On the integration of Manin pairs

TL;DR

This paper presents a unified geometric framework for integrating Manin pairs, elucidating how Poisson and Dirac structures relate to symplectic groupoids, and clarifies the role of source-connectedness in these integrations.

Contribution

It introduces a general approach to integrating Manin pairs, extending known results for Poisson and Dirac structures, and describes Hamiltonian spaces within this formalism.

Findings

Unified geometric approach to Manin pairs and integration results

Clarification of non-source-simply connected groupoids

Description of Hamiltonian spaces for Poisson and quasi-symplectic groupoids

Abstract

It is a remarkable fact that the integrability of a Poisson manifold to a symplectic groupoid depends only on the integrability of its cotangent Lie algebroid $A$: The source-simply connected Lie groupoid $G\rightrightarrows M$ integrating $A$ automatically acquires a multiplicative symplectic 2-form. More generally, a similar result holds for the integration of Lie bialgebroids to Poisson groupoids, as well as in the `quasi' settings of Dirac structures and quasi-Lie bialgebroids. In this article, we will place these results into a general context of Manin pairs $(\mathbb{E},A)$, thereby obtaining a simple geometric approach to these integration results. We also clarify the case where the groupoid $G$ integrating $A$ is not source-simply connected. Furthermore, we obtain a description of Hamiltonian spaces for Poisson groupoids and quasi-symplectic groupoids within this formalism.