Cluster-concealed algebras

TL;DR

This paper investigates the structure of cluster-concealed algebras, a class of cluster-tilted algebras, describing their indecomposable modules and their dimension vectors using root systems, with a focus on representation-finite cases.

Contribution

It introduces the concept of cluster-concealed algebras and characterizes the dimension vectors of their indecomposable modules via root systems, extending known results for representation-finite cases.

Findings

Indecomposable modules in representation-finite cluster-tilted algebras are determined by their dimension vectors.

For general cluster-tilted algebras, dimension vectors relate to roots of a quadratic form, including negative coordinates.

All representation-finite cluster-tilted algebras are cluster-concealed.

Abstract

The cluster-tilted algebras have been introduced by Buan, Marsh and Reiten, they are the endomorphism rings of cluster-tilting objects $T$ in cluster categories; we call such an algebra cluster-concealed in case $T$ is obtained from a preprojective tilting module. For example, all representation-finite cluster-tilted algebras are cluster-concealed. If $C$ is a representation-finite cluster-tilted algebra, then the indecomposable $C$-modules are shown to be determined by their dimension vectors. For a general cluster-tilted algebra $C$, we are going to describe the dimension vectors of the indecomposable $C$-modules in terms of the root system of a quadratic form. The roots may have both positive and negative coordinates and we have to take absolute values.