# Higher Structure Conferences Berest

## Representation homology and Koszul duality

Let A be an associative algebra, a Lie algebra or, more generally, an algebra over an arbitrary (k-linear quadratic) operad defined over a field k of characteristic zero. The set of all representations of A in a finite-dimensional vector space V can be given the structure of an affine k-scheme, called the represesentation scheme Rep_V(A). This geometric construction plays a fundamental role in representation theory of algebras. In this talk, I will discuss a derived version of Rep_V(A), which is obtained by deriving the representation functor Rep_V in the sense of abstract homotopy theory (Quillen’s homotopical algebra). The corresponding derived representation scheme DRep_V(A) is defined by a commutative DG algebra as an object in the homotopy category of DG algebras; its homology depends only on A (and V) and is called the representation homology of A. I will show how some of the geometric properties of Rep_V(A) can be interpreted in terms of representation homology. The main result I am going to explain is a relation between representation homology and the classical Chevalley-Eilenberg (co)homology of current Lie algebras.