Introduction 1
1 Poisson-Lie groups and Lie bialgebras 15
1.1 Poisson manifolds 16
A Definitions 16
B Functorial properties 18
C Symplectic leaves 18
1.2 Poisson-Lie groups 21
A Definitions 21
B Poisson homogeneous spaces 22
1.3 Lie bialgebras 24
A The Lie bialgebra of a Poisson-Lie group 24
B Manin triples 26
C Examples 28
D Derivations 32
1.4 Duals and doubles 33
A Duals of Lie bialgebras and Poisson-Lie groups 33
B The classical double 34
C Compact Poisson-Lie groups 35
1.5 Dressing actions and symplectic leaves 36
A Poisson actions 36
B Dressing transformations and symplectic leaves 37
C Symplectic leaves in compact Poisson-Lie groups 39
D The twisted case 41
1.6 Deformation of Poisson structures and quantization 43
A Deformations of Poisson algebras 43
B Weyl quantization 44
C Quantization as deformation 46
Bibliographical notes 48
2 Coboundary Poisson-Lie groups and the classical Yang-Baxter equation 50
2.1 Coboundary Lie bialgebras 50
A Defnitions 50
B The classical Yang-Baxter equation 54
C Examples 55
D The classical double 58
2.2 Coboundary Poisson-Lie groups 59
A The Sklyanin bracket 60
B r-matrices and 2-cocycles 62
C The classical R-matrix 67
2.3 Classical integrable systems 68
A Complete integrability 68
B Lax pairs 69
C Integrable systems from r-matrices 71
D Toda systems 75
Bibliographical notes 77
3 Solutions of the classical Yang-Baxter equation 79
3.1 Constant solutions of the CYBE 80
A The parameter space of non-skew solutions 80
B Description of the solutions 81
C Examples 82
D Skew solutions and quasi-Frobenius Lie algebras 84
3.2 Solutions of the CYBE with spectral parameters 87
A Classification of the solutions 87
B Elliptic solutions 90
C Trigonometric solutions 91
D Rational solutions 95
Bibliographical notes 98
4 Quasitriangular Hopf algebras 100
4.1 Hopf algebras 101
A Definitions 101
B Examples 105
C Representations of Hopf algebras 108
D Topological Hopf algebras and duality 111
E Integration on Hopf algebras 115
F Hopf ＊-algebras 117
4.2 Quasitriangular Hopf algebras 119
A Almost cocommutative Hopf algebras 119
B Quasitriangular Hopf algebras 123
C Ribbon Hopf algebras and quantum dimension 125
D The quantum double 127
E Twisting 129
F Sweedler's example 131
Bibliographical notes 133
5 Representations and quasitensor categories 135
5.1 Monoidal categories 136
A Abelian categories 136
B Monoidal categories 138
C Rigidity 139
D Examples 140
E Reconstruction theorems 147
5.2 Quasitensor categories 149
A Tensor categories 149
B Quasitensor categories 152
C Balancing 154
D Quasitensor categories and fusion rules 154
E Quasitensor categories in quantum field theory 157
5.3 Invariants of ribbon tangles 161
A Isotopy invariants and monoidal functors 161
B Tangle invariants 166
C Central elements 168
Bibliographical notes 168
6 Quantization of Lie bialgebras 170
6.1 Deformations of Hopf algebras 171
A Definitions 171
B Cohomology theory 173
C Rigidity theorems 176
6.2 Quantization 177
A (Co-)Poisson Hopf algebras 177
B Quantization 179
C Existence of quantizations 182
6.3 Quantized universal enveloping algebras 187
A Cocommutative QUE algebras 187
B Quasitriangular QUE algebras 188
C QUE duals and doubles 189
D The square of the antipode 190
6.4 The basic example 192
A Construction of the standard quantization 192
B Algebra structure 196
C PBW basis 199
D Quasitriangular structure 200
E Representations 203
F A non-standard quantization 206
6.5 Quantum Kac-Moody algebras 207
A The standard quantization 207
B The centre 212
C Multiparameter quantizations 212
Bibliographical notes 213
7 Quantized function algebras 215
7.1 The basic example 216
A Definition 216
B A basis of Fh(SL2(？)) 220
C The R-matrix formulation 222
D Duality 223
E Representations 227
7.2 R-matrix quantization 228
A From R-matrices to bialgebras 228
B From bialgebras to Hopf algebras:the quantum determinant 231
C Solutions of the QYBE 233
7.3 Examples of quantized function algebras 234
A The general definition 234
B The quantum special linear group 235
C The quantum orthogonal and symplectic groups 236
D Multiparameter quantized function algebras 238
7.4 Differential calculus on quantum groups 240
A The de Rham complex of the quantum plane 240
B The de Rham complex of the quantum m×m matrices 242
C The de Rham complex of the quantum general linear group 244
D Invariant forms on quantum GLm 245
7.5 Integrable lattice models 246
A Vertex models 246
B Transfer matrices 248
C Integrability 249
D Examples 251
Bibliographical notes 253
8 Structure of QUE algebras:the universal R-matrix 255
8.1 The braid group action 256
A The braid group 256
B Root vectors and the PBW basis 258
8.2 The quantum Weyl group 262
A The sl2 case 262
B The relation with the universal R-matrix 263
C The general case 265
8.3 The quasitriangular structure 266
A The quantum double construction 266
B The sl2 case 267
C The general case 271
D Multiplicative properties 274
E Uniqueness of the universal R-matrix 275
F The centre of Uh 275
G Matrix solutions of the quantum Yang-Baxter equation 276
Bibliographical notes 278
9 Specializations of QUE algebras 279
9.1 Rational forms 280
A The definition of Uq 280
B Some basic properties of Uq 282
C The Harish Chandra homomorphism and the centre of Uq 284
D A geometric realization 285
