Mikhail Gorsky (Bielefeld), Extended Hall algebras and localization of categories

Abstract:

Hall algebras play an important role in representation theory and algebraic geometry. The Hall algebra of an exact or a triangulated category captures information about the extensions between objects. It turns out that in some cases twisted and extended Hall algebras of triangulated categories are well-defined even when their non-deformed counterparts are not. I will explain how to associate a twisted extended Hall algebra to a triangulated category, when the latter arises as the homotopy category of a hereditary exact model category or as an orbit category of certain kind. This construction generalizes Bridgeland's Hall algebras categorifying quantum groups, modified Ringel-Hall algebras of Ming and Lu and semi-derived Hall algebras. I will report on the joint work in progress with Sarah Scherotzke on geometrical version of these algebras via perverse sheaves on graded and generalized quiver varieties. If time permits, i will also discuss conjectural relations between extended Hall algebras of cluster categories and quantum cluster algebras.