Explicit Quantization of Dynamical R-matrices for Finite Dimensional Semisimple Lie Algebras

1.1. Classical r-matrices. In the early eighties, Belavin and Drinfeld [BD] classified nonskewsymmetric classical r-matrices for simple Lie algebras. It turned out that such r-matrices, up to isomorphism and twisting by elements from the exterior square of the Cartan subalgebra, are classified by combinatorial objects which are now called Belavin-Drinfeld triples. By definition, a Belavin-Drinfeld triple for a simple Lie algebra g is a triple (Γ1,Γ2, T ), where Γ1,Γ2 are subsets of the Dynkin diagram Γ of g, and T : Γ1 → Γ2 is an isomorphism which preserves the inner product and satisfies the nilpotency condition: if α ∈ Γ1, then there exists k such that T k−1(α) ∈ Γ1 but T (α) / ∈ Γ1. The r-matrix corresponding to such a triple is given by a certain explicit formula. These results generalize in a straightforward way to semisimple Lie algebras. In [S], the third author generalized the work of Belavin and Drinfeld and classified classical nonskewsymmetric dynamical r-matrices for simple Lie algebras. It turns out that they have an even simpler classification: up to gauge transformations, they are classified by generalized Belavin-Drinfeld triples, which are defined as the usual Belavin-Drinfeld triples but without any nilpotency condition. The dynamical rmatrix corresponding to such a triple is given by a certain explicit formula. As before, these results can be generalized to semisimple Lie algebras.