Quasiderivations of the Algebra \(U\mathfrak{gl}_n\) and the Quantum Mischenko–Fomenko Algebras

IF 0.6 4区 数学 Q3 MATHEMATICS
Georgii Sharygin
{"title":"Quasiderivations of the Algebra \\(U\\mathfrak{gl}_n\\) and the Quantum Mischenko–Fomenko Algebras","authors":"Georgii Sharygin","doi":"10.1134/S0016266324030080","DOIUrl":null,"url":null,"abstract":"<p> Quasiderivations of the universal enveloping algebra <span>\\(U\\mathfrak{gl}_n\\)</span> were first introduced by D. Gurevich, P. Pyatov, and P. Saponov in their study of reflection equation algebras; they are linear operators on <span>\\(U\\mathfrak{gl}_n\\)</span> that satisfy certain algebraic relations, which generalise the usual Leibniz rule. In this note, we show that the iterated action of the operator equal to a linear combination of the quasiderivations on a certain set of generators of the center of <span>\\(U\\mathfrak{gl}_n\\)</span> (namely on the symmetrised coefficients of the characteristic polynomial) produces commuting elements. The resulting algebra coincides with the quantum Mischenko–Fomenko algebra in <span>\\(U\\mathfrak{gl}_n\\)</span>, introduced earlier by Tarasov, Rybnikov, Molev, and others. </p>","PeriodicalId":575,"journal":{"name":"Functional Analysis and Its Applications","volume":"58 3","pages":"326 - 339"},"PeriodicalIF":0.6000,"publicationDate":"2024-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Functional Analysis and Its Applications","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1134/S0016266324030080","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

Quasiderivations of the universal enveloping algebra \(U\mathfrak{gl}_n\) were first introduced by D. Gurevich, P. Pyatov, and P. Saponov in their study of reflection equation algebras; they are linear operators on \(U\mathfrak{gl}_n\) that satisfy certain algebraic relations, which generalise the usual Leibniz rule. In this note, we show that the iterated action of the operator equal to a linear combination of the quasiderivations on a certain set of generators of the center of \(U\mathfrak{gl}_n\) (namely on the symmetrised coefficients of the characteristic polynomial) produces commuting elements. The resulting algebra coincides with the quantum Mischenko–Fomenko algebra in \(U\mathfrak{gl}_n\), introduced earlier by Tarasov, Rybnikov, Molev, and others.

代数(U\mathfrak{gl}_n\)和量子 Mischenko-Fomenko 代数的类iderivations
古列维奇(D. Gurevich)、皮亚托夫(P. Pyatov)和萨波诺夫(P. Saponov)在研究反射方程代数时首次引入了普遍包络代数 \(U\mathfrak{gl}_n\)的类迭代;类迭代是 \(U\mathfrak{gl}_n\)上满足某些代数关系的线性算子,它们概括了通常的莱布尼兹规则。在这篇论文中,我们证明了等价于在\(U\mathfrak{gl}_n\)中心的某组生成子上(即在特征多项式的对称系数上)的类迭代作用的线性组合的算子的迭代作用会产生换元。由此产生的代数与塔拉索夫、雷布尼科夫、莫列夫等人早先引入的 \(U\mathfrak{gl}_n\) 中的量子米申科-弗门科代数相吻合。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 求助全文
来源期刊
CiteScore
0.90
自引率
0.00%
发文量
7
审稿时长
>12 weeks
期刊介绍: Functional Analysis and Its Applications publishes current problems of functional analysis, including representation theory, theory of abstract and functional spaces, theory of operators, spectral theory, theory of operator equations, and the theory of normed rings. The journal also covers the most important applications of functional analysis in mathematics, mechanics, and theoretical physics.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信