Poisson Geometry Related to Atiyah Sequences

TL;DR

This paper explores the structure of Poisson groupoids related to Atiyah sequences of principal bundles, revealing insights into symplectic leaves and applications in Hamiltonian mechanics.

Contribution

It constructs a canonical short exact sequence of Poisson VB-groupoids linked to Atiyah sequences, advancing the understanding of Poisson geometry in principal bundles.

Findings

Description of symplectic leaves of the Poisson groupoid

Construction of a Poisson VB-groupoid sequence related to Atiyah sequences

Discussion of the semidirect product case in Hamiltonian mechanics

Abstract

We construct and investigate a short exact sequence of Poisson $\mathcal{VB}$-groupoids which is canonically related to the Atiyah sequence of a $G$-principal bundle $P$. Our results include a description of the structure of the symplectic leaves of the Poisson groupoid $\frac{T^*P\times T^*P}{G}\rightrightarrows \frac{T^*P}{G}$. The semidirect product case, which is important for applications in Hamiltonian mechanics, is also discussed.