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

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 $H \subset G$ 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 the maximal compact subgroup K, and we get a family of K-homogeneous Poisson structures on $K/T$, where $T = K \cap H$ 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

Hopf algebroids and quantum groupoids

We introduce the notion of Hopf algebroids, in which neither the total algebras nor the base algebras are required to be commutative. We give a class of Hopf algebroids associated to module algebras of the Drinfeld doubles of Hopf algebras when the $R$-matrices act properly. When this construction is applied to quantum groups, we get examples of quantum groupoids, which are semi-classical limits of Poisson groupoids. The example of quantum $sl(2)$ is worked out in details.Comment: 30 pages, in Late

On a Dimension Formula for Twisted Spherical Conjugacy Classes in Semisimple Algebraic Groups

Let $G$ be a connected semisimple algebraic group over an algebraically closed field of characteristic zero, and let $\th$ be an automorphism of $G$. We give a characterization of $\th$-twisted spherical conjugacy classes in $G$ by a formula for their dimensions in terms of certain elements in the Weyl group of $G$, generalizing a result of N. Cantarini, G. Carnovale, and M. Costantini when $\th$ is the identity automorphism. For $G$ simple and $\th$ an outer automorphism of $G$, we also classify the Weyl group elements that appear in the dimension formula.Comment: 8 page