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Reduction of Cosymplectic groupoids by cosymplectic moment maps

Daniel López Garcia, Nicolas Martinez Alba

TL;DR

This work extends cosymplectic reduction to the setting of Lie groupoids, introducing cosymplectic moment maps that are groupoid morphisms and proving a Marsden–Weinstein type reduction theorem. It shows that, under free and proper actions and clean zero level, the reduced groupoid inherits a cosymplectic structure and that the infinitesimal reduction matches the reduction of the integrated groupoid. Moreover, cosymplectic reduction induces a compatible symplectic reduction on the symplectic leaf containing the units, and the reduced structure preserves the multiplicative Chern class under suitable hypotheses. The results bridge cosymplectic and symplectic reductions, clarify the behavior of infinitesimal data, and propose avenues for extending reduction theory to broader geometric contexts and higher structures.

Abstract

The Marsden-Weinstein-Meyer symplectic reduction has an analogous version for cosymplectic manifolds. In this paper we extend this cosymplectic reduction to the context of groupoids. Moreover, we prove how in the case of an algebroid associated to a cosymplectic groupoid, the integration commutes with the reduction (analogously to what happens in Poisson geometry). On the other hand, we show how the cosymplectic reduction of a groupoid induces a symplectic reduction on a canonical symplectic subgroupoid. Finally, we study what happens to the multiplicative Chern class associated with the $S^1$-central extensions of the reduced groupoid.

Reduction of Cosymplectic groupoids by cosymplectic moment maps

TL;DR

This work extends cosymplectic reduction to the setting of Lie groupoids, introducing cosymplectic moment maps that are groupoid morphisms and proving a Marsden–Weinstein type reduction theorem. It shows that, under free and proper actions and clean zero level, the reduced groupoid inherits a cosymplectic structure and that the infinitesimal reduction matches the reduction of the integrated groupoid. Moreover, cosymplectic reduction induces a compatible symplectic reduction on the symplectic leaf containing the units, and the reduced structure preserves the multiplicative Chern class under suitable hypotheses. The results bridge cosymplectic and symplectic reductions, clarify the behavior of infinitesimal data, and propose avenues for extending reduction theory to broader geometric contexts and higher structures.

Abstract

The Marsden-Weinstein-Meyer symplectic reduction has an analogous version for cosymplectic manifolds. In this paper we extend this cosymplectic reduction to the context of groupoids. Moreover, we prove how in the case of an algebroid associated to a cosymplectic groupoid, the integration commutes with the reduction (analogously to what happens in Poisson geometry). On the other hand, we show how the cosymplectic reduction of a groupoid induces a symplectic reduction on a canonical symplectic subgroupoid. Finally, we study what happens to the multiplicative Chern class associated with the -central extensions of the reduced groupoid.
Paper Structure (14 sections, 22 theorems, 40 equations)

This paper contains 14 sections, 22 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Cosymplectic structures
  4. Symmetries and reduction
  5. Oversymplectic groupoids
  6. Symmetries and reduction of oversymplectic groupoids
  7. Cosymplectic groupoids
  8. The infinitesimal counterpart
  9. Symmetries and reduction of cosymplectic groupoids
  10. Consequences of the main theorem
  11. Symmetries and reduction at the infinitesimal level
  12. Reduction of the symplectic leaf
  13. Reduction of the multiplicative Chern class
  14. Further direction of study

Key Result

Theorem 2.4

Let $G$ be a Lie group acting on a cosymplectic manifold $(Q, \omega, \eta)$ by cosymplectic automorphisms. Let $J:Q\to \mathfrak{g}^*$ be a cosymplectic moment map such that $\xi(\langle J,v\rangle)=0$ for all $v\in \mathfrak{g}$. If $0$ is a clean value of $J$, and $G$ acts freely and properly on where $i:J^{-1}(0)\to Q$ is the inclusion and $p:J^{-1}(0)\to Q_{red}$ is the projection.

Theorems & Definitions (48)

  • Remark 2.1
  • Example 2.2
  • Example 2.3
  • Theorem 2.4: Al
  • Remark 2.5
  • Proposition 2.6
  • proof
  • Proposition 2.7: Lemma 3.1 and Proposition 3.2 in BC2
  • Proposition 2.8: Lemma 4.1 and Proposition 4.3 in BC2
  • Proposition 2.9: Lemma 5.1 and Proposition 5.2 in BC2
  • ...and 38 more