2012-02-03

Derived Representation Schemes and Cyclic Homology

Ajay Ramadoss, Eidgenössische Technische Hochschule, Zürich

Many important varieties arising in algebra, geometry and physics can be realized as moduli spaces of finite-dimensional representations of associative algebras and groups. This talk deals with the affine scheme Rep_n(A) parametrizing the n-dimensional representations of an algebra A over a field k. For a fixed n, Rep_n defines a (non-additive) functor on the category of algebras. This functor can be derived in a suitable sense, yielding the derived representation scheme DRep_n(A). We give a simple algebraic construction of DRep_n(A), and study its stable homology as n goes to infinity. We construct canonical trace maps relating the homology of DRep_n(A) to the cyclic homology HC(A) of A: the existence of these maps allow us to compute (the invariant part of) the stable homology H_*[DRep_\infty(A)] in terms of HC(A). This result is analogous to the famous theorem of Loday, Quillen and Tsygan computing the stable homology of the Lie algebra gl_n(A) of nxn matrices with entries in A. Time permitting, we shall outline a variant of the main construction which we hope will be applicable to low dimensional topology and knot theory.