On the quiver with relations of a quasitilted algebra and applications

TL;DR

This paper characterizes quasitilted algebras using quivers with relations, providing conditions to identify them and applying these results to classify certain algebras of Dynkin type E.

Contribution

It offers new necessary and sufficient conditions for an algebra to be quasitilted based on quivers with relations, extending previous understanding.

Findings

Provided a sufficient condition for algebras with global dimension two to be quasitilted.

Showed that the sufficient condition is not necessary in general.

Determined the quivers with relations for all tilted and cluster tilted algebras of Dynkin type E.

Abstract

In this paper we discuss, in terms of quivers with relations, sufficient and necessary conditions for an algebra to be a quasitilted algebra. We start with an algebra with global dimension two and we give a sufficient condition for it to be a quasitilted algebra. We show that this condition is not necessary. In the case of a strongly simply connected schurian algebra, we discuss necessary conditions, and, combining both type of conditions, we are able to analyze if the given algebra is quasitilted. As an application we obtain the quiver with relations of all the tilted and cluster tilted algebras of Dynkin type $E_p$.