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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, PP. 419-427

Keywords: 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

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Abstract:

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？q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？q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(？q),我们得到sln(？q)在q是单位根时是基于有限维单李代数slmn(？q)的一个有扭双重loop代数.
Quantum tori play important roles in the study of extended a？ne Lie algebra of type
A. The？ quantum？ torus ？q 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 ？nite group of automorphism of the double loop algebra of matrices and realize the quantum torus ？q as its sub-algebra of ？xed points under？ this？ action.？ We？ further？ apply？ this？ result？ to？ the？ special？ linear Lie algebra sln(？q) coordinated by the quantum torus ？q, and conclude that the Lie algebra sln(？q) is also a twisted double loop Lie algebra based on the ？nite-dimensional simple Lie algebra slmn(？q) if q is a root of unit.

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