Seminar

If k is a field, the set of all representations of an associative k-algebra A on a finite-dimensional vector space V can be given the structure of an affine k-scheme, called the representation scheme Rep_V(A). According to a heuristic principle proposed by M.Kontsevich and A.Rosenberg, the family of schemes {Rep_V(A)} for a given non-commutative algebra A should be thought of as a substitute or `approximation' for `Spec(A)'. The idea is that every property or non-commutative geometric structure on A should naturally induce a corresponding geometric property or structure on Rep_V(A) for all V.
This viewpoint provides a litmus test for proposed definitions of non-commutative analogues of classical geometric notions, and in recent years many interesting structures in non-commutative geometry have originated from this idea. In practice, however, if an associative algebra A possesses a property of geometric nature (for example, A is a NC complete intersection, Cohen-Macaulay, Calabi-Yau, etc.), it may happen that, for some V, the scheme
Rep_V(A) fails to have a corresponding property in the usual algebro-geometric sense. The reason for this seems to be that the representation functor Rep_V is not `exact' and should be replaced by its derived functor DRep_V (in the sense of non-abelian homological algebra).
The higher homology of DRep_V(A), which we call representation homology, obstructs Rep_V(A) from having the desired property and thus measures the failure of the Kontsevich-Rosenberg `approximation.'

In this this talk, after reviewing the construction of DRep_V, we will present several results and a number of explicit examples confirming the above intuition. If time permits, we will also discuss other applications of representation homology, including its relation to Hochschild, cyclic homology and algebraic K-theory.