量子环代数和扬基的 Theta 系列

IF 2.2 1区 物理与天体物理 Q1 PHYSICS, MATHEMATICAL
Huafeng Zhang
{"title":"量子环代数和扬基的 Theta 系列","authors":"Huafeng Zhang","doi":"10.1007/s00220-024-05110-7","DOIUrl":null,"url":null,"abstract":"<div><p>We introduce and study a family of power series, which we call <i>Theta series</i>, whose coefficients are in the tensor square of a quantum loop algebra. They arise from a coproduct factorization of the T-series of Frenkel–Hernandez, which are leading terms of transfer matrices of certain infinite-dimensional irreducible modules over the upper Borel subalgebra in the category <span>\\({\\mathcal {O}}\\)</span> of Hernandez–Jimbo. We prove that each weight component of a Theta series is polynomial. As applications, we establish a decomposition formula and a polynomiality result for R-matrices between an irreducible module and a finite-dimensional irreducible module in category <span>\\({\\mathcal {O}}\\)</span>. We extend T-series and Theta series to Yangians by solving difference equations determined by the truncation series of Gerasimov–Kharchev–Lebedev–Oblezin. We prove polynomiality of Theta series by interpreting them as associators for triple tensor product modules over shifted Yangians.</p></div>","PeriodicalId":522,"journal":{"name":"Communications in Mathematical Physics","volume":"405 10","pages":""},"PeriodicalIF":2.2000,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Theta Series for Quantum Loop Algebras and Yangians\",\"authors\":\"Huafeng Zhang\",\"doi\":\"10.1007/s00220-024-05110-7\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We introduce and study a family of power series, which we call <i>Theta series</i>, whose coefficients are in the tensor square of a quantum loop algebra. They arise from a coproduct factorization of the T-series of Frenkel–Hernandez, which are leading terms of transfer matrices of certain infinite-dimensional irreducible modules over the upper Borel subalgebra in the category <span>\\\\({\\\\mathcal {O}}\\\\)</span> of Hernandez–Jimbo. We prove that each weight component of a Theta series is polynomial. As applications, we establish a decomposition formula and a polynomiality result for R-matrices between an irreducible module and a finite-dimensional irreducible module in category <span>\\\\({\\\\mathcal {O}}\\\\)</span>. We extend T-series and Theta series to Yangians by solving difference equations determined by the truncation series of Gerasimov–Kharchev–Lebedev–Oblezin. We prove polynomiality of Theta series by interpreting them as associators for triple tensor product modules over shifted Yangians.</p></div>\",\"PeriodicalId\":522,\"journal\":{\"name\":\"Communications in Mathematical Physics\",\"volume\":\"405 10\",\"pages\":\"\"},\"PeriodicalIF\":2.2000,\"publicationDate\":\"2024-09-14\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Communications in Mathematical Physics\",\"FirstCategoryId\":\"101\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s00220-024-05110-7\",\"RegionNum\":1,\"RegionCategory\":\"物理与天体物理\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"PHYSICS, MATHEMATICAL\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications in Mathematical Physics","FirstCategoryId":"101","ListUrlMain":"https://link.springer.com/article/10.1007/s00220-024-05110-7","RegionNum":1,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
引用次数: 0

摘要

我们引入并研究了一组幂级数,我们称之为 Theta 级数,其系数位于量子环代数的张量平方中。它们产生于 Frenkel-Hernandez 的 T 序列的共积因式分解,而 T 序列是 Hernandez-Jimbo 的 \({\mathcal {O}}\) 类别中上 Borel 子代数上的某些无限维不可还原模块的转移矩阵的前导项。我们证明了 Theta 级数的每个权重分量都是多项式的。作为应用,我们为范畴 \({\mathcal {O}}\) 中的不可还原模块和有限维不可还原模块之间的 R 矩建立了分解公式和多项式性结果。我们通过求解由 Gerasimov-Kharchev-Lebedev-Oblezin 的截断数列决定的差分方程,将 T 数列和 Theta 数列扩展到扬格数列。我们通过把 Theta 序列解释为在移位扬琴上的三重张量乘积模块的关联器,证明了 Theta 序列的多项式性。
本文章由计算机程序翻译,如有差异,请以英文原文为准。

Theta Series for Quantum Loop Algebras and Yangians

Theta Series for Quantum Loop Algebras and Yangians

We introduce and study a family of power series, which we call Theta series, whose coefficients are in the tensor square of a quantum loop algebra. They arise from a coproduct factorization of the T-series of Frenkel–Hernandez, which are leading terms of transfer matrices of certain infinite-dimensional irreducible modules over the upper Borel subalgebra in the category \({\mathcal {O}}\) of Hernandez–Jimbo. We prove that each weight component of a Theta series is polynomial. As applications, we establish a decomposition formula and a polynomiality result for R-matrices between an irreducible module and a finite-dimensional irreducible module in category \({\mathcal {O}}\). We extend T-series and Theta series to Yangians by solving difference equations determined by the truncation series of Gerasimov–Kharchev–Lebedev–Oblezin. We prove polynomiality of Theta series by interpreting them as associators for triple tensor product modules over shifted Yangians.

求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
Communications in Mathematical Physics
Communications in Mathematical Physics 物理-物理:数学物理
CiteScore
4.70
自引率
8.30%
发文量
226
审稿时长
3-6 weeks
期刊介绍: The mission of Communications in Mathematical Physics is to offer a high forum for works which are motivated by the vision and the challenges of modern physics and which at the same time meet the highest mathematical standards.
×
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