Cluster-tilted algebras

TL;DR

This paper introduces cluster-tilted algebras, a new class linked to cluster categories, and demonstrates their close relation to hereditary algebras, along with a generalized APR-tilting result.

Contribution

It defines cluster-tilted algebras and explores their properties, establishing their connection to hereditary algebras and extending tilting theory.

Findings

Cluster-tilted algebras are endomorphism algebras of tilting objects in cluster categories.

Representation theory of these algebras closely resembles that of hereditary algebras.

A generalized version of APR-tilting is proved.

Abstract

We introduce a new class of algebras, which we call cluster-tilted. They are by definition the endomorphism algebras of tilting objects in a cluster category. We show that their representation theory is very close to the representation theory of hereditary algebras. As an application of this, we prove a generalised version of so-called APR-tilting.