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.