Foliated Lie and Courant Algebroids

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 $A\mathrm{/}B$ 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 $B^{\perp }\mathrm{/}B$ is locally equivalent with Courant algebroids over the slice manifolds of $\mathcal{F}$. Examples that motivate the definition are given.