{"title":"Crystal Bases of Modified \\(\\imath \\)quantum Groups of Certain Quasi-Split Types","authors":"Hideya Watanabe","doi":"10.1007/s10468-023-10207-z","DOIUrl":null,"url":null,"abstract":"<div><p>In order to see the behavior of <span>\\(\\imath \\)</span>canonical bases at <span>\\(q = \\infty \\)</span>, we introduce the notion of <span>\\(\\imath \\)</span>crystals associated to an <span>\\(\\imath \\)</span>quantum group of certain quasi-split type. The theory of <span>\\(\\imath \\)</span>crystals clarifies why <span>\\(\\imath \\)</span>canonical basis elements are not always preserved under natural homomorphisms. Also, we construct a projective system of <span>\\(\\imath \\)</span>crystals whose projective limit can be thought of as the <span>\\(\\imath \\)</span>canonical basis of the modified <span>\\(\\imath \\)</span>quantum group at <span>\\(q = \\infty \\)</span>.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"27 1","pages":"1 - 76"},"PeriodicalIF":0.5000,"publicationDate":"2023-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Algebras and Representation Theory","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10468-023-10207-z","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
In order to see the behavior of \(\imath \)canonical bases at \(q = \infty \), we introduce the notion of \(\imath \)crystals associated to an \(\imath \)quantum group of certain quasi-split type. The theory of \(\imath \)crystals clarifies why \(\imath \)canonical basis elements are not always preserved under natural homomorphisms. Also, we construct a projective system of \(\imath \)crystals whose projective limit can be thought of as the \(\imath \)canonical basis of the modified \(\imath \)quantum group at \(q = \infty \).
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.