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Introduction1

1 Poisson-Lie groups and Lie bialgebras15

1.1 Poisson manifolds16

A Definitions16

B Functorial properties18

C Symplectic leaves18

1.2 Poisson-Lie groups21

A Definitions21

B Poisson homogeneous spaces22

1.3 Lie bialgebras24

A The Lie bialgebra of a Poisson-Lie group24

B Manin triples26

C Examples28

D Derivations32

1.4 Duals and doubles33

A Duals of Lie bialgebras and Poisson-Lie groups33

B The classical double34

C Compact Poisson-Lie groups35

1.5 Dressing actions and symplectic leaves36

A Poisson actions36

B Dressing transformations and symplectic leaves37

C Symplectic leaves in compact Poisson-Lie groups39

D The twisted case41

1.6 Deformation of Poisson structures and quantization43

A Deformations of Poisson algebras43

B Weyl quantization44

C Quantization as deformation46

Bibliographical notes48

2 Coboundary Poisson-Lie groups and the classical Yang-Baxter equation50

2.1 Coboundary Lie bialgebras50

A Defnitions50

B The classical Yang-Baxter equation54

C Examples55

D The classical double58

2.2 Coboundary Poisson-Lie groups59

A The Sklyanin bracket60

B r-matrices and 2-cocycles62

C The classical R-matrix67

2.3 Classical integrable systems68

A Complete integrability68

B Lax pairs69

C Integrable systems from r-matrices71

D Toda systems75

Bibliographical notes77

3 Solutions of the classical Yang-Baxter equation79

3.1 Constant solutions of the CYBE80

A The parameter space of non-skew solutions80

B Description of the solutions81

C Examples82

D Skew solutions and quasi-Frobenius Lie algebras84

3.2 Solutions of the CYBE with spectral parameters87

A Classification of the solutions87

B Elliptic solutions90

C Trigonometric solutions91

D Rational solutions95

Bibliographical notes98

4 Quasitriangular Hopf algebras100

4.1 Hopf algebras101

A Definitions101

B Examples105

C Representations of Hopf algebras108

D Topological Hopf algebras and duality111

E Integration on Hopf algebras115

F Hopf ＊-algebras117

4.2 Quasitriangular Hopf algebras119

A Almost cocommutative Hopf algebras119

B Quasitriangular Hopf algebras123

C Ribbon Hopf algebras and quantum dimension125

D The quantum double127

E Twisting129

F Sweedler's example131

Bibliographical notes133

5 Representations and quasitensor categories135

5.1 Monoidal categories136

A Abelian categories136

B Monoidal categories138

C Rigidity139

D Examples140

E Reconstruction theorems147

5.2 Quasitensor categories149

A Tensor categories149

B Quasitensor categories152

C Balancing154

D Quasitensor categories and fusion rules154

E Quasitensor categories in quantum field theory157

5.3 Invariants of ribbon tangles161

A Isotopy invariants and monoidal functors161

B Tangle invariants166

C Central elements168

Bibliographical notes168

6 Quantization of Lie bialgebras170

6.1 Deformations of Hopf algebras171

A Definitions171

B Cohomology theory173

C Rigidity theorems176

6.2 Quantization177

A (Co-)Poisson Hopf algebras177

B Quantization179

C Existence of quantizations182

6.3 Quantized universal enveloping algebras187

A Cocommutative QUE algebras187

B Quasitriangular QUE algebras188

C QUE duals and doubles189

D The square of the antipode190

6.4 The basic example192

A Construction of the standard quantization192

B Algebra structure196

C PBW basis199

D Quasitriangular structure200

E Representations203

F A non-standard quantization206

6.5 Quantum Kac-Moody algebras207

A The standard quantization207

B The centre212

C Multiparameter quantizations212

Bibliographical notes213

7 Quantized function algebras215

7.1 The basic example216

A Definition216

B A basis of Fh(SL2(？))220

C The R-matrix formulation222

D Duality223

E Representations227

7.2 R-matrix quantization228

A From R-matrices to bialgebras228

B From bialgebras to Hopf algebras:the quantum determinant231

C Solutions of the QYBE233

7.3 Examples of quantized function algebras234

A The general definition234

B The quantum special linear group235

C The quantum orthogonal and symplectic groups236

D Multiparameter quantized function algebras238

7.4 Differential calculus on quantum groups240

A The de Rham complex of the quantum plane240

B The de Rham complex of the quantum m×m matrices242

C The de Rham complex of the quantum general linear group244

D Invariant forms on quantum GLm245

7.5 Integrable lattice models246

A Vertex models246

B Transfer matrices248

C Integrability249

D Examples251

Bibliographical notes253

8 Structure of QUE algebras:the universal R-matrix255

8.1 The braid group action256

A The braid group256

B Root vectors and the PBW basis258

8.2 The quantum Weyl group262

A The sl2 case262

B The relation with the universal R-matrix263

C The general case265

8.3 The quasitriangular structure266

A The quantum double construction266

B The sl2 case267

C The general case271

D Multiplicative properties274

E Uniqueness of the universal R-matrix275

F The centre of Uh275

G Matrix solutions of the quantum Yang-Baxter equation276

Bibliographical notes278

9 Specializations of QUE algebras279

9.1 Rational forms280

A The definition of Uq280

B Some basic properties of Uq282

C The Harish Chandra homomorphism and the centre of Uq284

D A geometric realization285

