Relative homological algebra, equivariant de Rham theory, and Koszul duality

Abstract

Let G be a general (not necessarily finite dimensional compact) Lie group, let g be its Lie algebra, let Cg be the cone on g in the category of differential graded Lie algebras, and consider the functor which assigns to a chain complex V the V-valued total de Rham complex of G. We describe the G-equivariant de Rham cohomology in terms of a suitable relative differential graded Ext, defined on the appropriate category of (G,Cg)-modules. The meaning of "relative" is made precise via the dual standard construction associated with the monad involving the aforementioned functor and the associated forgetful functor. The corresponding infinitesimal equivariant cohomology is the relative differential Ext over Cg relative to g. The functor under discussion decomposes into two functors, the functor which determines differentiable cohomology in the sense of Hochschild-Mostow and the functor which determines the infinitesimal equivariant theory, suitably interpreted. This functor decomposition, in turn, entails an extension of a Decomposition Lemma due to Bott. Appropriate models for the differential graded Ext involving a comparison between a suitably defined simplicial Weil coalgebra and the Weil coalgebra dual to the familiar ordinary Weil algebra yield small models for equivariant de Rham cohomology including the standard Weil and Cartan models for the special case where the group G is compact and connected. Koszul duality in de Rham theory results from these considerations in a straightforward manner.