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 $$(na-1)$$ ( n a - 1 ) -representation infinite algebras. As a consequence, we see that an algebra is n -representation infinite if and only if its trivial extension is $$(n+1)$$ ( 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 $$(n+1)$$ ( 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.