The text is based on an established graduate course given at MIT that provides an introduction to the theory of the dynamical Yang-Baxter equation and its applications, which is an important area in representation theory and quantum groups. The book, which contains many detailed proofs and explicit calculations, will be accessible to graduate students of mathematics, who are familiar with the basics of representation theory of semi-simple Lie algebras.
Author(s): Pavel Etingof, Frederic Latour
Series: Oxford Lecture Series in Mathematics and Its Applications
Edition: illustrated edition
Publisher: Oxford University Press, USA
Year: 2005
Language: English
Pages: 151
Contents......Page 10
1.1.2 Examples of solutions of QDYBE......Page 14
1.1.4 Tensor category of representations......Page 15
1.1.6 Dynamical quantum groups......Page 16
1.1.7 The classical dynamical Yang–Baxter equation......Page 17
1.1.9 Classification of solutions for CDYBE......Page 18
1.2.1 Intertwining operators......Page 19
1.2.3 Fusion and exchange for quantum groups......Page 20
1.2.5 The universal fusion operator......Page 21
1.3.1 Trace functions......Page 22
1.3.4 Macdonald functions......Page 23
1.3.5 Dynamical Weyl groups......Page 25
2.1 Facts about sl[sub(2)]......Page 28
2.2 Semisimple finite-dimensional Lie algebras and roots......Page 29
2.3 Inner product on a simple Lie algebra......Page 30
2.4 Chevalley generators......Page 31
2.5 Representations of finite-dimensional semisimple Lie algebras......Page 32
2.6 Irreducible highest weight modules; Shapovalov form......Page 34
3.1 Intertwining operators......Page 39
3.2 The fusion operator......Page 40
3.3 The dynamical twist equation......Page 41
3.4 The exchange operator......Page 42
3.5 The ABRR equation......Page 47
3.6 The universal fusion and exchange operators......Page 51
4.1 Hopf algebras......Page 53
4.2 Representations of Hopf algebras......Page 54
4.3 The quantum group U[sub(q)](sl[sub(2)])......Page 55
4.4 The quantum group U[sub(q)](g)......Page 56
4.5 PBW for U[sub(q)](g)......Page 57
4.6 The Hopf algebra structure on U[sub(q)](g)......Page 58
4.7 Representation theory of U[sub(q)](g)......Page 59
4.8 Formal version of quantum groups......Page 60
4.9 Quasi-triangular Hopf algebras......Page 61
4.10 Quasi-triangular Hopf algebras and representation theory......Page 63
4.11 Quasi-triangularity and U[sub(q)](g)......Page 66
4.12 Twisting......Page 68
4.13 Quasi-classical limit for the QYBE......Page 70
4.14 Quasi-classical limit for the QDYBE......Page 71
5.1 Fusion operator for U[sub(q)](g)......Page 74
5.2 Exchange operator for U[sub(q)](g)......Page 76
5.3 The ABRR equation for U[sub(q)](g)......Page 77
5.4 Quasi-classical limit for ABRR equation for U[sub(q)](g)......Page 78
6.1 Classical mechanics vs. quantum mechanics......Page 83
6.2 Transfer matrix construction......Page 84
6.3 Dynamical transfer matrix construction......Page 85
7.1 Generalized Macdonald–Ruijsenaars operators......Page 91
7.3 Quantum spin Calogero–Moser Hamiltonian......Page 93
7.4 F[sub(V)] (λ, μ) for sl[sub(2)]......Page 98
7.5 Center of U[sub(q)](g) and quantum traces......Page 101
7.6 The functions Z[sub(V)] and X[sub(V)]......Page 105
7.7 The function G......Page 110
7.9 Dual Macdonald–Ruijsenaars equations......Page 120
7.10 The symmetry identity......Page 125
8.1 Macdonald polynomials......Page 127
8.2 Vector-valued characters......Page 131
9.1 Dynamical Weyl group (for g = sl[sub(2)])......Page 140
9.2 Dynamical Weyl group (for any finite-dim. simple g)......Page 144
References......Page 148
Index......Page 151
