Cluster-tilted algebras and slices

TL;DR

This paper provides a combinatorial criterion using local slices in the Auslander-Reiten quiver to determine when two tilted algebras share the same relation-extension, linking them to the same cluster-tilted algebra.

Contribution

It introduces a new combinatorial criterion based on local slices in the Auslander-Reiten quiver for identifying equivalent relation-extensions of tilted algebras.

Findings

The criterion effectively distinguishes when two tilted algebras have the same relation-extension.

Local slices serve as a key combinatorial tool in the analysis.

The approach simplifies the verification process for cluster-tilted algebra equivalence.

Abstract

We give a criterion allowing to verify whether or not two tilted algebras have the same relation-extension (thus correspond to the same cluster-tilted algebra). This criterion is in terms of a combinatorial configuration in the Auslander-Reiten quiver of the cluster-tilted algebra, which we call local slice.