Algebras of acyclic cluster type: Tree type and type Ã

Abstract

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 $\mathrm{Ã}$. 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 ${\mathrm{Ã}}_n$ for each possible orientation of ${\mathrm{Ã}}_n$. We give an explicit way to read off the derived equivalence class in which such an algebra lies, and we 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.