Twisted traces of intertwiners for Kac-Moody algebras and classical dynamical r-matrices corresponding to generalized Belavin-Drinfeld triples

Abstract

In early eighties, Belavin and Drinfeld showed that nonskewsymmetric classical r-matrices for simple Lie algebras are classified by combinatorial objects which are now called Belavin-Drinfeld triples. Later the second author of the present paper generalized this result to the case of dynamical r-matrices, and showed that they correspond to generalized Belavin-Drinfeld triples. The dynamical r-matrix corresponding a triple is given by a certain explicit formula, which works for any symmetrizable Kac-Moody algebra. In special cases, this formula gives the Felder and the Belavin r-matrix. In 1994, Felder associated with every dynamical r-matrix a system of differential equations called the KZB equations. In the case of the Felder and Belavin r-matrix, solutions of these equations are known to have a representation theoretical presentation, as conformal blocks (i.e. traces of products of intertwining operators between representations of Lie algebras). In this paper, we propose such a presentation for ANY dynamical r-matrix given by the second author's formula. In particular, the paper sheds light on the Belavin-Drinfeld classical r-matrices of 1984. This paper begins a series of papers. In the next paper, joint with T.Schedler, we will give an explicit quantization of the dynamical r-matrices for simple Lie algebras (in particular, Belavin-Drinfeld r-matrices), which has been an open problem for a number of years. In another paper, we plan to generalize the results of the present paper to quantum groups. This should yield quantum KZB equations which involve quantum R-matrices obtained by quantizing the classical r-matrices that appear in this paper.