Mutation classes of \(\widetilde A_n\)-quivers and derived equivalence classification of cluster tilted algebras of type \(\widetilde A_n\).
type: Item
instance of: scholarly article
title: Mutation classes of \(\widetilde A_n\)-quivers and derived equivalence classification of cluster tilted algebras of type \(\widetilde A_n\).
zbMATH Open document ID: 1263.16019
author: Janine Bastian
published in: Algebra \& Number Theory
publication date: 2012-07-11T00:00:00Z
full work available at URL: https://arxiv.org/abs/0901.1515
full work available at URL: p01.xhtml
reviewed by: Ryan Kinser
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 16E35
Mathematics Subject Classification ID: 18E30
Mathematics Subject Classification ID: 16E30
Mathematics Subject Classification ID: 18E35
zbMATH DE Number: 6054514
zbMATH Keywords: cluster tilted algebras
zbMATH Keywords: quiver mutations
zbMATH Keywords: derived equivalences
zbMATH Keywords: tilting complexes
zbMATH Keywords: gentle algebras
MaRDI profile type: Publication
OpenAlex ID: W2951843638
arXiv ID: 0901.1515
DOI: 10.2140/ANT.2011.5.567
review text: Let \(Q\) be a quiver which is mutation equivalent to a quiver of type \(\widetilde A_n\). When referring to type \(\widetilde A_n\) quivers in this review, we always assume that the orientation is not cyclic. This paper identifies four combinatorial parameters, which can be read off directly from a drawing of \(Q\), that determine its mutation class. The basic idea is that each oriented 3-cycle, and each arrow that is not contained in an oriented 3-cycle, are assigned a certain parity according to some rules (solid or dashed). The combinatorial parameters \(r_1\) and \(r_2\) count the number of solid arrows and 3-cycles, respectively, while \(s_1\) and \(s_2\) count the dashed ones. From these, one can classify derived equivalence classes of cluster tilted algebras of type \(\widetilde A_n\).NEWLINENEWLINE The proof builds on the classification of quivers mutation equivalent to a quiver of type \(A\) by \textit{A. B. Buan} and \textit{D. F. Vatne} [J. Algebra 319, No. 7, 2723-2738 (2008; Zbl 1155.16010)]. The author defines a class of quivers \(\mathcal Q_n\) that contains all quivers of type \(\widetilde A_n\), and prove that it is closed under mutation. Therefore \(\mathcal Q_n\) contains all quivers mutation equivalent to one of type \(\widetilde A_n\). Then the author gives an algorithm to take any quiver in this class and reduce it to a normal form, without changing the parameters \(r_1,r_2,s_1,s_2\) at any step. Finally, it is shown that a quiver in normal form is mutation equivalent to a quiver of type \(\widetilde A_n\) with \(r_1+2r_2\) arrows in one direction and \(s_1+2s_2\) arrows in the other direction. Thus \(\mathcal Q_n\) is exactly the set of quivers mutation equivalent to some quiver of type \(\widetilde A_n\).NEWLINENEWLINE Turning the focus to cluster tilted algebras, the main result is proven: namely, that two cluster tilted algebras of type \(\widetilde A_n\) are derived equivalent if and only if they have the same parameters \(r_1,r_2,s_1,s_2…
description: scientific article; zbMATH DE number 6054514