Stable representation homology and Koszul duality

This paper is a sequel to [2], where we study the derived affine scheme DRep(n)(A) parametrizing the n-dimensional representations of an associative k-algebra A. In [2], we have constructed canonical trace maps Tr-n(A)(center dot) : HC center dot(A) -> H-center dot[DRep(n)(A)](GLn) extending the usual characters of representations to higher cyclic homology. This raises the natural question whether a well-known theorem of Procesi [ 30] holds in the derived setting: namely, is the algebra homomorphism Lambda Tr-n(A)(center dot) : Lambda(k)[HC center dot(A)] -> H-center dot[DRep(n)(A)](GLn) defined by Tr-n(A)(center dot) surjective? In the present paper, we answer this question for augmented algebras. Given such an algebra, we construct a canonical dense subalgebra DRep(infinity)(A)(Tr) of the topological DG algebra lim DRep(n)(A)(GLn). Our main result is that on passing to the inverse limit, the family of maps (Lambda Tr-n) over left arrow (A)(center dot)'stabilizes' to an isomorphism Lambda(k)((HC center dot) over bar (A)) congruent to H-center dot[DRep(infinity)(A)(Tr)]. The derived version of Procesi's theorem does therefore hold in the limit as n -> infinity(However, for a fixed (finite) n, there exist homological obstructions to the surjectivity of Lambda Tr-n(A)(center dot), and we show on simple examples that these obstructions do not vanish in general. We compare our result with the classical theorem of Loday, Quillen and Tsygan on stable homology of matrix Lie algebras. We show that the Chevalley-Eilenberg complex C-center dot(gl(infinity)(A), gl(infinity)(k) ; k) equipped with a natural coalgebra structure is Koszul dual to the DG algebra DRep infinity(A)(Tr). We also extend our main results to bigraded DG algebras, in which case we show the equality DRep(infinity)(A)(Tr) = DRep(infinity)(A)(GL infinity). As an application, we compute the Euler characteristics of DRep(infinity)(A)(GL infinity) and (HC center dot) over bar (A) and derive some interesting combinatorial identities.