Basic Lie Algebra Of An Equivalent Characterizations

D.A. Towers([1]) brought up a question :"If L is a Lie algebra all of whose nilpotent subalgebras are abelian , does L split over each of its ideals?", and he gave a counterexample to tell that this problem was untrue in a general situation .This thesis explains the reason that this problem is not true by analysing the counterexample ,and lists the conditions that satify this problem .Finally this thesis gives an equivalent description of elementary Lie algebra in a general situation.It is organized as follows:Chapter 1 is an introduction. We present the background of the problem discussed in this paper and the basic concept and the research situation of the Frattini subalgebra , the Frattini ideal and elementary Lie algebra.Chapter 2 is preliminaries. We give the definitions of the Frattini subalgebra , elementary Lie algebra and splittable Lie algebra , and some relevant lemmas . Then we specially introduce Theorem 2.2 of E. I. Marshall in [4] and Theorem 2.12 of Lu Caihui in [3] .Chapter 3 is a deep thinking to the counterexample L1 given by D.A.Towers. Suppose that L1 is a finite dimensional Lie algebra over a field of algebraically closed and characteristic zero , we can prove that L1 is not an elementary Lie algebra ,although each of its nilpotent subalgebra is abelian . Furthermore we point out that L1 is not a splittable Lie algebra.Chapter 4 In this chapter we prove the main theorem of this paper Theorem 4.3:Suppose L is a finite dimensional and splittable Lie algebra over a field of algebraically closed and characteristic zero ,then L is an elementary Lie algebra if and only if each nilpotent subalgebra of L is abelian.