ON POISSON GROUPOIDS

Some important properties of Poisson groupoids are discussed. In particular, we obtain a useful formula for the Poisson tenser of an arbitrary Poisson groupoid, which generalizes the well-known multiplicativity condition for Poisson groups. Morphisms between Poisson groupoids and between Lie bialgebroids are also discussed. In particular, for a special class of Lie bialgebroid morphisms, we give an explicit lifting construction. As an application, we prove that a Poisson group action on a Poisson manifold lifts to a. Poisson action on its cu-simply connected symplectic groupoid.