Coadjoint orbits of Lie groupoids

TL;DR

This paper characterizes symplectic leaves of the Lie-Poisson structure on the dual of a Lie algebroid as orbits of a specific affine coadjoint action of a Lie groupoid, revealing their fiber bundle structure and applications to gauge theories.

Contribution

It introduces a new realization of symplectic leaves as coadjoint orbits of a Lie groupoid, extending classical Lie theory to the setting of Lie groupoids and algebroids.

Findings

Symplectic leaves are orbits of an affine coadjoint action of a Lie groupoid.

Each symplectic leaf has a fiber bundle structure.

In gauge groupoids, leaves serve as phase spaces for particles in Yang-Mills fields.

Abstract

For a Lie groupoid $\mathcal{G}$ with Lie algebroid $A$, we realize the symplectic leaves of the Lie-Poisson structure on $A^*$ as orbits of the affine coadjoint action of the Lie groupoid $\mathcal{J}\mathcal{G}\ltimes T^*M$ on $A^*$, which coincide with the groupoid orbits of the symplectic groupoid $T^*\mathcal{G}$ over $A^*$. It is also shown that there is a fiber bundle structure on each symplectic leaf. In the case of gauge groupoids, a symplectic leaf is the universal phase space for a classical particle in a Yang-Mills field.