Rights statement: The final, definitive version of this article has been published in the Journal, Communications in Algebra, 41 (4), 2013, © Informa Plc
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TY - JOUR
T1 - Further results on elementary Lie algebras and Lie A-algebras.
AU - Towers, David A.
AU - Varea, Vicente R.
N1 - The final, definitive version of this article has been published in the Journal, Communications in Algebra, 41 (4), 2013, © Informa Plc
PY - 2013
Y1 - 2013
N2 - 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.
AB - 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.
KW - Lie algebra
KW - elementary
KW - E-algebra
KW - A-algebra
KW - almost algebraic
KW - ad-semisimple
KW - parabolic subalgebra
U2 - 10.1080/00927872.2011.643667
DO - 10.1080/00927872.2011.643667
M3 - Journal article
VL - 41
SP - 1432
EP - 1441
JO - Communications in Algebra
JF - Communications in Algebra
SN - 0092-7872
IS - 4
ER -