# Equivariant cohomology over Lie groupoids and Lie-Rinehart algebras - Mathematics > Differential Geometry

Equivariant cohomology over Lie groupoids and Lie-Rinehart algebras - Mathematics > Differential Geometry - Download this document for free, or read online. Document in PDF available to download.

Abstract: Using the language and terminology of relative homological algebra, inparticular that of derived functors, we introduce equivariant cohomology over ageneral Lie-Rinehart algebra and equivariant de Rham cohomology over a locallytrivial Lie groupoid in terms of suitably defined monads also known astriples and the associated standard constructions. This extends acharacterization of equivariant de Rham cohomology in terms of derived functorsdeveloped earlier for the special case where the Lie groupoid is an ordinaryLie group, viewed as a Lie groupoid with a single object; in that theory over aLie group, the ordinary Bott-Dupont-Shulman-Stasheff complex arises as an aposteriori object. We prove that, given a locally trivial Lie groupoid G and asmooth G-manifold f over the space B of objects of G, the resultingG-equivariant de Rham theory of f boils down to the ordinary equivariant deRham theory of a vertex manifold relative to the corresponding vertex group,for any vertex in the space B of objects of G; this implies that theequivariant de Rham cohomology introduced here coincides with the stack de Rhamcohomology of the associated transformation groupoid whence this stack de Rhamcohomology can be characterized as a relative derived functor. We introduce anotion of cone on a Lie-Rinehart algebra and in particular that of cone on aLie algebroid. This cone is an indispensable tool for the description of therequisite monads.

Author: ** Johannes Huebschmann Universite de Lille 1**

Source: https://arxiv.org/