{"title":"Presentations of Braid Groups of Type A Arising from \\((m+2)\\)-angulations of Regular Polygons","authors":"Davide Morigi","doi":"10.1007/s10468-024-10257-x","DOIUrl":null,"url":null,"abstract":"<div><p>Coloured quiver mutation, introduced by Buan, A.B., Thomas, H (Adv. Math. <b>222</b>(3), 971–995 2009), gives a combinatorial interpretation of tilting in higher cluster categories. In type <i>A</i> work of Baur, K., Marsh, B. (Trans. Am. Math. Soc. <b>360</b>(11), 5789-5803 2008) shows that <i>m</i>-coloured quivers and <i>m</i>-coloured quiver mutations have a nice geometrical description, given in terms of <span>\\((m+2)\\)</span>-angulations of a regular polygon, and rotations of an <i>m</i>-diagonal. In this paper, using such correspondence, we describe presentations of braid groups of type <i>A</i> arising from coloured quivers of mutation type <i>A</i>.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"27 2","pages":"1237 - 1265"},"PeriodicalIF":0.5000,"publicationDate":"2024-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-024-10257-x.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Algebras and Representation Theory","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10468-024-10257-x","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Coloured quiver mutation, introduced by Buan, A.B., Thomas, H (Adv. Math. 222(3), 971–995 2009), gives a combinatorial interpretation of tilting in higher cluster categories. In type A work of Baur, K., Marsh, B. (Trans. Am. Math. Soc. 360(11), 5789-5803 2008) shows that m-coloured quivers and m-coloured quiver mutations have a nice geometrical description, given in terms of \((m+2)\)-angulations of a regular polygon, and rotations of an m-diagonal. In this paper, using such correspondence, we describe presentations of braid groups of type A arising from coloured quivers of mutation type A.
布安（Buan, A.B.）、托马斯（Thomas, H）（Adv. Math.222(3), 971-995 2009），给出了高簇类别中倾斜的组合解释。在鲍尔、K.、马什、B.的 A 型作品（Trans.Am.Math.360(11)，5789-5803 2008）表明，米色四元组和米色四元组突变有一个很好的几何描述，用正多边形的（(m+2)\）切线和米对角线的旋转给出。在本文中，我们利用这种对应关系描述了由突变类型 A 的彩色四边形产生的 A 型辫状群的呈现。
期刊介绍：
Algebras and Representation Theory features carefully refereed papers relating, in its broadest sense, to the structure and representation theory of algebras, including Lie algebras and superalgebras, rings of differential operators, group rings and algebras, C*-algebras and Hopf algebras, with particular emphasis on quantum groups.
The journal contains high level, significant and original research papers, as well as expository survey papers written by specialists who present the state-of-the-art of well-defined subjects or subdomains. Occasionally, special issues on specific subjects are published as well, the latter allowing specialists and non-specialists to quickly get acquainted with new developments and topics within the field of rings, algebras and their applications.
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