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. After giving an introduction to the relevant background material, 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. A generalized notion of almost Koszulity in the sense of Brenner–Butler–King yields similar results in the n-representation finite case.
This talk is based on joint work with Mads H. Sandøy.