Universal lifting theorem and quasi-Poisson groupoids

Abstract

We prove the universal lifting theorem: for an α-simply connected and α-connected Lie groupoid with Lie algebroid A, the graded Lie algebra of multi-differentials on A is isomorphic to that of multiplicative multi-vector fields on . 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 (d, g, h) induces a quasi-Poisson groupoid on the transformation groupoid G ×D/G ⇉ D/G. Its momentum map corresponds exactly with the D/G-momentum map of Alekseev and Kosmann-Schwarzbach.