Preprojective algebras of $d$-representation finite species with relations

TL;DR

This paper investigates the structure of preprojective algebras of representation finite species, generalizing known results, and introduces methods to explicitly compute their almost Koszul complexes, especially for tensor products of species.

Contribution

It generalizes the description of Nakayama automorphisms to species, shows these algebras are almost Koszul, and introduces the Segre product to analyze tensor products of species.

Findings

Preprojective algebras of species are almost Koszul.

Explicit description of almost Koszul complexes for tensor products.

Complete characterization of Nakayama automorphisms for species.

Abstract

In this article we study the properties of preprojective algebras of representation finite species. To understand the structure of a preprojective algebra, one often studies its Nakayama automorphism. A complete description of the Nakayama automorphism is given by Brenner, Butler and King when the algebra is given by a path algebra. We generalize this result to the species case. We show that the preprojective algebra of a representation finite species is an almost Koszul algebra. With this we know that almost Koszul complexes exist. It turns out that the almost Koszul complex for a representation finite species is given by a mapping cone of a certain chain map. We also study a higher dimensional analogue of representation finite hereditary algebras called $d$-representation finite algebras. One source of $d$-representation finite algebras comes from taking tensor products. By introducing a functor called the Segre product, we manage to give a complete description of the almost Koszul complex of the preprojective algebra of a tensor product of two species with relations with certain properties, in terms of the knowledge of the given species with relations. This allows us to compute the almost Koszul complex explicitly for certain species with relations more easily.