Coloured quiver mutation for higher cluster categories.
type: Item
instance of: scholarly article
title: Coloured quiver mutation for higher cluster categories.
zbMATH Open document ID: 1182.16010
author: Aslak Bakke Buan
author: Hugh Thomas
published in: Advances in Mathematics
publication date: 2009-09-29T00:00:00Z
full work available at URL: https://arxiv.org/abs/0809.0691
review text: \(m\)-cluster categories are \((m+1)\)-Calabi-Yau orbit categories of the derived category of a hereditary Abelian category. The authors give a combinatorial rule for the mutations in such a category. In a recent paper of \textit{O. Iyama} and \textit{Y. Yoshino} [Invent. Math. 172, No. 1, 117-168 (2008; Zbl 1140.18007)], it was shown that the existence of exchange triangles in cluster categories also holds for \(m\)-cluster categories. Furthermore, \textit{A. Wrålsen} [J. Algebra 321, No. 2, 532-547 (2009; Zbl 1177.18008)], and \textit{Y. Zhou, B. Zhu} [ J. Algebra 321, No. 10, 2898-2915 (2009; Zbl 1196.16013)] have proved that for an almost complete tilting object of an \(m\)-cluster category, there are \(m+1\) pairwise non-isomorphic complements given by \(m+1\) exchange triangles. The authors associate a quiver with coloured arrows of types \(\{0,\dots,m\}\) to each tilting object. For ordinary cluster categories (\(m=1\)), there is just the 0-coloured Gabriel quiver. Sequences of mutations can be performed thus in a purely combinatorial manner.
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 18E30
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 16E30
zbMATH DE Number: 5609504
zbMATH Keywords: cluster algebras
zbMATH Keywords: quiver mutations
zbMATH Keywords: cluster categories
zbMATH Keywords: Calabi-Yau categories
zbMATH Keywords: almost complete tilting objects
reviewed by: Wolfgang Rump
MaRDI profile type: Publication
OpenAlex ID: W2963837060
arXiv ID: 0809.0691
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DOI: 10.1016/J.AIM.2009.05.017
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description: scientific article