GEOMETRY AND CLASSIFICATION OF SOLUTIONS OF THE CLASSICAL DYNAMICAL YANG-BAXTER EQUATION

The classical Yang-Baxter equation(CYBE) is an algebraic equation cent ral in the theory of integrable systems. Its nondegenerate solutions w ere classified by Belavin and Drinfeld. Quantization of CYBE led to th e theory of quantum groups. A geometric interpretation of CYBE was giv en by Drinfeld and gave rise to the theory of Poisson-Lie groups, The classical dynamical Yang-Baxter equation (CDYBE) is an important diffe rential equation analogous to CYBE and introduced by Felder as the con sistency condition for the differential Knizhnik-Zamolodchikov-Bernard equations for correlation functions in conformal held theory on tori, Quantization of CDYBE allowed Felder to introduce an interesting elli ptic analog of quantum groups. It becomes clear that numerous importan t notions and results connected with CYBE have dynamical analogs. In t his paper we classify solutions to CDYBE and give geometric interpreta tion to CDYBE. The classification and interpretation are remarkably an alogous to the Belavin-Drinfeld picture.