REPRESENTATION THEORY AND RELATED TOPICS SEMINAR

ABSTRACT: The variety Rep_V(A) of representations of an associative algebra A in a finite-dimensional vector space V is a classical geometric invariant of A that plays a role in many areas of mathematics. In this talk, I will discuss a higher homological extension of this construction called the derived representation scheme DRep_V(A). The associated homology groups H_*(A,V) (called representation homology) are new interesting invariants of A which, however, are hard to compute in general. To remedy this problem we define two natural maps relating representation homology to more accessible invariants, such as cyclic homology and rings of graded symmetric polynomials (with differential). The first map is a homological extension of the character map. The second  is a derived version of the classical Harish-Chandra homomorphism. I will present some results and calculations in the case when A=Sym(V) is a polynomial algebra. This talk is (part of) joint work with Y.Berest, G.Felder, A,Ramadoss and Th.Willwacher.