Schur定理的推广  

Generalization of Schur’s Theorem

作　　者：王淑娟 刘舒畅 WANG Shujuan;LIU Shuchang(School of Mathematical Sciences,Heilongjiang University,Harbin 150080,Heilongjiang,China)

机构地区：[1]黑龙江大学数学科学学院,黑龙江哈尔滨150080

出　　处：《济南大学学报（自然科学版）》2021年第1期91-94,共4页Journal of University of Jinan(Science and Technology)

基　　金：国家自然科学基金项目(11701158)。

摘　　要：为了计算特征0的代数闭域上两两弱交换矩阵线性无关的极大维数,依据分块矩阵理论,采用数学归纳法,得到上三角矩阵空间的弱交换空间的极大维数,并且给出具有极大维数的弱交换空间的一组基底;利用Jacobson弱闭集定理,将一般线性Lie代数的交换子代数或特殊Jordan代数的交换子代数同时上三角化,即在相似意义下,这2种交换子代数的所有矩阵都可以看作上三角矩阵,进而得到2种交换子代数的极大维数。结果表明,Schur关于两两交换矩阵构成的线性空间极大维数的定理得到推广,并且统一得到了有限维交换Lie代数与交换Jordan代数忠实表示的极小维数。To determine the linearly independent maximum dimension of two weakly commutative matrices in the algebraic closed domain of feature 0,the maximal dimension for weakly commutative spaces contained in the upper triangular matrices space was obtained by using of the partitioned matrix theory and the inductive method.A basis with the maximum dimension was given for a weakly commutative space.From Jacobson’s theorem on weakly closed sets,any abelian subalgebra of general linear Lie algebras or special Jordan algebras was contained in the space consisting of upper triangular matrices.That is,in the sense of similarity,all matrices of these two commutative subalgebras could be regarded as upper triangular matrices.The maximal dimensions of above two abelian subalgebras were thus obtained.The results show that Schur’s theorem is generalized,which is about the maximal dimension of linear spaces consisting of mutually commutative matrices.The minimal dimension is given for faithful representations of any finite-dimensional abelian Lie or Jordan algebra by using a unified method.

关 键 词：LIE超代数 弱交换空间的维数 矩阵的相似变换 

分 类 号：O152.5[理学—数学]