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

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 A~\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.Comment: v2: 37 pages. Title is changed. A mistake in the previous version is now corrected (see Remark 3.14). Other changes making the paper coherent with the version 2 of 1003.491

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