三维单李代数sl(2)的双导子与交换映射
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摘要
论文完全决定了3维单李代数sl(2)的双导子与线性交换映射.特别地,证明了3维单李代数sl(2)的双导子都是内双导子.利用此结果,给出了每个3维单李代数sl(2)的线性交换映射的精确形式.特别地,证明了3维单李代数sl(2)的线性交换映射都是标准线性交换映射.
In the paper, the biderivations and linear commuting maps on the 3-dimentional simple Lie algebra sl(2) were completely determined. In particular, all the biderivations of the 3-dimentional simple Lie algebra sl(2) were shown to be inner biderivations. By the result, the explicit forms of each linear commuting map of the 3-dimentional simple Lie algebra sl(2) were given. In particular, all the linear commuting maps on the 3-dimentional simple Lie algebra sl(2) were shown to be standard linear commuting maps.
In the paper, the biderivations and linear commuting maps on the 3-dimentional simple Lie algebra sl(2) were completely determined. In particular, all the biderivations of the 3-dimentional simple Lie algebra sl(2) were shown to be inner biderivations. By the result, the explicit forms of each linear commuting map of the 3-dimentional simple Lie algebra sl(2) were given. In particular, all the linear commuting maps on the 3-dimentional simple Lie algebra sl(2) were shown to be standard linear commuting maps.
引文
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[2] BEIDAR K I, BRESAR M, CHEBOTAR M A, et al. Applying functional identities to some linear preserver problems[J]. Pacific J Math, 2002, 204: 257-271.
[3] HAN X, OU S K, XIA C G. Non-linear k-potence preserving maps on commuting symmetric matrices[J]. Journal of Anhui University (Natural Science Edition), 2011, 35(6): 15-19.
[4] BRESAR M. Commuting maps: a survey[J]. Taiwanese J Math, 2004, 8: 361-397.
[5] WANG D Y, YU X X. Biderivations and linear commuting maps on the Schrodinger-Virasoro Lie algebra[J]. Comm Algebra, 2013, 41: 2166-2173.
[6] HAN X, WANG D Y, XIA C G. Linear commuting maps and biderivations on the Lie algebras W(a, b)[J]. Journal of Lie Theory, 2016, 26: 777-786.
[7] XIA C G, WANG D Y, HAN X. Linear super-commuting maps and super-biderivations on the super-Virasoro algebras[J]. Comm Algebra, 2016, 44: 5342-5350.
[8] HUMPHREYS J E. Introduction to Lie algebras and representation theory[M]. Berlin: Springer, 1972.
[9] 万哲先. 李代数[M]. 北京: 科学出版社, 1964.
[10] 孟道骥. 复半单李代数引论[M]. 北京: 北京大学出版社, 1998.
[11] BRESAR M. On generalized biderivations and related maps[J]. J Algebra, 1995, 172: 764-786.
[12] BRESAR M, MARTINDAL E, MIERS C R. Centralizing maps on prime rings with involution[J]. J Algebra, 1993, 335: 525-546.
[13] WANG D Y, YU X X, CHEN Z X. Biderivations of the parabolic subalgebras of simple Lie algebras[J]. Comm Algebra, 2011, 39: 4097-4104.