2009/04/20 by David A. Towers, Towers, David A., Vicente R. Varea +1
Chemistry · Mathematics · #17B05 #17B20 #17B30 #17B50 (Primary) #20D10 #20D15 #20D25 (Secondary) #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Carbohydrate Chemistry and Synthesis #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.0904.3010
openalex publication_date 2009/04/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 `Elementary Lie Algebras and Lie A-algebras', J. Algebra 312 (2007), 891--901. If we denote by A, G, E, L, Φ the classes of A-algebras, almost algebraic algebras, E-algebras, elementary algebras and ϕ-free algebras respectively, then it is shown that: L ⊂ Φ⊂ G, L ⊂ A ⊂ E and G ∩ A = 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.