Representation homology, Lie algebra cohomology and the derived Harish-Chandra homomorphism

摘要

We study the derived representation scheme DRep(n)(A) parametrizing the n-dimensional representations of an associative algebra A over a field of characteristic zero. We show that the homology of DRep(n)(A) is isomorphic to the Chevalley-Eilenberg homology of the current Lie coalgebra gl*(n)((C) over bar) defined over a Koszul dual coalgebra of A. This gives a conceptual explanation to some of the main results of [BKR] and [BR1], relating them (via Koszul duality) to classical theorems on (co) homology of current Lie algebras gl(n)(A). We extend the above isomorphism to representation schemes of Lie algebras: for a finite-dimensional reductive Lie algebra g, we define the derived affine scheme DRep(g)(a) parametrizing the representations (in g) of a Lie algebra a; we show that the homology of DRep(g)(a) is isomorphic to the Chevalley-Eilenberg homology of the Lie coalgebra g(n)((C) over bar) where C is a cocommutative DG coalgebra Koszul dual to the Lie algebra a. We construct a canonical DG algebra map Phi(g)(a) : DRep(g)(a)(G) -> DRep(h)(a)(W), relating the G-invariant part of representation homology of a Lie algebra a in g to the W-invariant part of representation homology of a in a Cartan subalgebra of g. We call this map the derived HarishChandra homomorphism as it is a natural homological extension of the classical Harish-Chandra restriction map.