On classification of dynamical r-matrices

Abstract

Using the gauge transformations of the Classical Dynamical Yang-Baxter Equation introduced by P. Etingof and A. Varchenko in [EV], we reduce the classification of dynamical r-matrices r on a commutative subalgebra t of a Lie algebra g to a purely algebraic problem, under some assumption on the symmetric part of r. We then describe, for a simple complex Lie algebra g, all non skew-symmetric dynamical r-matrices on a commutative subalgebra t ⊂ g which contains a regular semisimple element. This interpolates results of P. Etingof and A. Varchenko ([EV], when t is a Cartan subalgebra) and results of A. Belavin and V. Drinfeld for constant r-matrices ([BD]). This classification is similar, and in some sense simpler than the Belavin-Drinfeld classification.