On -Lie algebras with abelian ideals and subalgebras

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• 作者：RuiPu Bai^a ; ^{bairp1@yahoo.com.cn} ; Lihong Zhang^a ; ^{zhanglihong8430@163.com} ; Yong Wu^a ; ^{wuyg1022@sina.com} ; Zhenheng Li^b ; ^{zhenhengl@usca.edu}
• 关键词：17B05 ; 17B60
• 刊名：Linear Algebra and its Applications
• 出版年：2013
• 出版时间：1 March, 2013
• 年：2013
• 卷：438
• 期：5
• 页码：2072-2082
• 全文大小：283 K

文摘

In this paper, we study the maximal dimension of abelian subalgebras and the maximal dimension of abelian ideals of m-dimensional -Lie algebras over an algebraically closed field. We show that these dimensions do not coincide if the field is of characteristic zero, even for nilpotent -Lie algebras. We then prove that 3-Lie algebras with are 2-step solvable (see definition in Section 2). Furthermore, we give a precise description of these 3-Lie algebras with one or two dimensional derived algebras. In addition, we provide a classification of -Lie algebras with . We also obtain the classification of -Lie algebras with and with their derived algebras of one dimension.