{"title":"Graded Triangular Bases","authors":"Jonathan Brundan","doi":"10.1007/s10468-025-10315-y","DOIUrl":null,"url":null,"abstract":"<div><p>This article develops a practical technique for studying representations of <span>\\(\\Bbbk \\)</span>-linear categories arising in the categorification of quantum groups. We work in terms of locally unital algebras which are <span>\\(\\mathbb {Z}\\)</span>-graded with graded pieces that are finite-dimensional and bounded below, developing a theory of <i>graded triangular bases</i> for such algebras. The definition is a graded extension of the notion of triangular basis as formulated in Brundan and Stroppel (Mem. Amer. Math. Soc. <b>293</b>(1459), vii+152 2024). However, in the general graded setting, finitely generated projective modules often fail to be Noetherian, so that existing results from the study of highest weight categories are not directly applicable. Nevertheless, we show that there is still a good theory of <i>standard modules</i>. In motivating examples arising from Kac-Moody 2-categories, these modules categorify the PBW bases for the modified forms of quantum groups constructed by Wang.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"28 1","pages":"239 - 280"},"PeriodicalIF":0.5000,"publicationDate":"2025-01-29","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/s10468-025-10315-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
This article develops a practical technique for studying representations of \(\Bbbk \)-linear categories arising in the categorification of quantum groups. We work in terms of locally unital algebras which are \(\mathbb {Z}\)-graded with graded pieces that are finite-dimensional and bounded below, developing a theory of graded triangular bases for such algebras. The definition is a graded extension of the notion of triangular basis as formulated in Brundan and Stroppel (Mem. Amer. Math. Soc. 293(1459), vii+152 2024). However, in the general graded setting, finitely generated projective modules often fail to be Noetherian, so that existing results from the study of highest weight categories are not directly applicable. Nevertheless, we show that there is still a good theory of standard modules. In motivating examples arising from Kac-Moody 2-categories, these modules categorify the PBW bases for the modified forms of quantum groups constructed by Wang.
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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