Calculus on Lie algebroids, Lie groupoids and Poisson manifolds

TL;DR

This paper reviews the theory of Lie algebroids and their duals, extending classical calculus concepts, and explores their applications to Poisson manifolds, providing detailed proofs of known results.

Contribution

It systematically develops calculus on Lie algebroids and their duals, and applies these concepts to Poisson manifolds, with detailed proofs of existing results.

Findings

Development of Lie derivatives, Schouten-Nijenhuis brackets, and exterior derivatives in Lie algebroid context

Application of Lie algebroid calculus to Poisson manifolds

Provision of detailed proofs for known theoretical results

Abstract

We begin with a short presentation of the basic concepts related to Lie groupoids and Lie algebroids, but the main part of this paper deals with Lie algebroids. A Lie algebroid over a manifold is a vector bundle over that manifold whose properties are very similar to those of a tangent bundle. Its dual bundle has properties very similar to those of a cotangent bundle: in the graded algebra of sections of its external powers, one can define an operator similar to the exterior derivative. We present the theory of Lie derivatives, Schouten-Nijenhuis brackets and exterior derivatives in the general setting of a Lie algebroid, its dual bundle and their exterior powers. All the results (which, for their most part, are already known) are given with detailed proofs. In the final sections, the results are applied to Poisson manifolds, whose links with Lie algebroids are very close.