JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2024, Vol. 59 ›› Issue (2): 14-21, 31.doi: 10.6040/j.issn.1671-9352.0.2022.632

Previous Articles     Next Articles

The module algebra structures of dihedral groups on the full matrix algebra of 2×2

Dong SU()   

  1. School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang 471023, Henan, China
  • Received:2022-11-28 Online:2024-02-20 Published:2024-02-20

RichHTML

0

PDF (PC)

45

Abstract:

By using the theories of weak similarity of matrices and weak similarity of matrix pairs, the module algebra structures of cyclic groups of Cm are researched over the full matrix algebra of 2×2 on the complex field $\mathscr{C}$, and all module algebra structures of dihedral groups D2m are characterized over the full matrix algebra of 2×2 up to isomorphism.

Key words: full matrix algebra, dihedral group, weak similarity, module algebra

CLC Number: 

  • O153.3
1 BEATTIE M . A direct sum decomposition for the Brauer group of H-module algebras[J]. Journal of Algebra, 1976, 43 (2): 686- 693.
doi: 10.1016/0021-8693(76)90134-4
2 MOLNAR R K . Semi-direct products of Hopf algebras[J]. Journal of Algebra, 1977, 47 (1): 29- 51.
doi: 10.1016/0021-8693(77)90208-3
3 CAENEPEEL S , VAN OYSTAEYEN F , ZHANG Y H . Quantum Yang-Baxter module algebras[J]. K-Theory, 1994, 8 (3): 231- 255.
doi: 10.1007/BF00960863
4 CAENEPEEL S , VAN OYSTAEYEN F , ZHANG Y H . The Brauer group of Yetter-Drinfeld module algebras[J]. Transactions of the American Mathematical Society, 1997, 349 (9): 3737- 3771.
doi: 10.1090/S0002-9947-97-01839-4
5 DUPLIJ S , SINELSHCHIKOV S . Classifications of Uq(sl2)-module algebra structures on the quantum plane[J]. Journal of Mathematical Physics, Analysis, Geometry, 2010, 6 (4): 406- 430.
6 CHEN Huixiang , ZHANG Yinhuo . Four-dimensional Yetter-Drinfeld module algebras over H4[J]. Journal of Algebra, 2006, 296 (2): 582- 634.
doi: 10.1016/j.jalgebra.2005.08.011
7 ZHAO Shiyin, WANG Yin, CHEN Xiaojuan, et al. The kS3-module algebra structures on M3(k)[EB/OL]. Beijing: Sciencepaper Online(2013-06-09)[2022-09-07]. http://www.paper.edu.cn/releasepaper/content/201306-142.
8 赵士银, 周克元. 二阶全矩阵代数的模代数结构[J]. 山东大学学报(理学版), 2013, 48 (8): 24- 29.
ZHAO Shiyin , ZHOU Keyuan . The module algebra structures on M2(k)[J]. Journal of Shandong University(Natural Science), 2013, 48 (8): 24- 29.
9 高凤霞, 高雪琴, 杨士林. 二阶全矩阵代数的H8-模代数结构[J]. 吉林大学学报(理学版), 2015, 53 (5): 887- 892.
GAO Fengxia , GAO Xueqin , YANG Shilin . H8-module algebra structures on M2(k)[J]. Journal of Jilin University(Science Edition), 2015, 53 (5): 887- 892.
10 高凤霞, 杨士林. 3阶全矩阵代数的H8-模代数结构[J]. 数学年刊, 2017, 38A (2): 215- 226.
GAO Fengxia , YANG Shilin . The H8-module algebra structures on M2(k)[J]. Chinese Annals of Mathematics, 2017, 38A (2): 215- 226.
11 NASTASESCU C , VAN OYSTAEYEN F . Methods of graded rings[M]. Berlin: Springer, 2004: 261- 269.
12 魏丰, 史荣昌, 闫晓霞. 矩阵分析学习指导[M]. 北京: 北京理工大学出版社, 2005: 51- 65.
WEI Feng , SHI Rongchang , YAN Xiaoxia . Matrix analysis learning guide[M]. Beijing: Beijing Institute of Technology Press, 2005: 51- 65.
13 高雪琴. 非交换非余交换8维半单Hopf代数的模代数结构[D]. 北京: 北京工业大学, 2015.
GAO Xueqin. The module algebra structures of non-commutative and non-cocommutative 8-dimensional semisimple Hopf algebra[D]. Beijing: Beijing Institute of Technology, 2015.
14 MONTGOMERY S . Hopf algebras and their actions on rings[M]. Providence: American Mathematical Society, 1993: 40- 43.
[1] Jie TAN,Liang ZHANG,Jidong GUO. The number of homomorphisms from dihedral groups to any finite groups [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2023, 58(6): 31-34.
[2] ZHANG Wei, LAI Ji-na, GUO Ji-dong, ZHANG Liang. The number of homomorphisms between a class of finite 2-groups and dihedral groups [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2023, 58(3): 7-13.
