David A. Towers

C-IDEALS OF LIE ALGEBRAS

A subalgebra B of a Lie algebra L is called a c-ideal of L if there is an ideal C of L such that L = B + C and B ∩ C ≤ B L , where B L is the largest ideal of L contained in B. This is analogous to the concept of c-normal subgroup, which has been studied by a number of authors. We obtain some properties of c-ideals and use them to give some characterisations of solvable and supersolvable Lie algebras. We also classify those Lie algebras in which every one-dimensional subalgebra is a c-ideal.

On Lie algebras in which modular pairs of subalgebras are permutable

the vector space sum of A and B. As we shall see, it is easy to show that (A, B) is a modular pair whenever A permutes with B. However, the converse is false. We propose to study here Lie algebras in which modular pairs of subalgebras are permutable. The corresponding problem for groups was considered by Iwasawa who proved [3, Satz 10, p. 1861 that a finite group is nilpotent if and only if every modular pair of subgroups is permutable. It will become apparent that the situation for Lie algebras is quite different. All Lie algebras considered will be finite dimensional over a field F. Denote by Fp the class of (finite-dimensional) Lie algebras L over a (fixed) field F and having the property that all modular pairs of subalgebras of L are permutable. It will prove convenient to define also a related class Q consisting of those Lie algebras L over F with the property that A permutes with B whenever A and B are subalgebras of L covered by A U B (i.e., maximal in A U B). It is clear that !I3 and Q are each closed under the taking of subalgebras and factor algebras. In Section 1 it is noted that any two maximal subalgebras form a modular 369 002 I -8693,‘8 1~020369-099602.001’0

Lie algebras all of whose proper subalgebras are nilpotent

Nonnilpotent Lie algebras all of whose proper subalgebras are nilpotent are studied. A fairly complete description is given of the nonperfect algebras in this class over a wide range of fields (including all perfect fields).

The index complex of a maximal subalgebra of a Lie algebra.

Let M be a maximal subalgebra of the Lie algebra L. A subalgebra C of L is said to be a completion for M if C is not contained in M but every proper subalgebra of C that is an ideal of L is contained in M. The set I(M) of all completions of M is called the index complex of M in L. We use this concept to investigate the influence of the maximal subalgebras on the structure of a Lie algebra, in particular finding new characterisations of solvable and supersolvable Lie algebras.

Complements of intervals and prefrattini subalgebras of solvable Lie algebras

In this paper we study a Lie-theoretic analogue of a generalisation of the prefrattini subgroups introduced by W. Gasch¨utz. The approach follows that of P. Hauck and H. Kurtzweil for groups, by first considering complements in subalgebra intervals. Conjugacy of these subalgebras is established for a large class of solvable Lie algebras.

Dualisms of Lie algebras

Relationships between the structure of a Lie algebra and that of its lattice of subalgebras have been studied by several authors; for instance, Kolman ([4] and [5J) studied modular, semi-modular and complemented Lie algebras, whilst Barnes [2] and Barnes and Wall [l] . investigated lattice isomorphisms between Lie algebras. In this paper we determine those Lie algebras which permit dual isomorphisms from their lattice of subalgebras onto that of another Lie algebra. Analogous questions for groups have been considered (see [7] for some results and references). One would expect the condition that L have a dual to be very restrictive, because the abundance of l-dimensional subalgebras possessed by the dual implies the existence of the same abundance of maximal subalgebras of L. We shall see that this expectation is realised; over algebraically closed fields of characteristic zero, the only Lie algebras permitting dualisms are very close to being abelian. If we restrict attention to solvable Lie algebras the conditions on the ground field are unnecessary.

Subalgebras that cover or avoid chief factors of Lie algebras

We call a subalgebra

Two generator subalgebras of Lie algebras

U

On almost nilpotent-by-abelian lie algebras

of a Lie algebra

The Frattini p-subalgebra of a solvable Lie p-algebra

L