酉群的有限子群所生成的代数  

Algebras Generated by Finite Subgroups of Unitary Groups

作　　者：罗来珍 李兴华 陶元红 LUO Lai-zhen;LI Xing-hua;TAO Yuan-hong(School of Science,Harbin University of Science and Technology,Harbin 150080,China;School of Big Data, Zhejiang University of Science and Technology, Hangzhou 310027, China)

机构地区：[1]哈尔滨理工大学理学院,哈尔滨150080 [2]浙江科技学院理学院/曙光大数据学院,杭州310027

出　　处：《哈尔滨理工大学学报》2021年第2期139-142,共4页Journal of Harbin University of Science and Technology

基　　金：国家自然基金(11761073);航空科学基金(201958058001)。

摘　　要：群表示理论是研究群的最有力的工具之一,酉群是群论的重要研究分支。针对酉群的表示问题,研究了一类由酉群生成的代数结构,证明了酉群U(d)的任何子群H生成的代数是von Neumann代数;证明了由有限群G生成的群代数■[G],对于n的不同值,与Ⅰn型von Neumann代数的直和同构,并进一步讨论了由酉群的有限子群所生成的Ⅰ型von Neumann代数。研究这类酉群的子群生成的代数结构,对于建立酉群和置换群的Schur-Weyl对偶有重要意义。Group representation theory is one of the most powerful tools to study groups. The unitary group is an important research branch of group theories. We study a class of algebraic structures generated by unitary groups, and we prove that Alg(H)is a von Neumann algebra for any subgroup H of the unitary group U(d). Then we show that group algebra ■[G], generated by a finite group G, is isomorphic to a direct sum of some von Neumann algebras of Type Ⅰn for distinct values of n. We further discuss what kinds of von Neumann algebra of Type Ⅰ can be generated by a finite unitary subgroup. It is of great significance to study the algebraic structure of the generation of subgroups of unitary groups for the establishment of Schur-Weyl duality of unitary groups and permutation groups.

关 键 词：von Neumann代数 酉群 有限群 群表示 

分 类 号：O177.2[理学—数学]