The Dynamical Yang-baxter Equation,

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 andexplicit calculations, will be accessible to graduate students of mathematics, who are familiar with the basics of representation theory of semisimple Lie algebras.

Introduction

1

(14)

The quantum dynamical Yang-Baxter equation

1

(5)

The equation

1

(1)

Examples of solutions of QDYBE

1

(1)

The QDYBE with spectral parameter

2

(1)

Tensor category of representations

2

(1)

Gauge transformations and classification

3

(1)

Dynamical quantum groups

3

(1)

The classical dynamical Yang-Baxter equation

4

(1)

Examples of solutions of CDYBE

5

(1)

Classification of solutions for CDYBE

5

(1)

The fusion and exchange construction

6

(3)

Intertwining operators

6

(1)

The fusion and exchange operators

7

(1)

Fusion and exchange for quantum groups

7

(1)

The ABRR equation

8

(1)

The universal fusion operator

8

(1)

The dynamical twist equation

9

(1)

Traces of intertwiners and Macdonald functions

9

(6)

Trace functions

9

(1)

Commuting difference operators

10

(1)

Difference equations for the trace functions

10

(1)

Macdonald functions

10

(2)

Dynamical Weyl groups

12

(3)

Background material

15

(11)

Facts about sl2

15

(1)

Semisimple finite-dimensional Lie algebras and roots

16

(1)

Inner product on a simple Lie algebra

17

(1)

Chevalley generators

18

(1)

Representations of finite-dimensional semisimple Lie algebras

19

(2)

Irreducible highest weight modules; Shapovalov form

21

(5)

Intertwiners, fusion and exchange operators for Lie algebras

26

(14)

Intertwining operators

26

(1)

The fusion operator

27

(1)

The dynamical twist equation

28

(1)

The exchange operator

29

(5)

The ABRR equation

34

(4)

The universal fusion and exchange operators

38

(2)

Quantum groups

40

(21)

Hopf algebras

40

(1)

Representations of Hopf algebras

41

(1)

The quantum group Uq (sl2)

42

(1)

The quantum group Uq (g)

43

(1)

PBW for Uq (g)

44

(1)

The Hopf algebra structure on Uq (g)

45

(1)

Representation theory of Uq (g)

46

(1)

Formal version of quantum groups

47

(1)

Quasi-triangular Hopf algebras

48

(2)

Quasi-triangular Hopf algebras and representation theory

50

(3)

Quasi-triangularity and Uq (g)

53

(2)

Twisting

55

(2)

Quasi-classical limit for the QYBE

57

(1)

Quasi-classical limit for the QDYBE

58

(3)

Intertwiners, fusion and exchange operators for Uq (g)

61

(9)

Fusion operator for Uq (g)

61

(2)

Exchange operator for Uq (g)

63

(1)

The ABRR equation for Uq (g)

64

(1)

Quasi-classical limit for ABRR equation for Uq (g)

65

(5)

Dynamical R-matrices and integrable systems

70

(8)

Classical mechanics vs. quantum mechanics

70

(1)

Transfer matrix construction

71

(1)

Dynamical transfer matrix construction

72

(6)

Traces of intertwiners for Uq (g)

78

(36)

Generalized Macdonald-Ruijsenaars operators

78

(2)

Construction of FV (λ, μ)

80

(1)

Quantum spin Calogero-Moser Hamiltonian

80

(5)

FV(λ, μ) for sl2

85

(3)

Center of Uq (g) and quantum traces

88

(4)

The functions Zv and Xv

92

(5)

The function G

97

(10)

Macdonald-Ruijsenaars equations

107

(1)

Dual Macdonald-Ruijsenaars equations

107

(5)

The symmetry identity

112

(2)

Traces of intertwiners and Macdonald polynomials

114

(13)

Macdonald polynomials

114

(4)

Vector-valued characters

118

(9)

Dynamical Weyl group

127

(8)

Dynamical Weyl group (for g = sl2)

127

(4)

Dynamical Weyl group (for any finite-dim. simple g)

131

(4)

References

135

(3)

Index

138

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The Dynamical Yang-baxter Equation, Representation Theory, And Quantum Integrable Systems

Summary

Table of Contents

Supplemental Materials