最简线状n-李代数

On the Simplest Filiform n-Lie Algebras

定义了一类特殊的幂零n-李代数,即最简线状n-李代数,它是最简线状李代数的推广.确定了m维最简线状n-李代数A的导子代数Der(A)和自同构群Aut(A),定义了n-李代数的全形h(A)=Der(A)(?)A,并证明了当A的基域F的特征p为零或p>m-n时,Der(A)是不可解的完备李代数,而h(A)的一个子代数是可解的完备李代数,当F的特征为零时,Aut(A)是无中心的不可解群.

A kind of nilpotent n-Lie algebras-the simplest filiform n-Lie algebras is defined, and the derivation algebra Der（A） and automorphism group Aut（A） of such an mdimensional algebra A are determined. It is proved that Der（A） and Aut（A） are unsolvable, but Der（A） is complete when the characteristic p of the base field F is zero or p 〉 m - n. Moreover, the authors give a definition of the holomorph h（A） = Der（A） ＋ A for an n-Lie algebra A, and show that h（A） has a solvable complete Lie algebra, if p = 0 or p 〉 m - n. And also prove that Aut（A） is centerless, when p = 0, which is different from the case of the usual simplest filiform Lie algebras.