华东师范大学学报(自然科学版) ›› 2003, Vol. 2003 ›› Issue (3): 1-8.

• 论文 •    下一篇

最简线状李代数

林磊   

  1. 华东师范大学 数学系，上海 200062
  • 收稿日期:1900-01-01 修回日期:1900-01-01 出版日期:2003-09-10 发布日期:2003-09-10

On the Simplest Filiform Lie Algebras

LIN Lei   

  1. Department of Mathematics, East China Normal University, Shanghai 200062,China
  • Received:1900-01-01 Revised:1900-01-01 Online:2003-09-10 Published:2003-09-10

摘要：

作者定义了一类线状李代数，即所谓的最简线状李代数，它是一类结构最简单的线状李代数，也是Luis Boza, Francisco J. Echarte 和 Juan Nunez在1994年对复数域上的10维线状李代数的分类中所提到的参数全为零的代数μ10130的推广。设g是域F上的n维最简单的线状李代数（n≧4）,确定了g的导子代数，并且证明了当F 的特征为0或p＞n-2时g的导子代数不可解的完备李代数。

还计算了g的自同构群，并证明了当∣F∣≥n时它是一个无中心的可解群。此外，对于素特征的的情形，还考虑了g 的可限制的充要条件，并对非可限制的情形确定了g 的极小p—包络。

关键词: 线状李代数, 完备李代数, 模李代数, 线状李代数, 完备李代数, 模李代数

Abstract: In this paper,the concept of simplest filiform Lie algebra is introduced and properties of n-dimensional simplest filiform Lie algebra g are discussed. A filiform Lie algebra g over any field F is referred to as simplest if its Lie multiplication between elements of an adapted basis is trivial except for ones defining the basis. Almost all discussions in the article are under the assumption o dim g≥4. A realization of such a Lie algebra g is given and derivation algebra Der g is determined. It is proved that if for the base field F, its characteristic is zero or p＞n-2 the derivation algebra Der g is a solvable complete Lie algebra.Automorphism groups of simplest filiform Lie algebra are determined and it is proved that it is proved that they are centerless and solvable if ∣F∣≥n. At the end of the paper ,it is shown that the simplest filiform moludar Lie algebra g is restrictable if and only if n＜p+2. And for the nonrestricable case a minimal p-envelope of g is given.

Key words: complete Lie algebra, modular Lie algebra , filiform Lie algebra, complete Lie algebra, modular Lie algebra