The Frattini Subalgebra of Restricted Lie Superalgebras  

作　　者：Liang Yun CHEN Dao Ji MENG Yong Zheng ZHANG 

机构地区：[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

出　　处：《Acta Mathematica Sinica,English Series》2006年第5期1343-1356,共14页数学学报（英文版）

基　　金：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.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：restricted Lie superalgebras E-p-restricted Lie superalgebras Frattini p-subalgebra Фp- free p-elementary 

分 类 号：O152.5[理学—数学]