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On the integration of Manin pairs

David Li-Bland, Eckhard Meinrenken

TL;DR

The paper unifies integration results across Poisson, Dirac, and quasi-structures by formulating Manin pairs $(\mathbb{E},A)$ within the VB- and CA-groupoid framework. It constructs a canonical, multiplicative morphism $R:(\mathbb{T}G,TG)\dasharrow (\mathbb{E}\times\overline{\mathbb{E}},A\times A)$ encoding the integrated multiplicative geometry, and it provides an equivariant, non-simply connected extension of the theory with a clear Hamiltonian-spaces interpretation. It recovers Mackenzie–Xu integration results for Lie bialgebroids, extends to quasi-Lie settings, and describes quasi-symplectic groupoids via exact Courant algebroids, all within a single geometric paradigm. The framework yields practical tools for Hamiltonian actions and Morita theory in Dirac/quasi settings, clarifying when integrability descends to quotients and how moment maps interact with multiplicative structures.

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.

On the integration of Manin pairs

TL;DR

The paper unifies integration results across Poisson, Dirac, and quasi-structures by formulating Manin pairs within the VB- and CA-groupoid framework. It constructs a canonical, multiplicative morphism encoding the integrated multiplicative geometry, and it provides an equivariant, non-simply connected extension of the theory with a clear Hamiltonian-spaces interpretation. It recovers Mackenzie–Xu integration results for Lie bialgebroids, extends to quasi-Lie settings, and describes quasi-symplectic groupoids via exact Courant algebroids, all within a single geometric paradigm. The framework yields practical tools for Hamiltonian actions and Morita theory in Dirac/quasi settings, clarifying when integrability descends to quotients and how moment maps interact with multiplicative structures.

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 : The source-simply connected Lie groupoid integrating 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 , thereby obtaining a simple geometric approach to these integration results. We also clarify the case where the groupoid integrating is not source-simply connected. Furthermore, we obtain a description of Hamiltonian spaces for Poisson groupoids and quasi-symplectic groupoids within this formalism.

Paper Structure

This paper contains 26 sections, 23 theorems, 161 equations.

Table of Contents

  1. Introduction
  2. $\mathcal{VB}$-groupoids
  3. Basic definitions
  4. Left and right actions
  5. The fat groupoid
  6. A canonical comorphism
  7. Integration of Manin pairs
  8. Manin pairs
  9. Multiplicative Manin pairs
  10. Equivariant Manin pairs
  11. Integration of Manin pairs
  12. Infinitesimally multiplicative version
  13. Special cases and applications
  14. The case $M=\operatorname{pt}$
  15. Lie group actions
  16. ...and 11 more sections

Key Result

Proposition 2.2

For every $\mathcal{VB}$-groupoid $J\rightrightarrows B$, with ${\operatorname{core}}(J)=C$, there is a canonical $\widehat{G}\times \widehat{G}$-equivariant multiplicative $\mathcal{VB}$-comorphism with base map $(\mathsf{t},\mathsf{s})\colon G\to \operatorname{Pair}(M)$. Explicitly, where $g$ is the base point of $x\in J$. If $J$ is an $\mathcal{LA}$-groupoid, then $S$ is a Lie algebroid comor

Theorems & Definitions (60)

  • Example 2.1
  • Proposition 2.2
  • proof
  • Example 2.3
  • Definition 3.1
  • Example 3.2
  • Lemma 3.3
  • proof
  • Definition 3.4
  • Proposition 3.5
  • ...and 50 more