Foliated Lie and Courant Algebroids

Abstract

If $A$ is a Lie algebroid over a foliated manifold $(M,\mathcal{F})$, a foliation of $A$ is a Lie subalgebroid $B$ with anchor image $T\mathcal{F}$ and such that $A/B$ is locally equivalent with Lie algebroids over the slice manifolds of $\mathcal{F}$. We give several examples and, for foliated Lie algebroids, we discuss the following subjects: the dual Poisson structure and Vaintrob's super-vector field, cohomology and deformations of the foliation, integration to a Lie groupoid. In the last section, we define a corresponding notion of a foliation of a Courant algebroid $A$ as a bracket-closed, isotropic subbundle $B$ with anchor image $T\mathcal{F}$ and such that $B^{\perp }/B$ is locally equivalent with Courant algebroids over the slice manifolds of $\mathcal{F}$. Examples that motivate the definition are given.

LaTex, 28 pages, contains additional results

References in corpus (2)

• Reduction of Courant algebroids and generalized complex structures
• Holomorphic Poisson Manifolds and Holomorphic Lie Algebroids

Cited by in corpus (1)

• A construction of Courant algebroids on foliated manifolds