . . . . . . . . .. 37 2.1.2 q-Binomial Coefficients. . . . . . . . . . . . . . . . . . . . . . . . . . .. 39 2.1.3 Basic Hypergeometric Functions. . . . . . . . . . . . . . . . . . .. 40 2.1.4 The Function 1 <1'0 (a; q, z). . . . . . . . . . . . . . . . . . . . . . . . .. 41 2.1.5 The Basic Hypergeometric Function 2<1'1 •••••••••••• 42 2.1.6 Transformation Formulas for 3<1'2 and 4<1'3 ••••••••••• 43 2.1.7 q-Analog of the Binomial Theorem. . . . . . . . . . . . . . . .. 44

2.2 q-Differentiation and q-Integration . . . . . . . . . . . . . . . . . . . . . . .. 44 2.2.1 q-Differentiation.................................. 44

X Table of Contents

2.2.2 q-Integral....................................... 46 2.2.3 q-Analog of the Exponential Function. . . . . . . . . . . . . .. 47 2.2.4 q-Analog of the Gamma Function .................. 48

2.3 q-Orthogonal Polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 49 2.3.1 Jacobi Matrices and Orthogonal Polynomials ........ 49 2.3.2 q-Hermite Polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . .. 50 2.3.3 Little q-Jacobi Polynomials. . . . . . . . . . . . . . . . . . . . . . .. 51 2.3.4 Big q-Jacobi Polynomials. . . . . . . . . . . . . . . . . . . . . . . . .. 52

2.4 Notes................................................. 52

3. The Quantum Algebra Uq (sI2) and Its Representations... 53 3.1 The Quantum Algebras Uq(sh) and Uh(sh) . . . . . . . . . . . . . . .. 53

3.1.1 The Algebra Uq(sI2) .............................. 53 3.1.2 The Hopf Algebra Uq(sI2) ......................... 55 3.1.3 The Classical Limit of the Hopf Algebra Uq(sh) . . . . .. 57 3.1.4 Real Forms of the Quantum Algebra Uq(sh) . . . . . . . .. 58 3.1.5 The h-Adic Hopf Algebra Uh(sh) . . . . . . . . . . . . . . . . . .. 60

3.2 Finite-Dimensional Representations of Uq(sI2) for q not a Root of Unity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 61 3.2.1 The Representations Twl .......................... 61 3.2.2 Weight Representations and Complete Reducibility . .. 63 3.2.3 Finite-Dimensional Representations of Uq(sh)

and Uh(sI2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 65 3.3 Representations of Uq(sh) for q a Root of Unity. . . . . . . . . . .. 66

3.3.1 The Center of Uq(sh) . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 66 3.3.2 Representations of Uq(sI2) . . . . . . . . . . . . . . . . . . . . . . . .. 67 3.3.3 Representations of u~es(sh) . . . . . . . . . . . . . . . . . . . . . . .. 71

3.4 Tensor Products of Representations. Clebsch-Gordan Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 72 3.4.1 Tensor Products of Representations 11 .............. 72 3.4.2 Clebsch-Gordan Coefficients. . . . . . . . . . . . . . . . . . . . . .. 74 3.4.3 Other Expressions for Clebsch-Gordan Coefficients ... 78 3.4.4 Symmetries of Clebsch-Gordan Coefficients. . . . . . . . .. 81

3.5 Racah Coefficients and 6j Symbols of Uq(SU2) . . . . . . . . . . . . .. 82 3.5.1 Definition of the Racah Coefficients. . . . . . . . . . . . . . . .. 82 3.5.2 Relations Between Racah

and Clebsch-Gordan Coefficients . . . . . . . . . . . . . . . . . .. 84 3.5.3 Symmetry Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 84 3.5.4 Calculation of Racah Coefficients. . . . . . . . . . . . . . . . . .. 85 3.5.5 The Biedenharn-Elliott Identity. . . . . . . . . . . . . . . . . . .. 88 3.5.6 The Hexagon Relation. . . . . . . . . . . . . . . . . . . . . . . . . . .. 90 3.5.7 Clebsch-Gordan Coefficients

as Limits of Racah Coefficients . . . . . . . . . . . . . . . . . . . .. 90 3.6 Tensor Operators and the Wigner-Eckart Theorem. . . . . . . .. 92

