{"title":"Rigidity of quantum algebras","authors":"Akaki Tikaradze","doi":"10.1112/jlms.70118","DOIUrl":null,"url":null,"abstract":"<p>Given an associative <span></span><math>\n <semantics>\n <mi>C</mi>\n <annotation>$\\mathbb {C}$</annotation>\n </semantics></math>-algebra <span></span><math>\n <semantics>\n <mi>A</mi>\n <annotation>$A$</annotation>\n </semantics></math>, we call <span></span><math>\n <semantics>\n <mi>A</mi>\n <annotation>$A$</annotation>\n </semantics></math> strongly rigid if for any pair of finite subgroups of its automorphism groups <span></span><math>\n <semantics>\n <mrow>\n <mi>G</mi>\n <mo>,</mo>\n <mi>H</mi>\n </mrow>\n <annotation>$G, H$</annotation>\n </semantics></math>, such that <span></span><math>\n <semantics>\n <mrow>\n <msup>\n <mi>A</mi>\n <mi>G</mi>\n </msup>\n <mo>≅</mo>\n <msup>\n <mi>A</mi>\n <mi>H</mi>\n </msup>\n </mrow>\n <annotation>$A^G\\cong A^H$</annotation>\n </semantics></math>, then <span></span><math>\n <semantics>\n <mi>G</mi>\n <annotation>$G$</annotation>\n </semantics></math> and <span></span><math>\n <semantics>\n <mi>H</mi>\n <annotation>$H$</annotation>\n </semantics></math> must be isomorphic. In this paper, we show that a large class of filtered quantizations are strongly rigid. We also solve the inverse Galois problem for a wide class of rational Cherednik algebras that includes all (simple) classical generalized Weyl algebras, and also for quantum tori. Finally, we show that the Picard group of an <span></span><math>\n <semantics>\n <mi>n</mi>\n <annotation>$n$</annotation>\n </semantics></math>-dimensional quantum torus is isomorphic to the group of its outer automorphisms.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 3","pages":""},"PeriodicalIF":1.0000,"publicationDate":"2025-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70118","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of the London Mathematical Society-Second Series","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/jlms.70118","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
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Abstract
Given an associative -algebra , we call strongly rigid if for any pair of finite subgroups of its automorphism groups , such that , then and must be isomorphic. In this paper, we show that a large class of filtered quantizations are strongly rigid. We also solve the inverse Galois problem for a wide class of rational Cherednik algebras that includes all (simple) classical generalized Weyl algebras, and also for quantum tori. Finally, we show that the Picard group of an -dimensional quantum torus is isomorphic to the group of its outer automorphisms.
期刊介绍：
The Journal of the London Mathematical Society has been publishing leading research in a broad range of mathematical subject areas since 1926. The Journal welcomes papers on subjects of general interest that represent a significant advance in mathematical knowledge, as well as submissions that are deemed to stimulate new interest and research activity.
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