Foliated Lie and Courant Algebroids

If A is a Lie algebroid over a foliated manifold (M, F), a foliation of A is a Lie subalgebroid B with anchor image TF and such that Lambda/B is locally equivalent with Lie algebroids over the slice manifolds of F. We give several examples and, for foliated Lie algebroids, we discuss the following subjects: the dual Poisson structure and Vaintrob's supervector 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 TF and such that B-perpendicular to/B is locally equivalent with Courant algebroids over the slice manifolds of F. Examples that motivate the definition are given.