3.6.1 Tensor Operators for Compact Lie Groups. . . . . . . . . .. 92

Table of Contents XI

3.6.2 Tensor Operators and the Wigner-Eckart Theorem for Uq(SU2) ...................................... 93

3.7 Applications........................................... 94 3.7.1 The Uq(sh) Rotator Model of Deformed Nuclei. . . . . .. 94 3.7.2 Electromagnetic Transitions in the Uq(sh) Model. . . .. 95

3.8 Notes................................................. 96

4. The Quantum Group SLq(2) and Its Representations ..... 97 4.1 The Hopf Algebra O(SLq(2)) ............................ 97

4.1.1 The Bialgebra O(Mq(2)) .......................... 97 4.1.2 The Hopf Algebra O(SLq(2)) . . . . . . . . . . . . . . . . . . . . .. 99 4.1.3 A Geometric Approach to SLq(2) .................. 101 4.1.4 Real Forms of O(SLq(2)) .......................... 102 4.1.5 The Diamond Lemma. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.2 Representations of the Quantum Group SLq(2) ............ 104 4.2.1 Finite-Dimensional Corepresentations of O(SLq(2)):

Main Results .................................... 104 4.2.2 A Decomposition of O(SLq(2)) . .................... 105 4.2.3 Finite-Dimensional Subcomodules of O(SLq(2)) ...... 106 4.2.4 Calculation of the Matrix Coefficients ............... 108 4.2.5 The Peter-Weyl Decomposition of O(SLq(2)) ........ 110 4.2.6 The Haar Functional of O(SLq(2)) ................. 111

4.3 The Compact Quantum Group SUq(2) and Its Representations ................................. 113 4.3.1 Unitary Representations

of the Quantum Group SUq (2) ..................... 113 4.3.2 The Haar State

and the Peter-Weyl Theorem for O(SUq(2)) ......... 114 4.3.3 The Fourier Transform on SUq (2) .................. 117 4.3.4 *-Representations and the C*-Algebra of O(SUq(2)) .. 117

4.4 Duality of the Hopf Algebras Uq(sh) and O(SLq(2)) ......................................... 119 4.4.1 Dual Pairing of the Hopf Algebras Uq(sh)

and O(SLq(2)) ................................... 119 4.4.2 Corepresentations of O(SLq(2))

and Representations of Uq (sI2) ..................... 123 4.5 Quantum 2-Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

4.5.1 A Family of Quantum Spaces for SLq(2) ............ 124 4.5.2 Decomposition of the Algebra O(S;p) ............... 126 4.5.3 Spherical Functions on S;p ........................ 129 4.5.4 An Infinitesimal Characterization of O(S;p) ......... 129

4.6 Notes ................................................. 132

XII Table of Contents

5. The q-Oscillator Algebras and Their Representations ..... 133 5.1 The q-Oscillator Algebras A~ and Aq ..................... 133

5.1.1 Definitions and Algebraic Properties ................ 133 5.1.2 Other Forms of the q-Oscillator Algebra ............. 136 5.1.3 The q-Oscillator Algebra

and the Quantum Algebra Uq (sI2) .................. 137 5.1.4 The q-Oscillator Algebras

and the Quantum Space Mq2 (2) .................... 140 5.2 Representations of q-Oscillator Algebras ................... 140

5.2.1 N-Finite Representations .......................... 140 5.2.2 Irreducible Representations

with Highest (Lowest) Weights ..................... 141 5.2.3 Representations Without Highest and Lowest Weights 143 5.2.4 Irreducible Representations of A~

for q a Root of Unity ............................. 145 5.2.5 Irreducible *-Representations of A~ and Aq .......... 147 5.2.6 Irreducible *-Representations

of Another q-Oscillator Algebra .................... 148 5.3 The Fock Representation of the q-Oscillator Algebra ........ 149

5.3.1 The Fock Representation .......................... 149 5.3.2 The Bargmann-Fock Realization ................... 150 5.3.3 Coherent States .................................. 152 5.3.4 Bargmann-Fock Space Realization

of Irreducible Representations of Uq(sh) ............. 153 5.4 Notes ................................................. 154

Part II. Quantized Universal Enveloping Algebras

6. Drinfeld-Jimbo Algebras ................................. 157 6.1 Definitions of Drinfeld-Jimbo Algebras .................... 157

