Cluster-tilted algebras without clusters

TL;DR

This paper explores the structure of cluster-tilted algebras, introducing reflections and an algorithm to analyze their module categories, especially for those of tree type, expanding understanding beyond traditional cluster frameworks.

Contribution

It introduces the notion of reflections of tilted algebras and provides an algorithm to construct the transjective component of the Auslander-Reiten quiver for cluster-tilted algebras of tree type.

Findings

Defined reflections of tilted algebras.

Developed an algorithm for the transjective component.

Applied to cluster-tilted algebras of tree type.

Abstract

Cluster-tilted algebras are trivial extensions of tilted algebras. This correspondence induces a surjective map from tilted algebras to cluster-tilted algebras. If B is a cluster-tilted algebra, we use the fibre of B under this map to study the module category of B. In particular, we introduce the notion of reflections of tilted algebras and define an algorithm that constructs the transjective component of the Auslander-Reiten quiver of cluster-tilted algebras of tree type.