Cluster Categories and Selfinjective Algebras: Type A

We show that the stable module categories of certain selfinjective algebras of finite representation type having tree class A n are actually u-cluster categories. Since their introduction in [6], [7], cluster categories have become a central topic in representation theory. They provide the framework for the representation-theoretic approach to the highly successful theory of cluster algebras, as introduced by Fomin and Zelevinsky [11]. As a generalization, u-cluster categories have been defined recently [16], [20], [21], which turn out to be closely connected to the generalized cluster complexes of Fomin and Reading [10]. For more details on the representation-theoretic aspects of u-cluster categories we refer to [2], In Section 1, we will recall the definition of cluster categories, and of the more general u-cluster categories. For the purposes of this introduction , let us just say that u-cluster categories are obtained from derived categories of hereditary algebras by identifying some of the objects.