Universal lifting theorem and quasi-Poisson groupoids

Abstract

We prove the universal lifting theorem: for an \alpha -simply connected and \alpha -connected Lie groupoid \Gamma with Lie algebroid A , the graded Lie algebra of multi-differentials on A is isomorphic to that of multiplicative multi-vector fields on \Gamma . As a consequence, we obtain the integration theorem for a quasi-Lie bialgebroid, which generalizes various integration theorems in the literature in special cases.The second goal of the paper is the study of basic properties of quasi-Poisson groupoids. In particular, we prove that a group pair (D, G) associated to a Manin quasi-triple (\mathfrak d, \mathfrak g, \mathfrak h) induces a quasi-Poisson groupoid on the transformation groupoid G\times D/G\rightrightarrows D/G . Its momentum map corresponds exactly with the D/G -momentum map of Alekseev and Kosmann-Schwarzbach.