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arXiv · 1009.4065

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

Claire Amiot·Steffen Oppermann
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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 $\widetilde{A}$. 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.

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Claire Amiot, Steffen Oppermann. 2012-03-11. Algebras of acyclic cluster type: tree type and type $\widetilde{A}$. https://doi.org/10.1215/00277630-2083124

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