Differential calculus on a Lie algebroid and Poisson manifolds

2008/04/15 by Charles-Michel Marle, Marle, Charles-Michel · 3 citations
Mathematics · #53D17 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Differential Geometry (math.DG) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #math.DG #msc:53D17

paper · pdf · doi:10.48550/arxiv.0804.2451

46 pages

openalex publication_date 2008/04/15 · arxiv created 2008/06/05 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Cited by

• Geometrical structures on the prolongation of a pre-Lie algebroid on fibered manifolds and application to Partial Finsler geometry on foliated anchored bundle
• Poisson structure on predual of Banach Lie algebroid
• On Poisson structures arising from a Lie group action

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