Differential calculus on a Lie algebroid and Poisson manifolds

TL;DR

This paper develops the theory of differential calculus on Lie algebroids and their duals, extending classical notions like Lie derivatives and brackets, and applies these concepts to Poisson manifolds.

Contribution

It generalizes differential calculus concepts to Lie algebroids and their duals, providing detailed proofs and applications to Poisson geometry.

Findings

Established Lie derivatives, Schouten-Nijenhuis brackets, and exterior derivatives for Lie algebroids.

Extended classical differential calculus to the setting of Lie algebroids.

Applied the developed theory to analyze structures on Poisson manifolds.

Abstract

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 in this paper 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.