1 8 M ay 2 00 0 Lectures on the dynamical Yang - Baxter equations

This paper 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. It 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. The quantum dynamical Yang-Baxter equation is a generalization of the ordinary quantum Yang-Baxter equation. It first appeared in physical literature in the work of Gervais and Neveu [GN], and was first considered from a mathematical viewpoint by Felder [F], who attached to every solution of this equation a quantum group, and an interesting system of difference equations, the quantum Knizhnik-Zamolodchikov-Bernard (qKZB) equation. Felder also considered the classical analogue of the quantum dynamical Yang-Baxter equation – the classical dynamical Yang-Baxter equation. Since then, this theory was systematically developed in many papers, some of which are listed below. By now, the theory of the classical and quantum dynamical Yang-Baxter equations and their solutions has many applications, in particular to integrable systems and representation theory. To discuss this theory and some of its applications is the goal of this paper. The structure of the paper is as follows. In Section 2 we consider the exchange construction, which is a natural construction in classical representation theory that leads one to discover the quantum dynamical Yang-Baxter equation and interesting solutions of this equation (dynamical R-matrices). In this section we define the main objects of the paper – the fusion and exchange matrices for Lie algebras and quantum groups, and compute them for the Lie algebra sl2 and quantum group Uq(sl2). In Section 3 we define the quantum dynamical Yang-Baxter equation, and see that the exchange matrices are solutions of this equation. We also study the quasiclassical limit of the quantum dynamical Yang-Baxter equation – the classical dynamical Yang-Baxter equation. We conjecture that any solution of this equation can be quantized. We compute classical limits of exchange matrices, which provides interesting examples of solutions of the classical dynamical Yang-Baxter equation, which we call basic solutions. In Section 4 we give a classification of solutions of the classical dynamical Yang-Baxter equation for simple Lie algebras defined on a Cartan subalgebra, satisfying the unitarity condition. The result is, roughly, that all such solutions can be obtained from the basic solutions.