Classical dynamical r-matrices, Poisson homogeneous spaces, and Lagrangian subalgebras

Abstract

Jiang-Hua Lu showed that any dynamical r-matrix for the pair $(g,u)$ naturally induces a Poisson homogeneous structure on $G/U$. She also proved that if $g$ is complex simple, $u$ is its Cartan subalgebra and $r$ is quasitriangular, then this correspondence is in fact 1-1. In the present paper we find some general conditions under which the Lu correspondence is 1-1. Then we apply this result to describe all triangular Poisson homogeneous structures on $G/U$ for a simple complex group $G$ and its reductive subgroup $U$ containing a Cartan subgroup.