6.1.1 Semisimple Lie Algebras .......................... 157 6.1.2 The Drinfeld-Jimbo Algebras Uq(g) . ................ 161 6.1.3 The h-Adic Drinfeld-Jimbo Algebras Uh(g) .......... 165 6.1.4 Some Algebra Automorphisms

of Drinfeld-Jimbo Algebras ........................ 167 6.1.5 Triangular Decomposition of Uq(g) ................. 168 6.1.6 Hopf Algebra Automorphisms of Uq(g) .............. 171 6.1. 7 Real Forms of Drinfeld-Jimbo Algebras ............. 172

6.2 Poincare-Birkhoff-Witt Theorem and Verma Modules ....... 173 6.2.1 Braid Groups .................................... 173 6.2.2 Action of Braid Groups on Drinfeld-Jimbo Algebras .. 174 6.2.3 Root Vectors and Poincare-Birkhoff-Witt Theorem ... 175 6.2.4 Representations with Highest Weights ............... 177 6.2.5 Verma Modules .................................. 179

Table of Contents XIII

6.2.6 Irreducible Representations with Highest Weights .... 180 6.2.7 The Left Adjoint Action of Uq(g) ................... 181

6.3 The Quantum Killing Form and the Center of Uq(g) ........ 184 6.3.1 A Dual Pairing of the Hopf Algebras Uq(b+)

and Uq(b_)OP .................................... 184 6.3.2 The Quantum Killing Form on Uq(g) ............... 187 6.3.3 A Quantum Casimir Element ...................... 189 6.3.4 The Center of Uq(g)

and the Harish-Chandra Homomorphism ............ 192 6.3.5 The Center of Uq(g) for q a Root of Unity ........... 194

6.4 Notes ................................................. 196

7. Finite-Dimensional Representations of Drinfeld-Jimbo Algebras ............................... 197 7.1 General Properties of Finite-Dimensional Representations

of Uq(g) ............................................... 197 7.1.1 Weight Structure and Classification ................. 197 7.1.2 Properties of Representations ...................... 200 7.1.3 Representations of h-Adic Drinfeld-Jimbo Algebras ... 202 7.1.4 Characters of Representations and Multiplicities

of Weights. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 7.1.5 Separation of Elements of Uq(g) .................... 204 7.1.6 The Quantum Trace

of Finite-Dimensional Representations .............. 205 7.2 Tensor Products of Representations ....................... 207

7.2.1 Multiplicities in Tensor Products of Representations .. 208 7.2.2 Clebsch-Gordan Coefficients ....................... 211

7.3 Representations of Uq(gln) for q not a Root of Unity ........ 212 7.3.1 The Hopf Algebra Uq(gln) ......................... 212 7.3.2 Finite-Dimensional Representations of Uq(gln) ....... 213 7.3.3 Gel'fand-Tsetlin Bases and Explicit Formulas

for Representations ............................... 214 7.3.4 Representations of Class 1 ......................... 217 7.3.5 Tensor Products of Representations ................. 218 7.3.6 Tensor Operators and the Wigner-Eckart Theorem ... 219 7.3.7 Clebsch-Gordan Coefficients

for the Tensor Product T m ® Tl ... j • • • • • • • • • • • • • • • • • 220 7.3.8 Clebsch-Gordan Coefficients

for the Tensor Product Tm ® Tp .................... 221 7.3.9 The Tensor Product Tm ® Tl for q±l --+ 0 ........... 224

7.4 Crystal Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 7.4.1 Crystal Bases of Finite-Dimensional Modules ........ 226 7.4.2 Existence and Uniqueness of Crystal Bases .......... 227 7.4.3 Crystal Bases of Tensor Product Modules ........... 228

XIV Table of Contents

7.4.4 Globalization of Crystal Bases ..................... 229 7.4.5 Crystal Bases of U~(n_) ........................... 230

7.5 Representations of Uq(g) for q a Root of Unity ............. 232 7.5.1 General Results .................................. 232 7.5.2 Cyclic Representations ............................ 234 7.5.3 Cyclic Representations of the Algebra U.(SII+1) ...... 235 7.5.4 Representations of Minimal Dimensions ............. 237 7.5.5 Representations of U.(SII+1) in Gel'fand-Tsetlin Bases 238

