REPRESENTATION THEORY AND RELATED TOPICS SEMINAR

ABSTRACT: In this talk, we discuss a connection between two areas of independent interest in representation theory, namely Koszul duality and higher homological algebra. This is studied through a generalization of the notion of T-Koszul algebras, as introduced by Madsen and Green–Reiten–Solberg. We present a higher version of classical Koszul duality and sketch some applications for n-hereditary algebras. In particular, we see that an important class of our generalized Koszul algebras can be characterized in terms of n-representation infinite algebras. As a consequence, we show that an algebra is n-representation infinite if and only if its trivial extension is (n+1)-Koszul with respect to its degree 0 part. Furthermore, we see that when an n-representation infinite algebra is n-representation tame, then the bounded derived categories of graded modules over the trivial extension and over the associated (n+1)-preprojective algebra are equivalent. A generalized notion of almost Koszulity in the sense of Brenner–Butler–King yields similar results in the n-representation finite case. This is based on joint work with Mads H. Sandøy.