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Further results on elementary Lie algebras and Lie A-algebras.

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<mark>Journal publication date</mark>2013
<mark>Journal</mark>Communications in Algebra
Issue number4
Number of pages10
Pages (from-to)1432-1441
Publication StatusPublished
<mark>Original language</mark>English


A finite-dimensional Lie algebra L over a field F of characteristic zero is called elementary if each of its subalgebras has trivial Frattini ideal; it is an A-algebra if every nilpotent subalgebra is abelian. This paper is a continuation of the study of these algebras initiated by the authors in a previous paper. If we denote by $\mathcal{A}$, $\mathcal{G}$, $\mathcal{E}$, $\mathcal{L}$, $\Phi$ the classes of A-algebras, almost algebraic algebras, E-algebras, elementary algebras and $\phi$-free algebras respectively, then it is shown that: \mathcal{L} \subset \Phi \subset \mathcal{G} \mathcal{L} \subset \mathcal{A} \subset \mathcal{E} \mathcal{G} \cap \mathcal{A} = \mathcal{L}. It is also shown that if L is a semisimple Lie algebra all of whose minimal parabolic subalgebras are $\phi$-free then L is an A-algebra, and hence elementary. This requires a number of quite delicate properties of parabolic subalgebras. Finally characterisations are given of $E$-algebras and of Lie algebras all of whose proper subalgebras are elementary.

Bibliographic note

The final, definitive version of this article has been published in the Journal, Communications in Algebra, 41 (4), 2013, © Informa Plc