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 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 \({T{\mathcal {F}}}\) and such that \({B^{ \bot } /B}\) is locally equivalent with Courant algebroids over the slice manifolds of \({\mathcal F}\). Examples that motivate the definition are given.