实Clifford代数及其单位群的实矩阵表示

吉林大学学报(理学版) ›› 2021, Vol. 59 ›› Issue (3): 519-524.

实Clifford代数及其单位群的实矩阵表示

宋元凤¹, 杨柳², 李武明¹

1. 通化师范学院数学学院, 吉林通化 134002; 2. 吉林大学数学学院, 长春 130012

收稿日期:2020-10-19 出版日期:2021-05-26 发布日期:2021-05-23
通讯作者: 杨柳 E-mail:yangliu@jlu.edu.cn

Real Matrix Representations of Real Clifford Algebras and Their Unit Groups

SONG Yuanfeng¹, YANG Liu², LI Wuming¹

1. School of Mathematics, Tonghua Normal University, Tonghua 134002, Jilin Province, China; 2. College of Mathematics, Jilin University, Changchun 130012, China

Received:2020-10-19 Online:2021-05-26 Published:2021-05-23

摘要/Abstract

摘要： 首先, 用实Clifford代数的线性变换构造实Clifford代数的单位群Cl*_0,3,Cl*_2,1,Cl*_3,0的忠实实矩阵表示, 发现其为8级实矩阵群的子群; 其次, 借助实Clifford代数Cl_p,q(p+q=3)基元素相互关系及其对应的矩阵关系构建实Clifford代数Cl_2,1和Cl_3,0的忠实实矩阵表示, 发现其为4级实矩阵代数的子代数, 并给出其非忠实实矩阵表示.

关键词: 单位群, 忠实表示, 实Clifford代数, 矩阵表示

Abstract: Firstly, we used the linear transformation of real Clifford algebras to construct the faithful real matrix representations of unit groups of real Clifford algebras Cl*_0,3,Cl*_2,1,Cl*_3,0, and found that they were the subgroups of 8 order real matrix groups. Secondly, we constructed the faithful real matrix representations of real Clifford algebras Cl_2,1 and Cl_3,0 by the relations of basis elements of real Clifford algebras Cl_p,q(p+q=3) and their corresponding matrix relations, we found that they were the subalgebras of 4 order real matrix algebras, and gave their unfaithful real matrix representations.

Key words: unit group, faithful representation, real Clifford algebra, matrix representation

中图分类号:

O151.24

宋元凤, 杨柳, 李武明. 实Clifford代数及其单位群的实矩阵表示[J]. 吉林大学学报(理学版), 2021, 59(3): 519-524.

SONG Yuanfeng, YANG Liu, LI Wuming. Real Matrix Representations of Real Clifford Algebras and Their Unit Groups[J]. Journal of Jilin University Science Edition, 2021, 59(3): 519-524.