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

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

2010-09-21

arxiv.org/abs/1009.4065v2

Mathematics

Representation Theory

v2: 37 pages. Title is changed. A mistake in the previous version is now corrected (see Remark 3.14). Other changes making the

Scientific paper

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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