Cluster tilted algebras with a cyclically oriented quiver

TL;DR

This paper characterizes certain algebras of global dimension two whose endomorphism algebras are cluster-tilted with cyclically oriented quivers, linking them to hereditary algebras of Dynkin or extended Dynkin types.

Contribution

It provides a characterization of algebras with global dimension two leading to specific cluster-tilted endomorphism algebras and establishes derived equivalences in special cases.

Findings

Characterization of algebras with cluster-tilted endomorphism algebras

Identification of conditions for cyclically oriented quivers

Derived equivalence to hereditary algebras in Dynkin cases

Abstract

In association with a finite dimensional algebra A of global dimension two, we consider the endomorphism algebra of A, viewed as an object in the triangulated hull of the orbit category of the bounded derived category, in the sense of Amiot. We characterize the algebras A of global dimension two such that its endomorphism algebra is isomorphic to a cluster-tilted algebra with a cyclically oriented quiver.Furthermore, in the case that the cluster tilted algebra with a cyclically oriented quiver is of Dynkin or extended Dynkin type then A is derived equivalent to a hereditary algebra of the same type.