Further results on elementary Lie algebras and Lie A-algebras

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dc.creatorTowers, David A.
dc.creatorVarea, Vicente R.
dc.date2009-04-20
dc.date.accessioned2026-07-07T13:05:57Z
dc.date.available2026-07-07T13:05:57Z
dc.descriptionA 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 `Elementary Lie Algebras and Lie A-algebras', J. Algebra 312 (2007), 891--901. If we denote by $\mathcal{A}$, $\mathcal{G}$, $\mathcal{E}$, $\mathcal{L}$, $Φ$ the classes of $A$-algebras, almost algebraic algebras, $E$-algebras, elementary algebras and $ϕ$-free algebras respectively, then it is shown that: \mathcal{L} \subset Φ\subset \mathcal{G}, \mathcal{L} \subset \mathcal{A} \subset \mathcal{E} and \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 $ϕ$-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.
dc.identifierhttps://arxiv.org/abs/0904.3010
dc.identifierhttp://arxiv.org/abs/0904.3010
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/227655
dc.subjectRings and Algebras
dc.subjectRepresentation Theory
dc.subject17B05, 17B20, 17B30, 17B50 (Primary); 20D10, 20D15, 20D25 (Secondary)
dc.titleFurther results on elementary Lie algebras and Lie A-algebras
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