On the Moduli Space of Classical Dynamical r-matrices

Introduction. A classical dynamical r-matrix is an l-equivariant function r : l∗ → g ⊗ g (where l, g are Lie algebras), such that r21 + r = Ω is g-invariant, which satisfies the classical dynamical Yang-Baxter equation (CDYBE). CDYBE is a differential equation, which generalizes the usual classical Yang-Baxter equation. It was introduced in 1994 by G.Felder [Fe], in the context of conformal field theory. Solutions of CDYBE and their quantizations appear naturally in several mathematical theories: the theory of integrable systems, special functions, representation theory (see [ES] for a review). Since classical dynamical r-matrices were introduced, several authors tried to study and classify them ([EV],[S],[Xu]). The goal of this paper is to describe the moduli space of classical dynamical r-matrices modulo gauge transformations. In particular, we improve and generalize the results of [EV], [S], as well as correct some errors that occurred in these papers (See remarks 3 and 5). The main achievement of this paper, compared to the previous ones, is that its results are valid for dynamical r-matrices for a nonabelian Lie algebra l. It turns out that this generalization not only brings in new interesting examples (see [EV],[AM]) but also makes the general theory much more clear and natural. The composition of the paper is as follows. In Section 1, we recall the definition of a dynamical r-matrix. In Section 2, we extend to the nonabelian case the notion of a gauge transformation of dynamical r-matrices, introduced in [EV]. In Section 3, we decribe the space of dynamical r-matrices modulo gauge transformations (the moduli space). Here we formulate our main theorem, stating that under some technical conditions, the moduli space can be identified with a certain explicitly given affine variety. For instance, if l = g, this variety consists of one point, which is the Alekseev-Meinrenken solution [AM] (for semisimple Lie algebras, it was also constructed in [EV]).