7.6 Applications ........................................... 240 7.7 Notes ................................................. 242

8. Quasitriangularity and Universal R-Matrices ............. 243 8.1 Quasitriangular Hopf Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

8.1.1 Definition and Basic Properties ......... , ........... 243 8.1.2 R-Matrices for Representations ..................... 246 8.1.3 Square and Inverse of the Antipode. . . . . . . . . . . . . . . . . 247

8.2 The Quantum Double and Universal R-Matrices ............ 250 8.2.1 The Quantum Double of Skew-Paired Bialgebras ..... 250 8.2.2 Quasitriangularity of Quantum Doubles

of Finite-Dimensional Hopf Algebras ................ 254 8.2.3 The Rosso Form of the Quantum Double ............ 257 8.2.4 Drinfeld-Jimbo Algebras as Quotients

of Quantum Doubles .............................. 258 8.3 Explicit Form of Universal R-Matrices .................... 259

8.3.1 The Universal R-Matrix for Uh(sh) ................. 259 8.3.2 The Universal R-Matrix for Uh(g) .................. 261 8.3.3 R-Matrices for Representations of Uq(g) ............. 264

8.4 Vector Representations and R-Matrices ................... 267 8.4.1 Vector Representations of Drinfeld-Jimbo Algebras ... 267 8.4.2 R-Matrices for Vector Representations .............. 269 8.4.3 Spectral Decompositions of R-Matrices

for Vector Representations ......................... 272 8.5 L-Operators and L-Functionals ........................... 275

8.5.1 L-Operators and L-Functionals ..................... 275 8.5.2 L-Functionals for Vector Representations ............ 277 8.5.3 The Extended Hopf Algebras u;xt(g) ............... 281 8.5.4 L-Functionals for Vector Representations of Uq(g) .... 283 8.5.5 The Hopf Algebras U(R) and Uf(g) ................ 285

8.6 An Analog of the Brauer-Schur-Weyl Duality .............. 288 8.6.1 The Algebras Uq(SON) ............................ 288 8.6.2 Tensor Products of Vector Representations .......... 289 8.6.3 The Brauer-Schur-Weyl Duality

for Drinfeld-Jimbo Algebras ....................... 291 8.6.4 Hecke and Birman-Wenzl-Murakami Algebras ....... 293

8.7 Applications........................................... 294

Table of Contents XV

8.7.1 Baxterization .................................... 295 8.7.2 Elliptic Solutions

of the Quantum Yang-Baxter Equation ............. 297 8.7.3 R-Matrices and Integrable Systems ................. 298

8.8 Notes ................................................. 300

Part III. Quantized Algebras of Functions

9. Coordinate Algebras of Quantum Groups and Quantum Vector Spaces .............................. 303 9.1 The Approach of Faddeev-Reshetikhin-Takhtajan .......... 303

9.1.1 The FRT Bialgebra A(R) ......................... 303 9.1.2 The Quantum Vector Spaces Xd!j R) and XR(fj R) .. 307

9.2 The Quantum Groups GLq(N) and SLq(N) ............... 309 9.2.1 The Quantum Matrix Space Mq(N)

and the Quantum Vector Space C: ................. 310 9.2.2 Quantum Determinants ........................... 311 9.2.3 The Quantum Groups GLq(N) and SLq(N) ......... 313 9.2.4 Real Forms of GLq(N) and SLq(N)

and *-Quantum Spaces ............................ 316 9.3 The Quantum Groups Oq(N) and Spq(N) ................. 317

9.3.1 The Hopf Algebras O(Oq(N)) and O(Spq(N)) ....... 318 9.3.2 The Quantum Vector Space

for the Quantum Group Oq(N) .................... 320 9.3.3 The Quantum Group SOq(N) ..................... 323 9.3.4 The Quantum Vector Space

for the Quantum Group Spq(N) .. .................. 324 9.3.5 Real Forms of Oq(N) and Spq(N)

and *-Quantum Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 9.4 Dual Pairings of Drinfeld-Jimbo Algebras

and Coordinate Hopf Algebras ....... . . . . . . . . . . . . . . . . . . . . 327 9.5 Notes ....... · .......................................... 330

10. Coquasitriangularity and Crossed Product Constructions. 331 10.1 Coquasitriangular Hopf Algebras ......................... 331

