A Guide to Quantum Groups

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Paperback | 25 September 1995

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Since they first arose in the 1970s and early 1980s, quantum groups have proved to be of great interest to mathematicians and theoretical physicists. The theory of quantum groups is now well established as a fascinating chapter of representation theory, and has thrown new light on many different topics, notably low-dimensional topology and conformal field theory. The goal of this book is to give a comprehensive view of quantum groups and their applications. The authors build on a self-contained account of the foundations of the subject and go on to treat the more advanced aspects concisely and with detailed references to the literature. Thus this book can serve both as an introduction for the newcomer, and as a guide for the more experienced reader. All who have an interest in the subject will welcome this unique treatment of quantum groups.

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"An ideal guide for both experts and newcomers." American Mathematical Monthly

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Non-Fiction Mathematics Groups & Group Theory Science Physics Quantum Physics & Quantum Mechanics & Quantum Field Theory Geometry Algebra

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

A Guide to Quantum Groups

At a Glance

Paperback

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