Geometry and classification of solutions of the Classical Dynamical Yang-Baxter Equation

Abstract

The classical Yang-Baxter equation (CYBE) is an algebraic equation central in the theory of integrable systems. Its solutions were classified by Belavin and Drinfeld. Quantization of CYBE led to the theory of quantum groups. A geometric interpretation of CDYB was given by Drinfeld and gave rise to the theory of Poisson-Lie groups. The classical dynamical Yang-Baxter equation (CDYBE) is an important differential equation analagous to CYBE and introduced by Felder as the consistency condition for the Knizhnik-Zamolodchikov-Bernard equations for correlation functions in conformal field theory on tori. Quantization of CDYBE allowed Felder to introduce an interesting elliptic analog of quantum groups. It becomes clear that numerous important notions and results connected with CYBE have dynamical analogs. In this paper we classify solutions to CDYBE and give geometric interpretation to CDYBE. The classification and interpretation are remarkably analogous to the Belavin-Drinfeld picture.