Minimal Nonnilpotent Solvable Lie Algebras

Abstract

Abstract.We shall say that a solvable Lie algebra L is a minimal nonnilpotent Lie algebra if L is nonnilpotent but all proper subalgebras of L are nilpotent. It is shown here that if L is a minimal nonnilpotent Lie algebra, then L is the vector space direct sum of A and F where A is an ideal in L, F is a one-dimensional subalgebra of L, either A is a minimal ideal of L or the center of A coincides with the derived algebra, A', of A and in either case facts irreducibly on A/A'. P. Hall and G. Higman have shown in [5] that a nonnilpotent finite group G all of whose proper subgroups are nilpotent can be considered as the product of subgroups P and Q where P is cyclic of prime power order, pa, Q is an invariant g-subgroup of G, q^p, 4>(P) ^Z(G) and Q is either elementary abelian or <j>(Q) = Z(Q) = [Q, Q] where in either case P acts irreducibly on Q/<j>(Q). We shall find a result on solvable Lie algebras which is roughly analogous to this result.Let L be a finite-dimensional solvable Lie algebra and let M be a self-normalizing maximal subalgebra of L. The maximal ideal of L contained in M will be called the core of M. The intersection of all maximal subalgebras of L will be denoted by <p(L) and is an ideal in L by Lemma 3.4 of [2]. The derived algebra of L will be denoted by L' and the center of L by Z(L). We first show the following result, the group theory analogue of which is shown in [4]. Proposition.Let L be a solvable Lie algebra and let H be a selfnormalizing maximal subalgebra of L. Let K be the core of H. Then(1) L/K contains a unique minimal ideal A/K,

Two generator subalgebras of Lie algebras

“…Then every two-dimensional subspace of L is a subalgebra of L, from which it follows that every subspace of L is a subalgebra of L, and hence that L is quasi-abelian. The structure of solvable minimal non-abelian Lie algebras has been fully described by Stitzinger in [13,Theorem 1]. If L is such a Lie algebra, then L = A+F x, where A is an abelian ideal of L and ad x acts irreducibly on A. Non-solvable minimal non-abelian Lie algebras have been studied by Farnsteiner in [5] and Gein in [6] (see also Elduque [3]).…”

“…Minimal non-solvable Lie algebras over an algebraically closed field of prime characteristic have been studied by Varea in [21] Minimal non-nilpotent Lie algebras have been studied by Stitzinger in [13], Gein and Kuznecov [9], Towers [16] and Farnsteiner [5]. In Gein [8], it is proved that if L is simple and minimal non-nilpotent then the intersection of two distinct maximal subalgebras of L is zero and L has no non-zero ad-nilpotent elements.…”

Minimal Nonnilpotent Leibniz Algebras

“…In Lie algebras [12,13] prove that A 3 = 0. We recover this result for the case where L is a Lie algebra and generalize to A 3 ≤ Leib(L) in the non-Lie case.…”

“…An algebra L is called minimal nonnilpotent if L is nonnilpotent, solvable, and all proper subalgebras of L are nilpotent. Minimal nonnilpotent Lie algebras were studied by Stitzinger in [12]. Later Towers classified all such Lie algebras in [13].…”