9.2 The non-restricted specialization288

A The non-restricted integral form289

B The centre290

C The quantum coadjoint action293

9.3 The restricted specialization296

A The restricted integral form297

B A remarkable finite-dimensional Hopf algebra301

C A Frobenius map in characteristic zero304

D The quiver approach307

9.4 Automorphisms and real forms309

A Automorphisms309

B Real forms309

Bibliographical notes311

10 Representations of QUE algebras:the generic case313

10.1 Classification of finite-dimensional representations313

A Highest weight modules313

B The determinant formula319

C Specialization:the non-root of unity case324

D R-matrices associated to representations of Uq327

E Unitary representations329

10.2 Quantum invariant theory332

A Hecke and Birman-Murakami-Wenzl algebras332

B Quantum Brauer-Frobenius-Schur duality334

C Another realization of Hecke algebras336

Bibliographical notes337

11 Representations of QUE algebras:the root of unity case338

11.1 The non-restricted case339

A Parametrization of the irreducible representations of U？339

B Some explicit constructions344

C Intertwiners and the QYBE348

11.2 The restricted case351

A Highest weight representations351

B A tensor product theorem357

C Quasitensor structure359

D Some conjectures359

11.3 Tilting modules and the fusion tensor product361

A Tilting modules361

B Quantum dimensions365

C Tensor products367

D The categorical formulation370

Bibliographical notes372

12 Infinite-dimensional quantum groups374

12.1 Yangians and their representations375

A Three realizations375

B Basic properties380

C Classification of the finite-dimensional representations383

D Evaluation representations386

E The sl2 case388

12.2 Quantum affine algebras392

A Another realization:quantum loop algebras392

B Finite-dimensional representations of quantum loop algebras394

C Evaluation representations399

12.3 Frobenius-Schur duality for Yangians and quantum affine algebras403

A Affine Hecke algebras and their degenerations403

B Representations of affine Hecke algebras405

C Duality for U？(sln+1(？))-revisited408

D Quantum affine algebras and affine Hecke algebras410

E Yangians and degenerate affine Hecke algebras413

12.4 Yangians and infinite-dimensional classical groups414

A Tame representations415

B The relation with Yangians416

12.5 Rational and trigonometric solutions of the QYBE417

A Yangians and rational solutions418

B Quantum affine algebras and trigonometric solutions423

Bibliographical notes426

13 Quantum harmonic analysis428

13.1 Compact quantum groups and their representations430

A Definitions430

B Highest weight representations433

C The sl2 case435

D The general case:tensor products437

E The twisted case and quantum tori439

F Representations at roots of unity442

13.2 Quantum homogeneous spaces445

A Quantum G-spaces445

B Quantum flag manifolds and Schubert varieties447

C Quantum spheres448

13.3 Compact matrix quantum groups451

A C＊ completions and compact matrix quantum groups451

B The Haar integral on compact quantum groups454

13.4 A non-compact quantum group459

A The quantum euclidean group459

B Representation theory462

C Invariant integration on the quantum euclidean group463

13.5 q-special functions465

A Little q-Jacobi polynomials and quantum SU2466

B Big q-Jacobi polynomials and quantum spheres467

C q-Bessel functions and the quantum euclidean group469

Bibliographical notes473

14 Canonical bases475

14.1 Crystal bases476

A Gelfand-Tsetlin bases476

B Crystal bases478

C Globalization480

D Crystal graphs and tensor products481

14.2 Lusztig's canonical bases486

A The algebraic construction486

B The topological construction488

C Some combinatorial formulas490

Bibliographical notes492

15 Quantum group invariants of knots and 3-manifolds494

15.1 Knots and 3-manifolds:a quick review495

A From braids to links496

B From links to 3-manifolds502

15.2 Link invariants from quantum groups504

A Link invariants from R-matrices504

B Link invariants from vertex models510

15.3 Modular Hopf algebras and 3-manifold invariants517

A Modular Hopf algebras517

B The construction of 3-manifold invariants522

Bibliographical notes525

16 Quasi-Hopf algebras and the Knizhnik-Zamolodchikov equation527

16.1 Quasi-Hopf algebras528

A Definitions529

B An example from conformal field theory533

C Quasi-Hopf QUE algebras534

16.2 The Kohno-Drinfel'd monodromy theorem537

A Braid groups and configuration spaces537

B The Knizhnik-Zamolodchikov equation539

C The KZ equation and affine Lie algebras541

D Quantization and the KZ equation543

E The monodromy theorem549

16.3 Affine Lie algebras and quantum groups550

A The category O？551

B The tensor product552

C The equivalence theorem555

16.4 Quasi-Hopf algebras and Grothendieck's esquisse556

A Gal(？／？)and pro-finite fundamental groups557

B The Grothendieck-Teichmüller group and quasitriangular quasi-Hopf algebras559

Bibliographical notes560

Appendix Kac-Moody algebras562

A 1 Generalized Cartan matrices562

A 2 Kac-Moody algebras562

A 3 The invariant bilinear form563

A 4 Roots563

A 5 The Weyl group564

A 6 Root vectors565

A 7 Affine Lie algebras565

A 8 Highest weight modules566

References567

Index of notation638

General index643