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Algebras of acyclic cluster type: Tree type and type Ã

Published online by Cambridge University Press:  11 January 2016

Claire Amiot  and
Steffen Oppermann
Claire Amiot
Affiliation:
Institut Fourier–UMR 5582, 100 rue des maths, 38402 Saint Martin d’Hères, France, Claire.Amiot@ujf-grenoble.fr
Steffen Oppermann
Affiliation:
Institutt for matematiske fag, Norwegian University of Science and Technology, 7491 Trondheim, Norway, steffen.oppermann@math.ntnu.no
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Abstract

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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 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 Ãn for each possible orientation of Ãn. 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.

Keywords

16E3516G2016G7016E1016W50
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 211 , September 2013 , pp. 1 - 50
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2013

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