Foliated Lie and Courant Algebroids

Abstract

ABSTRACT. If A is a Lie algebroid over a foliated manifold (M, F ), a foliation of A is a Lie subalgebroid B with anchor image T F and such that A/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 T F and such that B ⊥ /B is locally equivalent with Courant algebroids over the slice manifolds of F . Examples that motivate the definition are given.

Foliated groupoids and infinitesimal ideal systems

“…Foliated algebroids in the sense of Vaisman. In [34], foliated Lie algebroids are defined as follows. A foliated Lie algebroid is a Lie algebroid A → M together with a subalgebroid B of A and an involutive subbundle Recall our definition of ideal system on a Lie algebroid (Definition 1.1).…”

“…We show that kernels of Dirac structures, usual ideals in Lie algebroids, Bott connections associated to involutive subbundle and kernels of transitive Lie algebroid morphisms are examples of infinitesimal ideal systems. We compare also our notion of foliated algebroids with the ones of [34], as well as the infinitesimal descriptions in both approaches.…”

Lie and Courant algebroids on foliated manifolds

“…This is an expository paper based on [19,20]. The idea to generalize the differential-geometric part of foliation theory to Lie algebroids was inspired by papers of C. Laurent-Gengoux, M. Stiénon and Ping Xu [7,8] where a systematic study of Lie algebroids in the holomorphic category was developed.…”

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This is an exposition of the subject, which was developed in the author's papers [19,20]. Various results from the theory of foliations (cohomology, characteristic classes, deformations, etc.)

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Poisson structures on almost complex Lie algebroids

“…In fact a Lie algebroid is an anchored vector bundle with a Lie bracket on module of sections. The cotangent bundle of a Poisson manifold has a natural structure of a Lie algebroid and between Poisson structures and Lie algebroids are many other connections, as for instance for every Lie algebroid structure on an anchored vector bundle there is a specific linear Poisson structure on the corresponding dual vector bundle and conversely, see [43,44].…”