9.2 The non-restricted specialization 288
A The non-restricted integral form 289
B The centre 290
C The quantum coadjoint action 293
9.3 The restricted specialization 296
A The restricted integral form 297
B A remarkable finite-dimensional Hopf algebra 301
C A Frobenius map in characteristic zero 304
D The quiver approach 307
9.4 Automorphisms and real forms 309
A Automorphisms 309
B Real forms 309
Bibliographical notes 311
10 Representations of QUE algebras:the generic case 313
10.1 Classification of finite-dimensional representations 313
A Highest weight modules 313
B The determinant formula 319
C Specialization:the non-root of unity case 324
D R-matrices associated to representations of Uq 327
E Unitary representations 329
10.2 Quantum invariant theory 332
A Hecke and Birman-Murakami-Wenzl algebras 332
B Quantum Brauer-Frobenius-Schur duality 334
C Another realization of Hecke algebras 336
Bibliographical notes 337
11 Representations of QUE algebras:the root of unity case 338
11.1 The non-restricted case 339
A Parametrization of the irreducible representations of U？ 339
B Some explicit constructions 344
C Intertwiners and the QYBE 348
11.2 The restricted case 351
A Highest weight representations 351
B A tensor product theorem 357
C Quasitensor structure 359
D Some conjectures 359
11.3 Tilting modules and the fusion tensor product 361
A Tilting modules 361
B Quantum dimensions 365
C Tensor products 367
D The categorical formulation 370
Bibliographical notes 372
12 Infinite-dimensional quantum groups 374
12.1 Yangians and their representations 375
A Three realizations 375
B Basic properties 380
C Classification of the finite-dimensional representations 383
D Evaluation representations 386
E The sl2 case 388
12.2 Quantum affine algebras 392
A Another realization:quantum loop algebras 392
B Finite-dimensional representations of quantum loop algebras 394
C Evaluation representations 399
12.3 Frobenius-Schur duality for Yangians and quantum affine algebras 403
A Affine Hecke algebras and their degenerations 403
B Representations of affine Hecke algebras 405
C Duality for U？(sln+1(？))-revisited 408
D Quantum affine algebras and affine Hecke algebras 410
E Yangians and degenerate affine Hecke algebras 413
12.4 Yangians and infinite-dimensional classical groups 414
A Tame representations 415
B The relation with Yangians 416
12.5 Rational and trigonometric solutions of the QYBE 417
A Yangians and rational solutions 418
B Quantum affine algebras and trigonometric solutions 423
Bibliographical notes 426
13 Quantum harmonic analysis 428
13.1 Compact quantum groups and their representations 430
A Definitions 430
B Highest weight representations 433
C The sl2 case 435
D The general case:tensor products 437
E The twisted case and quantum tori 439
F Representations at roots of unity 442
13.2 Quantum homogeneous spaces 445
A Quantum G-spaces 445
B Quantum flag manifolds and Schubert varieties 447
C Quantum spheres 448
13.3 Compact matrix quantum groups 451
A C＊ completions and compact matrix quantum groups 451
B The Haar integral on compact quantum groups 454
13.4 A non-compact quantum group 459
A The quantum euclidean group 459
B Representation theory 462
C Invariant integration on the quantum euclidean group 463
13.5 q-special functions 465
A Little q-Jacobi polynomials and quantum SU2 466
B Big q-Jacobi polynomials and quantum spheres 467
C q-Bessel functions and the quantum euclidean group 469
Bibliographical notes 473
14 Canonical bases 475
14.1 Crystal bases 476
A Gelfand-Tsetlin bases 476
B Crystal bases 478
C Globalization 480
D Crystal graphs and tensor products 481
14.2 Lusztig's canonical bases 486
A The algebraic construction 486
B The topological construction 488
C Some combinatorial formulas 490
Bibliographical notes 492
15 Quantum group invariants of knots and 3-manifolds 494
15.1 Knots and 3-manifolds:a quick review 495
A From braids to links 496
B From links to 3-manifolds 502
15.2 Link invariants from quantum groups 504
A Link invariants from R-matrices 504
B Link invariants from vertex models 510
15.3 Modular Hopf algebras and 3-manifold invariants 517
A Modular Hopf algebras 517
B The construction of 3-manifold invariants 522
Bibliographical notes 525
16 Quasi-Hopf algebras and the Knizhnik-Zamolodchikov equation 527
16.1 Quasi-Hopf algebras 528
A Definitions 529
B An example from conformal field theory 533
C Quasi-Hopf QUE algebras 534
16.2 The Kohno-Drinfel'd monodromy theorem 537
A Braid groups and configuration spaces 537
B The Knizhnik-Zamolodchikov equation 539
C The KZ equation and affine Lie algebras 541
D Quantization and the KZ equation 543
E The monodromy theorem 549
16.3 Affine Lie algebras and quantum groups 550
A The category O？ 551
B The tensor product 552
C The equivalence theorem 555
16.4 Quasi-Hopf algebras and Grothendieck's esquisse 556
A Gal(？／？)and pro-finite fundamental groups 557
B The Grothendieck-Teichmüller group and quasitriangular quasi-Hopf algebras 559
Bibliographical notes 560
Appendix Kac-Moody algebras 562
A 1 Generalized Cartan matrices 562
A 2 Kac-Moody algebras 562
A 3 The invariant bilinear form 563
A 4 Roots 563
A 5 The Weyl group 564
A 6 Root vectors 565
A 7 Affine Lie algebras 565
A 8 Highest weight modules 566
References 567
Index of notation 638
General index 643