Classical dynamical r-matrices and homogeneous Poisson structures on G/H and K/T

Abstract

Let G be a finite dimensional simple complex group equipped with the standard Poisson Lie group structure. We show that all G-homogeneous (holomorphic) Poisson structures on G/H, where H ⊂ G is a Cartan subgroup, come from solutions to the Classical Dynamical Yang-Baxter equations which are classified by Etingof and Varchenko. A similar result holds for a maximal compact subgroup K, and we get a family of K-homogeneous Poisson structures on K/T, where T = K ∩ H is a maximal torus of K. This family exhausts all K-homogeneous Poisson structures on K/T up to isomorphisms. We study some Poisson geometrical properties of members of this family such as their symplectic leaves, their modular classes, and the moment maps for the T-action.