本原元只含不高于六阶若当块矩阵群的幂单性
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摘要
针对两个幂单矩阵生成的矩阵是否幂单的问题,先利用矩阵对数工具得到了自由群生成元的新的组合性质。从这些新的组合性质出发,证明了由一个若当块不高于二阶和若当块不高于五阶矩阵生成的群,在本原元若当块不高于六阶的情况下,当本原元素均幂单时,生成的群是幂单群。这样就可以得出线性表示像满足同样条件的自由群也是幂单群。
In order to solve the problem of whether the matrices generated by two unipotent matrices are unipotency, a new combinatorial property of generators of free groups is obtained by using the matrix logarithmic tool.From the point of the combination of these new properties, it is proved that a Jordan block is not higher than two and if the order when the block is not higher than five order matrix generated group, when the block is not higher than six order, the primitive element is unipotency, the group generated is unipotent group.
In order to solve the problem of whether the matrices generated by two unipotent matrices are unipotency, a new combinatorial property of generators of free groups is obtained by using the matrix logarithmic tool.From the point of the combination of these new properties, it is proved that a Jordan block is not higher than two and if the order when the block is not higher than five order matrix generated group, when the block is not higher than six order, the primitive element is unipotency, the group generated is unipotent group.
引文
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[3] TAVGEN O I, DU JUNHUA, LIU CHUNYAN. TheRepresentation Image Unipotency of the GroupF2by mApping Primitive Elements into Unipotent Matrices with Small Jordan Blocks [J]. Problems of Physics, Mathematics and Technics, 2011, 3(8): 81.
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[5] 杜君花, 谭鹏顺, 杨新松. 八阶线性群的幂单性质[J]. 哈尔滨商业大学自然科学学报, 2012, 28(4): 461.
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[7] AN Jianbei, Charles William Eaton, Nilpotent Blocks of Quasisimple Groups for the Prime Two[J], Algebras and Representation Theory, 2013,16(1):1.
[8] LIN Yanjun,SONG Xueling,XIONG Huan.Almost Simple DD-groups[J],北京大学学报(自然科学版),2013, 49(5):741.
[9] DAVID Seviri, NICO Groenewald. Generalization of Nilpotency of Ring Elements to Module Elements[J], Communications in Algebra, 2014,42(2):571.
[10] 杨新松, 刘欢. 二元生成自由群的幂单性质[J]. 哈尔滨理工大学学报, 2014, 19(6): 16.
[11] ISAAC.A.Garcia, HECTOR Giacomini, JAUME Gine, et al. Analytic Nilpotent Centers as Limits of Nondegenerate Centers Revisited[J]. Journal of Mathematical Analysis and Applications, 2015:893.
[12] JASON Bell, JAIRO Goncalves, Free Algebras and Free Groups in Ore Extensions and Free Group Algebras in Division Rings[J]. Journal of Algebra, 2015:235.
[13] YU Fengyao, BIN Shu, Nilpotent Orbits of Certain Simple Lie Algebras over Truncated Polynomial Rings[J]. Journal of Algebra, 2015:1.
[14] BILLIG. Yuly, FUTORNY. Vyacheslav, Classification of Irreducible Representations of Lie Algebra of Vector Fields on a Torus[J]. Journal of Algebra, 2016(11):199.
[15] BURDE. D, MOENS. W. Commutative Post-Lie Algebra Structures on Lie Algebras[J], Journal of Algebra,2016(12):183.
[16] 杨新松, 王那英. 标准型不高于五阶若当块矩阵群的幂单性[J]. 哈尔滨理工大学学报, 2016, 21(2): 112.
[17] NEFF. Patrizio, EIDEL. Bernhard, MARTIN, et al. Geometry of Logarithmic Strain Measures in Solid Mechanics[J]. Archive for Rational Mechanics and Analysis, 2016.11.507.
[18] DANCHEV. Peter Vassilev, LAM. Tsit Yuen. Rings with Unipotent Units[J], Publications Mathematicae Debrecen, 2016.449.
