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.