NDLI: Equivariant Cohomology over Lie Groupoids and Lie

Content Provider	Springer Nature Link
Author	Huebschmann, Johannes
Copyright Year	2009
Abstract	Using the language and terminology of relative homological algebra, in particular that of derived functors, we introduce equivariant cohomology over a general Lie–Rinehart algebra and equivariant de Rham cohomology over a locally trivial Lie groupoid in terms of suitably defined monads (also known as triples) and the associated standard constructions. This extends a characterization of equivariant de Rham cohomology in terms of derived functors developed earlier for the special case where the Lie groupoid is an ordinary Lie group, viewed as a Lie groupoid with a single object; in that theory over a Lie group, the ordinary Bott–Dupont–Shulman–Stasheff complex arises as an a posteriori object. We prove that, given a locally trivial Lie groupoid Ω and a smooth Ω-manifold f : M → B $_{Ω}$ over the space B $_{Ω}$ of objects of Ω, the resulting Ω-equivariant de Rham theory of f reduces to the ordinary equivariant de Rham theory of a vertex manifold f $^{−1}$(q) relative to the vertex group $$\Omega^q_q$$ , for any vertex q in the space B $_{Ω}$ of objects of Ω; this implies that the equivariant de Rham cohomology introduced here coincides with the stack de Rham cohomology of the associated transformation groupoid; thus this stack de Rham cohomology can be characterized as a relative derived functor. We introduce a notion of cone on a Lie–Rinehart algebra and in particular that of cone on a Lie algebroid. This cone is an indispensable tool for the description of the requisite monads.
Ending Page	251
Page Count	51
Starting Page	201
File Format	PDF
ISSN	03779017
e-ISSN	15730530
Journal	Letters in Mathematical Physics
Issue Number	1-3
Volume Number	90
Language	English
Publisher	Springer Netherlands
Publisher Date	2009-10-27
Publisher Place	Dordrecht
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	comonad and standard construction Infinite-dimensional Lie groups and their Lie algebras: general properties Theoretical, Mathematical and Computational Physics monad and dual standard construction Statistical Physics, Dynamical Systems and Complexity Lie–Rinehart cohomology equivariant cohomology equivariant de Rham cohomology relative to a groupoid Atiyah sequence relative derived functor Resolutions; derived functors differentiable cohomology Topological groupoids (including differentiable and Lie groupoids) Lie algebras of vector fields and related (super) algebras Pseudogroups and differentiable groupoids Lie groupoid Lie–Rinehart algebra cone on a Lie–Rinehart algebra de Rham cohomology Borel construction Equivariant homology and cohomology equivariant cohomology relative to a Lie–Rinehart algebra Geometry Infinite-dimensional Lie (super) algebras Lie algebroid cone on a Lie algebroid Triples (= standard construction, monad or triad), algebras for a triple, homology and derived functors for triples Group Theory and Generalizations
Content Type	Text
Resource Type	Article
Subject	Statistical and Nonlinear Physics Mathematical Physics

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