[19] ANDRADA. A, VILLACAMPA. R, Abelian Balanced Hermitian Structures on Unimodular Lie Algebras[J]. Transformation Groups, 2016.12.903.
[20] POGUDIN. G, RAZMYSLOV. Y, Prime Lie Algebras Satisfying the Standard Lie Identity of Degree 5[J]. Journal of Algebra, 2016(12):182.
[2] ТАВГЕНВ О И, САМСОНОВ Ю Б. Унипотентность Образа ПредставленияF2(x,y)Матрицами из GL(n,C), n=2,3,4, При Условии Отображения Образующих и Примитивных Элементов в Унипотентные Матрицы [J]. Докл. НАН Беларуси, 2001, 45(6): 29.
[3] TAVGEN O I, DU JUNHUA, LIU CHUNYAN. TheRepresentation Image Unipotency of the GroupF2by mApping Primitive Elements into Unipotent Matrices with Small Jordan Blocks [J]. Problems of Physics, Mathematics and Technics, 2011, 3(8): 81.
[4] ЯН СИНЬСУН. Линейные Прудставления Свободных Групп [M]. РИВШ: Минск, 2011: 11.
[5] 杜君花, 谭鹏顺, 杨新松. 八阶线性群的幂单性质[J]. 哈尔滨商业大学自然科学学报, 2012, 28(4): 461.
[6] 谭鹏顺. 本原元幂单的二元生成自由群的幂单性[D]. 哈尔滨:哈尔滨理工大学, 2013.
[7] AN Jianbei, Charles William Eaton, Nilpotent Blocks of Quasisimple Groups for the Prime Two[J], Algebras and Representation Theory, 2013,16(1):1.
[8] LIN Yanjun,SONG Xueling,XIONG Huan.Almost Simple DD-groups[J],北京大学学报(自然科学版),2013, 49(5):741.
[9] DAVID Seviri, NICO Groenewald. Generalization of Nilpotency of Ring Elements to Module Elements[J], Communications in Algebra, 2014,42(2):571.
[10] 杨新松, 刘欢. 二元生成自由群的幂单性质[J]. 哈尔滨理工大学学报, 2014, 19(6): 16.
[11] ISAAC.A.Garcia, HECTOR Giacomini, JAUME Gine, et al. Analytic Nilpotent Centers as Limits of Nondegenerate Centers Revisited[J]. Journal of Mathematical Analysis and Applications, 2015:893.
[12] JASON Bell, JAIRO Goncalves, Free Algebras and Free Groups in Ore Extensions and Free Group Algebras in Division Rings[J]. Journal of Algebra, 2015:235.
[13] YU Fengyao, BIN Shu, Nilpotent Orbits of Certain Simple Lie Algebras over Truncated Polynomial Rings[J]. Journal of Algebra, 2015:1.
[14] BILLIG. Yuly, FUTORNY. Vyacheslav, Classification of Irreducible Representations of Lie Algebra of Vector Fields on a Torus[J]. Journal of Algebra, 2016(11):199.
[15] BURDE. D, MOENS. W. Commutative Post-Lie Algebra Structures on Lie Algebras[J], Journal of Algebra,2016(12):183.
[16] 杨新松, 王那英. 标准型不高于五阶若当块矩阵群的幂单性[J]. 哈尔滨理工大学学报, 2016, 21(2): 112.
[17] NEFF. Patrizio, EIDEL. Bernhard, MARTIN, et al. Geometry of Logarithmic Strain Measures in Solid Mechanics[J]. Archive for Rational Mechanics and Analysis, 2016.11.507.
[18] DANCHEV. Peter Vassilev, LAM. Tsit Yuen. Rings with Unipotent Units[J], Publications Mathematicae Debrecen, 2016.449.
[19] ANDRADA. A, VILLACAMPA. R, Abelian Balanced Hermitian Structures on Unimodular Lie Algebras[J]. Transformation Groups, 2016.12.903.
[20] POGUDIN. G, RAZMYSLOV. Y, Prime Lie Algebras Satisfying the Standard Lie Identity of Degree 5[J]. Journal of Algebra, 2016(12):182.