Calculus on Lie algebroids, Lie groupoids and Poisson manifolds

Charles-Michel Marle

  • 2008

Abstract

top
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 exterior powers, one can define an operator d E 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 the 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.

How to cite

top

Charles-Michel Marle. Calculus on Lie algebroids, Lie groupoids and Poisson manifolds. 2008. <http://eudml.org/doc/285998>.

@book{Charles2008,
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 exterior powers, one can define an operator $d_\{E\}$ 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 the 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.},
author = {Charles-Michel Marle},
keywords = {Lie groupoids; Lie algebroids; Schouten-Nijenhuis bracket; Poisson manifolds},
language = {eng},
title = {Calculus on Lie algebroids, Lie groupoids and Poisson manifolds},
url = {http://eudml.org/doc/285998},
year = {2008},
}

TY - BOOK
AU - Charles-Michel Marle
TI - Calculus on Lie algebroids, Lie groupoids and Poisson manifolds
PY - 2008
AB - 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 exterior powers, one can define an operator $d_{E}$ 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 the 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.
LA - eng
KW - Lie groupoids; Lie algebroids; Schouten-Nijenhuis bracket; Poisson manifolds
UR - http://eudml.org/doc/285998
ER -

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.