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

by Etingof, Pavel; Latour, Frederic

• ISBN13: 9780198530688
• ISBN10: 0198530684
• Format: Hardcover
• Copyright: 2005-06-02
• Publisher: Oxford University Press

Summary

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.

Table of Contents

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