10.1.1 Definition and Basic Properties .................... 331 10.1.2 Coquasitriangularity of FRT Bialgebras A(R)

and Coordinate Hopf Algebras O(Gq) ............... 337 10.1.3 L-Functionals of Coquasitriangular Hopf Algebras .... 342

10.2 Crossed Product Constructions of Hopf Algebras ........... 349 10.2.1 Crossed Product Algebras ......................... 349 10.2.2 Crossed Coproduct Coalgebras ..................... 352 10.2.3 Twisting of Algebra Structures by 2-Cocycles

and Quantum Doubles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354

XVI Table of Contents

10.2.4 Twisting of Coalgebra Structures by 2-Cocycles and Quantum Co doubles .......................... 357

10.2.5 Double Crossed Product Bialgebras and Quantum Doubles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359

10.2.6 Double Crossed Coproduct Bialgebras and Quantum Co doubles .......................... 362

10.2.7 Realifications of Quantum Groups .................. 363 10.3 Braided Hopf Algebras .................................. 365

10.3.1 Covariantized Products for Coquasitriangular Bialgebras ................... 365

10.3.2 Braided Hopf Algebras Associated with Coquasitriangular Hopf Algebras .... 370

10.3.3 Braided Hopf Algebras Associated with Quasitriangular Hopf Algebras ...... 376

10.3.4 Braided Tensor Categories and Braided Hopf Algebras 377 10.3.5 Braided Vector Algebras .......................... 380 10.3.6 Bosonization of Braided Hopf Algebras .............. 382 10.3.7 *-Structures on Bosonized Hopf Algebras ............ 386 10.3.8 Inhomogeneous Quantum Groups ................... 388 10.3.9 *-Structures for Inhomogeneous Quantum Groups .... 390

10.4 Notes ................................................. 394

11. Corepresentation Theory and Compact Quantum Groups. 395 11.1 Corepresentations of Hopf Algebras ....................... 395

11.1.1 Corepresentations ................................ 395 11.1.2 Intertwiners ..................................... 397 11.1.3 Constructions of New Corepresentations ............. 397 11.1.4 Irreducible Corepresentations ...................... 398 11.1.5 Unitary Corepresentations ......................... 401

11.2 Cosemisimple Hopf Algebras ............................. 402 11.2.1 Definition and Characterizations ................... 402 11.2.2 The Haar Functional of a Cosemisimple Hopf Algebra. 404 11.2.3 Peter-Weyl Decomposition

of Coordinate Hopf Algebras. . . . . . . . . . . . . . . . . . . . . . . 408 11.3 Compact Quantum Group Algebras ....................... 415

11.3.1 Definitions and Characterizations of CQG Algebras ... 415 11.3.2 The Haar State of a CQG Algebra .................. 419 11.3.3 C*-Algebra Completions of CQG Algebras ........... 420 11.3.4 Modular Properties of the Haar State ............... 422 11.3.5 Polar Decomposition of the Antipode ............... 426 11.3.6 Multiplicative Unit aries of CQG Algebras ........... 427

11.4 Compact Quantum Group C*-Algebras ................... 429 11.4.1 CQG C*-Algebras and Their CQG Algebras ......... 429 11.4.2 Existence of the Haar State of a CQG C*-Algebra .... 431 11.4.3 Proof of Theorem 39 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433

Table of Contents XVII

11.4.4 Another Definition of CQG C*-Algebras ............ 434 11.5 Finite-Dimensional Representations of GLq(N) ............. 435

11.5.1 Some Quantum Subgroups of GLq(N) .............. 435 11.5.2 Submodules of Relative Invariant Elements .......... 436 11.5.3 Irreducible Representations of GLq(N) .............. 437 11.5.4 Peter-Weyl Decomposition of O(GLq(N)) ........... 439 11.5.5 Representations of the Quantum Group Uq(N) ....... 441

11.6 Quantum Homogeneous Spaces ........................... 442 11.6.1 Definition of a Quantum Homogeneous Space ........ 442 11.6.2 Quantum Homogeneous Spaces

Associated with Quantum Subgroups ............... 443 11.6.3 Quantum Gel'fand Pairs .......................... 445 11.6.4 The Quantum Homogeneous Space Uq(N-1)\Uq (N) .. 447 11.6.5 Quantum Homogeneous Spaces

of Infinitesimally Invariant Elements . . . . . . . . . . . . . . . . 451 11.6.6 Quantum Projective Spaces ........................ 452

