Lie Algebroids Associated to Poisson Actions

Abstract

This work is motivated by a result of Drinfeld on Poisson homogeneous spaces. For each Poisson manifold $P$ with a Poisson action by a Poisson Lie group $G$, we describe a Lie algebroid structure on the direct sum vector bundle $P\times \mathfrak{g}\oplus T^{\ast }P$, where $\mathfrak{g}$ is the Lie algebra of $G$. It is built out of the transformation Lie algebroid $P\times \mathfrak{g}$ and the cotangent bundle Lie algebroid $T^{\ast }P$ together with a pair of representations of them on each other. When the action of $G$ on $P$ is transitive, the kernel of the anchor map of this Lie algebroid gives a Lie algebra bundle over $P$, the fibers of which are given by Drinfeld. As applications, we describe the symplectic leaves and the $G$-invariant Poisson cohomology of Poisson homogeneous $G$-spaces.

35 pages

References in corpus (1)

• Poisson homogeneous spaces

Cited by in corpus (3)

• L-infinity algebra actions
• Manin Triples for Lie Bialgebroids
• Anchored vector bundles and algebroids