[3] CAO Liu-feng. Maschke theorem for the Grothendieck algebra of dihedral group [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2023, 58(2): 44-50.
[4] XIE Wei, GUO Ji-dong. Number of homomorphisms between central-dihedral group and dihedral group [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2020, 55(8): 80-86.
[5] LAI Ji-na, GUO Ji-dong. Number of homomorphisms from a class of non-abelian groups into dihedral groups [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2020, 55(12): 30-36.
[6] LI Hong-xia, GUO Ji-dong, HAI Jin-ke. The number of homomorphisms from dihedral groups to a class of metacyclic groups [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(6): 34-40.
[7] CHEN Hua-xi, XU Qing-bing. The fundamental theorem forAMHH in Yetter-Drinfeld module categories [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(8): 107-110.
[8] ZHOU Nan, ZHANG Tao, LU Dao-wei. L-R-smash products over monoidal Hom-bialgebras [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(2): 5-8.
[9] GUO Shuang-jian, LI Yi-zheng. When is BHQ a pre-braided category over quasi-Hopf algebras [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(12): 10-15.
[10] WU Zi-juan, CHEN Yuan-yuan, ZHANG Liang-yun. Quasidimodule algebras over Hopf quasigroups and Yetter-Drinfeld quasimodule algebras [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2016, 51(10): 28-33.
[11] JIA Ling, CHEN Xiao-yuan. A duality theorem for a Yetter-Drinfeld Hopf algebra [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2015, 50(12): 98-101.
[12] ZHAO Shi-yin1,2, ZHOU Ke-yuan1. The module algebra structures on M2(k) [J]. J4, 2013, 48(8): 24-29.
[13] CHEN Hua-xi1, ZHANG Xiao-hui2, XU Qing-bing3. The Structure Theorem of weak comodule algebras in
Yetter-Drinfeld module categories [J]. J4, 2013, 48(12): 14-17.
[14] YU Wei-yan1,2， ZHANG Jian-hua1. Generalized Jordan derivations of full matrix algebras [J]. J4, 2010, 45(4): 86-89.
[15] LV Lin-yan . Improvement of relative Hopf module structure theorem in YetterDrinfeld category [J]. J4, 2006, 41(6): 90-92 .
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
[1] JIANG Xue-lian, SHI Hong-bo*. The learning algorithm of a generative and discriminative combination classifier[J]. J4, 2010, 45(7): 7 -12 .
[2] PENG Yan-fen,LI Bao-zong,LIU Tian-bao . Relationships between the structures and the anesthetic[J]. J4, 2006, 41(5): 148 -150 .
[3] YANG Lun, XU Zheng-gang, WANG Hui*, CHEN Qi-mei, CHEN Wei, HU Yan-xia, SHI Yuan, ZHU Hong-lei, ZENG Yong-qing*. Silence of PID1 gene expression using RNA interference in C2C12 cell line[J]. J4, 2013, 48(1): 36 -42 .
[4] LIU Yan-ping, WU Qun-ying. Almost sure limit theorems for the maximum of Gaussian sequences#br# with optimized weight[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2014, 49(05): 50 -53 .
[5] DU Ji-xiang1,2, YU Qing1, ZHAI Chuan-ming1. Age estimation of facial images based on non-negative matrix factorization with sparseness constraints[J]. J4, 2010, 45(7): 65 -69 .
[6] Ming-Chit Liu. THE TWO GOLDBACH CONJECTURES[J]. J4, 2013, 48(2): 1 -14 .
[7] WANG Kai-rong, GAO Pei-ting. Two mixed conjugate gradient methods based on DY[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2016, 51(6): 16 -23 .
[8] WU Dai-yong, ZHANG Hai. Stability and bifurcation analysis for a single population discrete model with Allee effect and delay[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2014, 49(07): 88 -94 .
[9] SUN Shou-bin,MENG Guang-wu ,ZHAO Feng . Dα-Continuity of order homomorphism[J]. J4, 2007, 42(7): 49 -53 .
[10] GUO Ting,BAO Xiao-ming . Influences of sitedirected mutagenesis on the enzymeactivity and thethermostability of the xylose isomerase from Thermus thermphilus[J]. J4, 2006, 41(6): 145 -148 .
Copyright 2018 © Editorial office of JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)
Supported by Beijing Magtech Co.Ltd