Coadjoint orbits of Lie groupoids.

For a Lie groupoid 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 J G ⋉ T ∗ M on A ∗ , which coincide with the groupoid orbits of the symplectic groupoid T ∗ 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. [ABSTRACT FROM AUTHOR]