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数学进展 - 2016, Vol. 45(2): 211-220
研究论文
3-李代数的包络 \mbox{\boldmath$B$}-代数
Enveloping \mbox{\boldmath$B$}-algebras of \mbox{\boldmath$3$}-Lie Algebras}

白瑞蒲1,安宏伟2,郭委委1,林丽鑫1
BAI Ruipu1, AN Hongwei2, GUO Weiwei1, LIN Lixin1

1. 河北大学数学与信息科学学院, 保定, 河北, 071002;
2. 石家庄铁道学院四方学院, 石家庄, 河北, 051132
1. College of Mathematics and Information Science,Hebei University, Baoding, Hebei, 071002, P. R.China;
2. Sifang College, Shijiazhuang Tiedao University, Shijiazhuang, Hebei, 051132, P. R. China

收稿日期: 2014-11-12
2016, Vol. 45(2): 211-220
DOI: 10.11845/sxjz.2014182b


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摘要 定义并构造了3-李代数的包络$B$-代数, 研究了3-李代数的标准包络$B$-代数的结构, 证明了3-李代数是单3-李代数的充分必要条件是其标准的包络$B$-代数${\rm SB}(L)_{\theta是单的.
Abstract:$B$-algebras and enveloping $B$-algebras of $3$-Lie algebras are introduced, and enveloping $B$-algebras of $3$-Lie algebras are constructed. The structures of standard enveloping $B$-algebras of $3$-Lie algebras are studied, and it is proved that the $3$-Lie algebra $L$ is simple if and only if its standard enveloping $B$-algebra ${\rm SB}(L)_{\theta is simple.
PACS:  O152.5  
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[2] Burdujan, I., A universal enveloping algebra for a Lie triple system, In: Proceedings of the 3rd International Colloquium Mathematics in Engineering and Numerical Physics, Bucharest, Romania, 2004, 59-71.
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[4] Filippov, V.T., Enveloping anticommutative triple systems and $3$-Lie algebras, Algebra Logika, 1990, 29(2): 241-257.
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[6] Lister, W.G., A structure theory of Lie triple system, Trans.Aer. Math. Soc., 1952, 72(2): 217-242.
[7] Zhang, Z.X., Shi, Y.Q. and Zhao, L.N., Invariant symmetric bilinear forms on Lie triple systems, Comm. Algebra, 2002, 30(11): 5563-5573.
[1] 白瑞蒲, 郭委委, 陈双双, 林丽鑫. ${3}$-李代数不可分解的~${\bm {T^*_{\theta}}}$-扩张[J]. 数学进展, 2016, 45(3): 365-370.
[2] 余德民,卢才辉. 无限维项链李代数的若干有限维李子代数[J]. 数学进展, 2015, 44(6): 816-826.
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