11.7 Notes ................................................. 454

Part IV. Noncommutative Differential Calculus

12. Covariant Differential Calculus on Quantum Spaces ....... 457 12.1 Covariant First Order Differential Calculus ................ 457

12.1.1 First Order Differential Calculi on Algebras .......... 457 12.1.2 Covariant First Order Calculi on Quantum Spaces .... 459

12.2 Covariant Higher Order Differential Calculus ............... 461 12.2.1 Differential Calculi on Algebras .................... 461 12.2.2 The Differential Envelope of an Algebra ............. 462 12.2.3 Covariant Differential Calculi on Quantum Spaces .... 463

12.3 Construction of Covariant Differential Calculi on Quantum Spaces .................................... 464 12.3.1 General Method .................................. 464 12.3.2 Covariant Differential Calculi

on Quantum Vector Spaces ........................ 467 12.3.3 Covariant Differential Calculus on C:

and the Quantum Weyl Algebra. . . . . . . . . . . . . . . . . . . . 468 12.3.4 Covariant Differential Calculi

on the Quantum Hyperboloid . . . . . . . . . . . . . . . . . . . . . . 471 12.4 Notes ................................................. 472

13. Hopf Bimodules and Exterior Algebras ................... 473 13.1 Covariant Bimodules .................................... 473

13.1.1 Left-Covariant Bimodules ......................... 473 13.1.2 Right-Covariant Bimodules ........................ 477 13.1.3 Bicovariant Bimodules (Hopf Bimodules) ............ 477

XVIII Table of Contents

13.1.4 Woronowicz' Braiding of Bicovariant Bimodules ...... 480 13.1.5 Bicovariant Bimodules and Representations

of the Quantum Double ........................... 483 13.2 Tensor Algebras and Exterior Algebras

of Bicovariant Bimodules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485 13.2.1 The Tensor Algebra of a Bicovariant Bimodule ....... 485 13.2.2 The Exterior Algebra of a Bicovariant Bimodule ...... 488

13.3 Notes ................................................. 490

14. Covariant Differential Calculus on Quantum Groups . ..... 491 14.1 Left-Covariant First Order Differential Calculi ............. 491

14.1.1 Left-Covariant First Order Calculi and Their Right Ideals ............................ 491

14.1.2 The Quantum Tangent Space ...................... 494 14.1.3 An Example: The 3D-Calculus on SLq {2} ........... 496 14.1.4 Another Left-Covariant Differential Calculus

on SLq {2} ....................................... 498 14.2 Bicovariant First Order Differential Calculi ................ 498

14.2.1 Right-Covariant First Order Differential Calculi ...... 498 14.2.2 Bicovariant First Order Differential Calculi .......... 499 14.2.3 Quantum Lie Algebras

of Bicovariant First Order Calculi .................. 500 14.2.4 The 4D+- and the 4D_-Calculus on SLq {2} .......... 504 14.2.5 Examples of Bicovariant First Order Calculi

on Simple Lie Groups .................... ' ......... 505 14.3 Higher Order Left-Covariant Differential Calculi ............ 506

14.3.1 The Maurer-Cartan Formula ...................... 506 14.3.2 The Differential Envelope of a Hopf Algebra ......... 507 14.3.3 The Universal DC of a Left-Covariant FODC ........ 508

14.4 Higher Order Bicovariant Differential Calculi ............... 511 14.4.1 Bicovariant Differential Calculi

and Differential Hopf Algebras ..................... 511 14.4.2 Quantum Lie Derivatives and Contraction Operators .. 514

14.5 Bicovariant Differential Calculi on Coquasitriangular Hopf Algebras ...................... 517

14.6 Bicovariant Differential Calculi on Quantized Simple Lie Groups ......................... 521 14.6.1 A Family of Bicovariant First Order

Differential Calculi ............................... 521 14.6.2 Braiding and Structure Constants of the FODC F±,z . 524 14.6.3 A Canonical Basis for the Left-Invariant I-Forms ..... 525 14.6.4 Classification of Bicovariant First Order

Differential Calculi ............................... 527 14.7 Notes ................................................. 528

Table of Contents XIX

Bibliography . ................................................. 529

Index ......................................................... 545