On classification of dynamical r-matrices

Using the gauge transformations of the Classical Dynamical YangBaxter Equation introduced by P. Etingof and A. Varchenko in [EV], we reduce the classification of dynamical r-matrices on a commutative subalgebra l of a Lie algebra g to a purely algebraic problem under the assumption that l admits a g-invariant complement, where g is the centralizer of l. We then describe, for a simple complex Lie algebra g, all non skew-symmetric dynamical r-matrices on a commutative subalgebra l ⊂ g which contains a regular semisimple element. This interpolates results of P. Etingof and A. Varchenko (when l 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. Acknowledgments: I heartily thank Pavel Etingof for his constant encouragements and his great, communicative enthousiasm for mathematics. I am also grateful to A. Varchenko and P. Etingof for sharing with me their work before publication, and Hung Yean Loke, Vadik Vologodsky for interesting discussions. 1 The Classical Yang-Baxter Equation Let g be a Lie algebra. The CYBE is the following algebraic equation for an element r ∈ g ⊗ g: [r, r] + [r, r] + [r, r] = 0. (1) Solutions of this equation are called r-matrices. In the theory of quantum groups, one is mainly interested in r-matrices satisfying r + r ∈ (Sg). (2) See [CP] for the links with the theory of quantum groups, and [Che] for links with Conformal Field Theory and the Wess-Zumino-Witten model on P. The geometric interpretation of the CYBE was given by Drinfeld in terms of PoissonLie groups ([Dr1]).