Algebras of acyclic cluster type: tree type and type $\widetilde{A}$

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 global dimension at most 2 whose generalized cluster category is equivalent to the cluster category of an acyclic quiver which is either a tree or of type $\widetilde{A}$. We are particularly interested in their derived equivalence classification. We prove that each algebra which is cluster equivalent to a tree quiver is derived equivalent to the path algebra of this tree. Then we describe 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 such an algebra lies, and describe the Auslander-Reiten quiver of its derived category. Together, these results in particular provide a complete classification of algebras which are cluster equivalent to tame acyclic quivers.

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

Domaines: Théorie des représentations [math.RT]

Claire Amiot, Steffen Oppermann. Algebras of acyclic cluster type: tree type and type $\widetilde{A}$. 2010. ⟨hal-00519712v2⟩