Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Reduction of symplectic groupoids and quotients of quasi-Poisson manifolds

D. Álvarez

Abstract

In this work we study the integrability of quotients of quasi-Poisson manifolds. Our approach allows us to put several classical results about the integrability of Poisson quotients in a common framework. By categorifying one of the already known methods of reducing symplectic groupoids we also describe double symplectic groupoids which integrate the recently introduced Poisson groupoid structures on gauge groupoids.

Reduction of symplectic groupoids and quotients of quasi-Poisson manifolds

Abstract

In this work we study the integrability of quotients of quasi-Poisson manifolds. Our approach allows us to put several classical results about the integrability of Poisson quotients in a common framework. By categorifying one of the already known methods of reducing symplectic groupoids we also describe double symplectic groupoids which integrate the recently introduced Poisson groupoid structures on gauge groupoids.

Paper Structure

This paper contains 12 sections, 8 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lie groupoids and Lie algebroids
  4. Poisson structures, closed IM 2-forms and symplectic groupoids
  5. Integrability of quotients of quasi-Poisson manifolds
  6. Poisson structures and quasi-Poisson manifolds
  7. The integrability criterion
  8. Integrability of quotients of Poisson actions
  9. Integrability of quotients of q-Poisson $G$-manifolds
  10. Double symplectic groupoids and gauge Poisson groupoids
  11. Double symplectic groupoids
  12. Integrability of gauge Poisson groupoids in the complete case

Key Result

Theorem 1.1

The Poisson manifold $(S/G,\sigma)$ is integrable if and only if the Lie algebroid $C$ is integrable. Moreover, if $\mathcal{G} (C)$ is the source-simply-connected integration of $C$, then there is a lifted $G$-action on $\mathcal{G} (C)$ such that the orbit space $\mathcal{G} (C)/G$ is a symplectic

Theorems & Definitions (33)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 3.1: roycou
  • Example 3.2
  • Definition 3.3: quapoi
  • Example 3.4
  • Example 3.5
  • Proposition 3.6
  • proof
  • Remark 3.7
  • ...and 23 more