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The Frattini Subalgebra of Restricted Lie Superalgebras
作者姓名:Liang  Yun  CHEN  Dao  Ji  MENG  Yong  Zheng  ZHANG
作者单位:[1]Department of Mathematics, Harbin Institute of Technology, Harbin 150001, P. R. China [2]Department of Mathematics, Northeast Normal University, Changchun 130024, P. R. China [3]Department of Mathematics, Nankai University, Tianjin 300071, P. R. China
基金项目:Supported by National Natural Science Foundation of China (Grant No. 10271076).
摘    要:In the present paper, we study the Frattini subalgebra of a restricted Lie superalgebra (L, p]). We show first that if L = A1 + A2 +… +An, then Фp(L) = Фp(A1) +Фp(A2) +…+Фp(An), where each Ai is a p-ideal of L. We then obtain two results: F(L) = Ф(L) = J(L) = L if and only if L is nilpotent; Fp(L) and F(L) are nilpotent ideals of L if L is solvable. In addition, necessary and sufficient conditions are found for Фp-free restricted Lie superalgebras. Finally, we discuss the relationships of E-p-restricted Lie superalgebras and E-restricted Lie superalgebras.

关 键 词:限制李超级代数  E-p限制李超级代数  Frattini  p-子代数  p-元素
收稿时间:2003-12-09
修稿时间:2003-12-092004-09-14

The Frattini Subalgebra of Restricted Lie Superalgebras
Liang Yun CHEN Dao Ji MENG Yong Zheng ZHANG.The Frattini Subalgebra of Restricted Lie Superalgebras[J].Acta Mathematica Sinica,2006,22(5):1343-1356.
Authors:Liang Yun Chen  Dao Ji Meng  Yong Zheng Zhang
Institution:(1) Department of Mathematics, Harbin Institute of Technology, Harbin 150001, P. R. China;(2) Department of Mathematics, Northeast Normal University, Changchun 130024, P. R. China;(3) Department of Mathematics, Nankai University, Tianjin 300071, P. R. China;(4) Department of Mathematics, Northeast Normal University, Changchun 130024, P. R. China
Abstract:In the present paper, we study the Frattini subalgebra of a restricted Lie superalgebra (L, p]). We show first that if L = A 1A 2 ⊕⋯⊕ A n An, then ϕp(L) = ϕp(A 1) + ϕp(A 2) +⋯+ ϕp(A n ), where each A i is a p–ideal of L. We then obtain two results: F(L) = ϕ(L) = J(L) = L (1) if and only if L is nilpotent; F p (L) and F(L) are nilpotent ideals of L if L is solvable. In addition, necessary and sufficient conditions are found for ϕp–free restricted Lie superalgebras. Finally, we discuss the relationships of Ep–restricted Lie superalgebras and E–restricted Lie superalgebras. Supported by National Natural Science Foundation of China (Grant No. 10271076)
Keywords:restricted Lie superalgebras            E  p  restricted Lie superalgebras  Frattini p  subalgebra            φ  p  free            p  elementary
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