量子环面代数及其上的李代数
Quantum Tori and Lie Algebras over Quantum Tori
DOI: 10.12677/PM.2021.114054, PDF, HTML, 下载: 395  浏览: 540  科研立项经费支持
作者: 陆狄雷, 常智华：华南理工大学数学学院，广东 广州
关键词: 量子环面；有扭双重 loop 代数；扩张仿射李代数；Quantum Torus； Twisted Double Loop Algebra； Extended A?ne Lie Algebra
摘要: 量子环面代数在A型扩张仿射李代数的研究中起到重要作用。两个变量的量子环面代数ℂq在q是一个m次本原单位根时，同构于m阶矩阵代数的一个有扭双重 loop代数。为证明这一结果,本文具体地构造了矩阵代数的双重loop代数的一个有限自同构群并将量子环面代数ℂq实现为矩阵代数的双重loop代数在这一有限群作用下的不动点子代数。进一步将量子环面代数的结果应用于以其为坐标环的特殊线性李代数sln(ℂq),我们得到sln(ℂq)在q是单位根时是基于有限维单李代数slmn(ℂq)的一个有扭双重loop代数.
Abstract: Quantum tori play important roles in the study of extended affine Lie algebra of type A. The  quantum  torus ℂq in  two  variables  is  isomorphic  to  a  twisted  double  loop algebra of the m × m-matrices provided that q is  a  m-th  primitive  root  of  unit.  In order to prove this result, we concretely construct a finite group of automorphism of the double loop algebra of matrices and realize the quantum torus ℂq as its sub-algebra of fixed points under  this  action.  We  further  apply  this  result  to  the  special  linear Lie algebra sln(ℂq) coordinated by the quantum torus ℂq, and conclude that the Lie algebra sln(ℂq) is also a twisted double loop Lie algebra based on the finite-dimensional simple Lie algebra slmn(ℂq) if q is a root of unit.
文章引用：陆狄雷, 常智华. 量子环面代数及其上的李代数[J]. 理论数学, 2021, 11(4): 419-427. https://doi.org/10.12677/PM.2021.114054

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