Lectures on the dynamical Yang-Baxter equations

Abstract

This paper contains a systematic and elementary introduction to a new area of the theory of quantum groups -- the theory of the classical and quantum dynamical Yang-Baxter equations. It arose from a minicourse given by the first author at MIT in the Spring of 1999, when the second author extended and improved his lecture notes of this minicourse. The quantum dynamical Yang-Baxter equation is a generalization of the ordinary quantum Yang-Baxter equation, considered in a physical context by Gervais and Neveu, and later from a mathematical viewpoint by Felder. Felder attached to every solution of this equation a quantum group, and also considered the classical analogue of the quantum dynamical Yang-Baxter equation -- the classical dynamical Yang-Baxter equation. Since then, the theory of dynamical Yang-Baxter equations and the corresponding quantum groups was systematically developed in many papers. By now, this theory has many applications, in particular to integrable systems and representation theory. The goal of this paper is to discuss this theory and some of its applications.

39 pages, latex; in the revised version, misprints found by T.Koornwinder were corrected; two new appendices were added, and the review of literature was updated

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Cited by in corpus (40)

• Gauge Theory and Integrability, II
• Finite Quantum Groupoids and Their Applications
• Mickelsson algebras and Zhelobenko operators
• Chiral Extensions of the WZNW Phase Space, Poisson-Lie Symmetries and Groupoids
• Characteristic Classes of SL(N)-Bundles and Quantum Dynamical Elliptic R-Matrices
• Spectral extension of the quantum group cotangent bundle
• Characteristic Classes and Integrable Systems. General Construction
• A noncommutative weight-dependent generalization of the binomial theorem
• Generalizations of Felder's elliptic dynamical r-matrices associated with twisted loop algebras of self-dual Lie algebras
• Elliptic Dynamical R-Matrices from the Monodromy of the q-Knizhnik-Zamolodchikov Equations for the Standard Representation of Uq(sl(n+1))
• A note on a canonical dynamical r-matrix
• On the Chiral WZNW Phase Space, Exchange r-Matrices and Poisson-Lie Groupoids
• Algebraic Topology Foundations of Supersymmetry and Symmetry Breaking in Quantum Field Theory and Quantum Gravity: A Review
• Orthogonal basis for the Shapovalov form on $A_n$
• Gauge fixing and classical dynamical r-matrices in ISO(2,1)-Chern-Simons theory
• On a Poisson-Lie analogue of the classical dynamical Yang-Baxter equation for self-dual Lie algebras
• Diagonal reduction algebra and reflection equation
• Classical dynamical r-matrices, Poisson homogeneous spaces, and Lagrangian subalgebras
• Dynamical r-matrices and the chiral WZNW phase space
• On the structure of weak Hopf algebras
• Dynamical quantum groups at roots of 1
• Chern-Simons Theory with Sources and Dynamical Quantum Groups I: Canonical Analysis and Algebraic Structures
• The non-Abelian momentum map for Poisson-Lie symmetries on the chiral WZNW phase space
• Contravariant form for reduction algebras and Pieri rule
• Dynamical Yang-Baxter equations, quasi-Poisson homogeneous spaces, and quantization
• Differential calculus on $\mathbf{h}$-deformed spaces
• Felder's elliptic quantum group and elliptic hypergeometric series on the root system A_n
• Classification of Non-Affine Non-Hecke Dynamical R-Matrices
• Dynamical r-matrices and Poisson-Lie symmetries in the chiral WZNW model
• On the quantum Kazhdan-Lusztig functor
• Fusion and exchange matrices for quantized sl(2) and associated q-special functions
• Dynamical Yang-Baxter equation and quantum vector bundles
• Solutions of Trigonometric KZ Equations satisfy Dynamical Difference Equations
• Quantum groupoids and dynamical categories
• Vertex-IRF transformations and quantization of dynamical r-matrices
• Dynamical twists in group algebras
• Pairings and actions for dynamical quantum groups
• Quantum Groups for Restricted SOS Models
• Dynamical quantum determinants and Pfaffians
• On Calogero-Francoise-type Lax matrices and their dynamical r-matrices