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

Lu has shown 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 one- toone. In this Letter we nd some general conditions under which the Lu correspondence is one- to- one. 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.