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

TL;DR

This paper develops a higher version of Koszul duality connecting it with $n$-hereditary algebras, providing new characterizations and equivalences in higher homological algebra.

Contribution

It generalizes classical Koszul duality to a higher setting and links it with $n$-hereditary and $n$-representation infinite algebras, introducing new concepts like almost $n$-$T$-Koszul algebras.

Findings

Characterization of $n$-$T$-Koszul algebras of highest degree in terms of $(na-1)$-representation infinite algebras.

An algebra is $n$-representation infinite iff its trivial extension is $(n+1)$-Koszul.

Derived category equivalences for $n$-representation tame algebras.

Abstract

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.