{"title":"Superpotentials and Quiver Algebras for Semisimple Hopf Actions","authors":"Simon Crawford","doi":"10.1007/s10468-022-10165-y","DOIUrl":null,"url":null,"abstract":"<div><p>We consider the action of a semisimple Hopf algebra <i>H</i> on an <i>m</i>-Koszul Artin–Schelter regular algebra <i>A</i>. Such an algebra <i>A</i> is a derivation-quotient algebra for some twisted superpotential <i><span>w</span></i>, and we show that the homological determinant of the action of <i>H</i> on <i>A</i> can be easily calculated using <i><span>w</span></i>. Using this, we show that the smash product <i>A</i><i>#</i><i>H</i> is also a derivation-quotient algebra, and use this to explicitly determine a quiver algebra Λ to which <i>A</i><i>#</i><i>H</i> is Morita equivalent, generalising a result of Bocklandt–Schedler–Wemyss. We also show how Λ can be used to determine whether the Auslander map is an isomorphism. We compute a number of examples, and show how several results for the quantum Kleinian singularities studied by Chan–Kirkman–Walton–Zhang follow using our techniques.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"26 6","pages":"2709 - 2752"},"PeriodicalIF":0.5000,"publicationDate":"2022-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-022-10165-y.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Algebras and Representation Theory","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10468-022-10165-y","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
We consider the action of a semisimple Hopf algebra H on an m-Koszul Artin–Schelter regular algebra A. Such an algebra A is a derivation-quotient algebra for some twisted superpotential w, and we show that the homological determinant of the action of H on A can be easily calculated using w. Using this, we show that the smash product A#H is also a derivation-quotient algebra, and use this to explicitly determine a quiver algebra Λ to which A#H is Morita equivalent, generalising a result of Bocklandt–Schedler–Wemyss. We also show how Λ can be used to determine whether the Auslander map is an isomorphism. We compute a number of examples, and show how several results for the quantum Kleinian singularities studied by Chan–Kirkman–Walton–Zhang follow using our techniques.
我们考虑一个半简单霍普夫代数 H 对一个 m-Koszul Artin-Schelter 正则代数 A 的作用。这样一个代数 A 对于某个扭曲超势 w 来说是一个导数-商代数，我们证明 H 对 A 的作用的同调行列式可以很容易地用 w 计算出来。利用这一点，我们证明了粉碎积 A#H 也是一个导数-商代数，并利用这一点明确地确定了 A#H 与之莫里塔等价的四元组代数Λ，从而推广了博克兰-谢勒-韦米斯（Bocklandt-Schedler-Wemyss）的一个结果。我们还展示了如何利用Λ来确定奥斯兰德映射是否是同构。我们计算了一些例子，并展示了如何利用我们的技术得出 Chan-Kirkman-Walton-Zhang 所研究的量子克莱因奇点的几个结果。
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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