Cluster-tilted and quasi-tilted algebras

TL;DR

This paper explores the relationship between quasi-tilted and cluster-tilted algebras, proving that relation-extensions of quasi-tilted algebras are 2-Calabi-Yau tilted and providing methods to describe their module categories.

Contribution

It introduces a characterization of quasi-tilted algebras whose relation-extensions are cluster-tilted of euclidean type and develops an algorithmic approach for constructing their tubes.

Findings

Relation-extensions of quasi-tilted algebras are 2-Calabi-Yau tilted.

A new notion of reflection helps describe module categories of cluster-tilted algebras.

An algorithm for constructing tubes of cluster-tilted algebras of euclidean type.

Abstract

In this paper, we prove that relation-extensions of quasi-tilted algebras are 2-Calabi-Yau tilted. With the objective of describing the module category of a cluster-tilted algebra of euclidean type, we define the notion of reflection so that any two local slices can be reached one from the other by a sequence of reflections and coreflections. We then give an algorithmic procedure for constructing the tubes of a cluster-tilted algebra of euclidean type. Our main result characterizes quasi-tilted algebras whose relation-extensions are cluster-tilted of euclidean type.