量子Loop代数Uq(L(sl2))的单权模

2024-05-15 17:43吴青云谭易兰夏利猛
吉林大学学报(理学版) 2024年2期

吴青云 谭易兰 夏利猛

摘要: 用构造的方法解决量子Loop代数Uq(L(sl2))具有一个一维权空间的单权模的结构问题, 得到了任意一个具有一维权空间的单权模必同构于Uq(L(sl2))的四类单权模之一. 此外, 还构造了一类权空间维数为2的既非最高权也非最低权的量子Loop代数Uq(L(sl2))的单权模.

关键词: 量子Loop代数; 权模; 单模; Dense模

中图分类号: O152.5文献标志码: A文章编号: 1671-5489(2024)02-0256-07

Simple Weight Modules of Quantum Loop Algebra Uq(L(sl2))

WU Qingyun, TAN Yilan, XIA Limeng

(School of Mathematical Sciences, Jiangsu University, Zhenjiang 212013, Jiangsu Province, China)

Abstract: The structural problem of simple weight modules with a one-dimensional weight space in the quantum Loop algebra Uq(L(sl2)) was solved by using a construction method, and it was obtained that any simple weight module with a one-dimensional weight space must be  isomorphic to one of the four classes of simple weight modules of Uq(L(sl2)). In addition, a class of simple weight modules of  the quantum Loop algebra Uq(L(sl2)) with  weight space dimension of 2, which was neither the highest weight nor the lowest weight, was constructed.

Keywords: quantum Loop algebra; weight module; simple module; Dense module

1 引言与主要结果

设g是复数域上的有限维单李代数, Yangian代數Y(g)和量子仿射代数Uq(g^)组成了两族重要的仿射型量子群. Uq(g^)是由一族生成元和一系列生成关系构成的具有单位元的结合代数, Uq(g^)商去由中心元C生成的双边理想后得到的商代数记为Uq(L(g)). 其代数结构和表示理论在数学和物理中都有重要的理论意义和应用价值. 例如, Uq(L(g))模可用于构造量子Yang-Baxter方程的三角解[1].

关于Uq(L(g))模的研究是量子群表示理论的重要问题之一[2-3]. 目前, Uq(L(g))模的研究主要集中在最高权模, 包括有限维不可约模、 局部Weyl模、 KR(Kirillow-Reshetikhin)模和素表示(Prime representations)[4-8]. 而Uq(L(sl2))的表示理论在Uq(L(g))的研究中具有重要作用.

本文目标是分类Uq(L(sl2))的一类单权模. 从Uq(sl2)的Dense模出发[9], 构造一类既不是最高权也不是最低权的无限维Uq(L(sl2))单权模, 然后分类具有一个一维权空间的Uq(L(sl2))单权模. 本文主要结果如下:

参考文献

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[9]杨冬梅. 无限维不可分解的Uq(sl2)-模的分类 [D]. 北京: 北京工业大学, 2005. (YANG D M. Classification of Infinitely Dimensional in Decomposable Uq(sl2) Modules [D]. Beijing: Beijing University of Technology, 2005.)

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(責任编辑: 李 琦)

收稿日期: 2023-08-20.

第一作者简介: 吴青云(1999—), 女, 汉族, 硕士研究生, 从事李理论的研究, E-mail: 2212102047@stmail.ujs.edu.cn.

通信作者简介: 谭易兰(1981—), 男, 汉族, 博士, 副教授, 从事李理论的研究, E-mail: tanyanlan@ujs.edu.cn.

基金项目: 国家自然科学基金(批准号: 12171155).