Vicente R. Varea

Two generator subalgebras of Lie algebras

In [Thompson, J., 1968, Non-solvable finite groups all of whose local subgroups are solvable. Bulletin of the American Mathematical Society, 74, 383–437.], Thompson showed that a finite group G is solvable if and only if every two-generated subgroup is solvable (Corollary 2, p. 388). Recently, Grunevald et al. [Grunewald et al., 2000, Two-variable identities in groups and Lie algebras. Rossiiskaya Akademiya Nauk POMI, 272, 161–176; 2003. Journal of Mathematical Sciences (New York), 116, 2972–2981.] have shown that the analogue holds for finite-dimensional Lie algebras over infinite fields of characteristic greater than 5. It is a natural question to ask to what extent the two-generated subalgebras determine the structure of the algebra. It is to this question that this article is addressed. Here, we consider the classes of strongly-solvable and of supersolvable Lie algebras, and the property of triangulability.

Further Results on Elementary Lie Algebras and Lie A-Algebras

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 article is a continuation of the study of these algebras initiated by the authors in [10]. 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: 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.

On upper modular subalgebras of a Lie algebra.

This paper is a further contribution to the extensive study by a number of authors of the subalgebra lattice of a Lie algebra. We give some necessary and some sufficient conditions for a subalgebra to be upper modular. For algebraically closed fields of any characteristic these enable us to determine the structure of Lie algebras having abelian upper modular subalgebras which are not ideals. We then study the structure of solvable Lie algebras having an abelian upper modular subalgebra which is not an ideal and which has trivial intersection with the derived algebra; in particular the structure is determined for algebras over the real field. Next we classify non-solvable Lie algebras over fields of characteristic zero having an upper modular atom which is not an ideal. Finally it is shown that every Lie algebra over a field of characteristic different from two and three in which every atom is upper modular is either quasi-abelian or a μ-algebra.

The lattice of ideals of a nilpotent lie Algebra

Let L be a nilpotent Lie algebra over a field of characteristic zero. We are interested in the relationship between the structure of L and that the lattice I(L) of all ideals of L. We prove that the upper and the lower central series of L, as well as the dimensionality of each one oftheir factors, are determined by the lattice I(L). We show that if the ideal lattice of L is isomorphic to that of a non-nilpotent Lie algebra M, then the structures of L and M are both severely constrained. We are able to classify such Lie algebras L and M provided that the index of nilpoteney of L is ≤ 3 and the ground field is the real number field.

On Lie Algebras All of Whose Minimal Subalgebras Are Lower Modular

Abstract The main purpose of this paper is to study Lie algebras L such that if a subalgebra U of L has a maximal subalgebra of dimension one then every maximal subalgebra of U has dimension one. Such an L is called lm(0)-algebra. This class of Lie algebras emerges when it is imposed on the lattice of subalgebras of a Lie algebra the condition that every atom is lower modular. We see that the effect of that condition is highly sensitive to the ground field F. If F is algebraically closed, then every Lie algebra is lm(0). By contrast, for every algebraically non-closed field there exist simple Lie algebras which are not lm(0). For the real field, the semisimple lm(0)-algebras are just the Lie algebras whose Killing form is negative-definite. Also, we study when the simple Lie algebras having a maximal subalgebra of codimension one are lm(0), provided that char(F) ≠ 2. Moreover, lm(0)-algebras lead us to consider certain other classes of Lie algebras and the largest ideal of an arbitrary Lie algebra L on which the action of every element of L is split, which might have some interest by themselves.