Lectures on the dynamical Yang-Baxter equations

TL;DR

This paper provides a systematic introduction to the classical and quantum dynamical Yang-Baxter equations, highlighting their development, mathematical structure, and applications in integrable systems and representation theory.

Contribution

It offers an elementary overview of the dynamical Yang-Baxter equations and discusses recent developments and applications in quantum groups and integrable systems.

Findings

Development of the theory of dynamical Yang-Baxter equations

Connection to quantum groups and integrable systems

Applications in representation theory

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.