Higher Koszul duality and connections with $n$-hereditary algebras

We establish a connection between two areas of independent interest in representation theory, namely Koszul duality and higher homological algebra. This is done through a generalization of the notion of T -Koszul algebras, for which we obtain a higher version of classical Koszul duality. Our approach is motivated by and has applications for n-hereditary algebras. In particular, we characterize an important class of n-T -Koszul algebras of highest degree a in terms of (na− 1)-representation infinite algebras. As a consequence, we see 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 show 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. In the n-representation finite case, we introduce the notion of almost n-T -Koszul algebras and obtain similar results.