Calculus on Lie algebroids, Lie groupoids and Poisson manifolds

Details Calculus on Lie algebroids, Lie groupoids and Poisson manifolds Calculus on Lie algebroids, Lie groupoids and Poisson manifolds

2008-06-05

arxiv.org/abs/0806.0919v3

Dissertationes Mathematicae 457, Institute of mathematics, Polish Academy of Sciences, Warszawa 2008

Mathematics

Differential Geometry

To appear in Dissertationes Mathematicae. 57 pages, 2 figures. Subsection 3.2.6 about integration of Lie algebroids rewritten,

Scientific paper

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.

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