Algebras of tame acyclic cluster type

Claire Amiot (1) , Steffen Oppermann (2)

1 IRMA - Institut de Recherche Mathématique Avancée
2 Institutt for Matematiske Fag [Trondheim]

In this paper, we study algebras of tame acyclic cluster type. These are algebras of global dimension at most 2 whose generalized cluster category is equivalent to a cluster category of an acyclic quiver of tame representation type. We are particularly interested in their derived equivalence classification. We prove that the algebras which are cluster equivalent to a tree quiver are derived equivalent to the path algebra of this tree. Then we complete the classification by describing explicitly the algebras of cluster type $\A_n$ for each possible orientation of $A_n$. We give an explicit way to read off in which derived equivalence class these algebras are, and describe the Auslander-Reiten quiver of their derived categories.

Mots clés: quiver mutation; cluster category; cluster-tilting theory; derived equivalence

Domaines: Théorie des représentations [math.RT]

Claire Amiot, Steffen Oppermann. Algebras of tame acyclic cluster type. 2010. ⟨hal-00519712v1⟩