Foliated Lie and Courant Algebroids

TL;DR

This paper explores the structure and properties of foliated Lie and Courant algebroids, including examples, dual structures, cohomology, deformations, and integration to Lie groupoids, advancing the understanding of their geometric and algebraic features.

Contribution

It introduces the concept of foliated Lie and Courant algebroids, providing examples and analyzing their dual structures, cohomology, deformations, and integration, which were not previously comprehensively studied.

Findings

Defined foliated Lie algebroids and Courant algebroids with examples.

Analyzed dual Poisson structures and Vaintrob's super-vector fields.

Discussed cohomology, deformations, and integration to Lie groupoids.

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.