Reduced Poisson actions

Abstract

A sufficient and necessary condition is given for the action of the quotient of a Poisson-Lie group G on the quotient of a Poisson G -space P to be a Poisson action, where both the Poisson structures on the quotient group and the quotient manifold are induced by Dirac structures. The left invariant Dirac structure and the left invariant tensor descriptions of Poisson homogeneous spaces are proved to be equivalent.