{"title":"Frobenius Kernels of Algebraic Supergroups and Steinberg’s Tensor Product Theorem","authors":"Taiki Shibata","doi":"10.1007/s10468-023-10240-y","DOIUrl":null,"url":null,"abstract":"<div><p>For a split quasireductive supergroup <span>\\(\\mathbbm {G}\\)</span> defined over a field, we study structure and representation of Frobenius kernels <span>\\(\\mathbbm {G}_r\\)</span> of <span>\\(\\mathbbm {G}\\)</span> and we give a necessary and sufficient condition for <span>\\(\\mathbbm {G}_r\\)</span> to be unimodular in terms of the root system of <span>\\(\\mathbbm {G}\\)</span>. We also establish Steinberg’s tensor product theorem for <span>\\(\\mathbbm {G}\\)</span> under some natural assumptions.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"27 1","pages":"927 - 959"},"PeriodicalIF":0.5000,"publicationDate":"2023-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-023-10240-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/s10468-023-10240-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
For a split quasireductive supergroup \(\mathbbm {G}\) defined over a field, we study structure and representation of Frobenius kernels \(\mathbbm {G}_r\) of \(\mathbbm {G}\) and we give a necessary and sufficient condition for \(\mathbbm {G}_r\) to be unimodular in terms of the root system of \(\mathbbm {G}\). We also establish Steinberg’s tensor product theorem for \(\mathbbm {G}\) under some natural assumptions.
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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