由Tubular代数的Hall代数实现相应的单李代数  

Realization of Simple Lie Algebras Via Hall Algebras of Tubular Algebras

作　　者：林增强[1] 

机构地区：[1]厦门大学数学科学学院,福建厦门361005

出　　处：《厦门大学学报（自然科学版）》2006年第1期5-9,共5页Journal of Xiamen University：Natural Science

基　　金：国家自然科学基金(10371101)资助

摘　　要：与李代数的交叉与渗透是近年来有限维代数表示理论发展的重要特点之一.用Hall代数的方法实现李代数是一个有趣的问题.按照Asashiba的思路,本文利用Tubular代数的根范畴的Ringel-Hall李代数与2-Toroidal李代数的同构对应,在T(2,2,2,2),T(3,3,3),T(4,4,2),T(6,3,2)型Tubular代数的退化合成李代数上构造商代数,并证明它们同构于相应的D4,E6,E7,E8型单李代数,而且李运算完全由Hall积给出.作为例子文中还通过计算系数给出D4型单李代数的具体实现.In recent years, representation theory of finite-dimensional algebras has intersected and permeated through Lte algebras, which is one of the important features. It is an interesting problem on realization of Lie algebras via Hall algebras, According to Asashiba, it takes advantage of the isomorphic correspondance between Ringel-Hall Lie algebras which are realized by root categories of Tubular algebras and 2-Toroidal Lie algebras to construct quotient algebras of degenerate composition Lie algebras of Tubular algebras of type T（2,2,2,2）,T（3,3,3）,T（4,4,2）,T（6,3,2）. They are proved to be isomorphic to complex simple Lie algebras of type D4 ,E6 ,E7 ,E8. Moreover the Lie bracket is only given by the Hall multiplication. Then it is given an explicit realization of simple Lie algebras of type D4 by computing the coefficients .

关 键 词：Tubular代数 HALL代数 退化合成李代数 单李代数 

分 类 号：O153.1[理学—数学] O152.5[理学—基础数学]