2009/07/31 by Johannes Huebschmann · 1 citation
Mathematics · #Advanced Operator Algebra Research #Algebraic structures and combinatorial models #Chern–Weil homomorphism #Cohomology #De Rham cohomology #Equivariant cohomology #Equivariant map #Group cohomology #Homotopy and Cohomology in Algebraic Topology #Lie conformal algebra #Sheaf cohomology #math.DG #math.SG #msc:14F40 #msc:17B65 #msc:17B66 #msc:18C15 #msc:18G10 #msc:22A22 #msc:22E65 #msc:55N91 #msc:58H05 #Čech cohomology
paper · pdf · doi:10.1007/s11005-009-0356-x
published in Letters in Mathematical Physics 90(1-3) (Springer Science+Business Media) · 47 pages
arxiv created 2009/07/31 · openalex publication_date 2009/10/26 · arxiv updated 2013/03/12 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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 Ωqq , 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.