Algebras of tame acyclic cluster type
Résumé
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.
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