The index complex of a maximal subalgebra of a Lie algebra

Abstract

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.

c-Sections of Lie algebras

“…Let M be a maximal subalgebra of L. We say that a chief factor C/D of L supplements M in L if L = C + M and D ⊆ C ∩ M ; if D = C ∩ M we say that C/D complements M in L. In [11] we defined the ideal index of a maximal subalgebra M of L, denoted by η(L : M ), to be the well-defined dimension of a chief factor C/D where C is an ideal minimal with respect to supplementing M in L. Here we introduce a further concept which is related to the previous two.…”

On abelian subalgebras and ideals of maximal dimension in supersolvable Lie algebras

“…If M is an ideal of L we have finished. So suppose that M is self-idealising, in which case L 2 is one-dimensional and φ(L) = 0, by [19,Proposition 3.2]. Write L 2 = F b and note that AsocL = L 2 ⊕ Z(L).…”

On theta pair for a proper subalgebra

“…The relation between the properties of subalgebras of a Lie algebra L and the structure of L has been studied extensively. In [8] and [10], Towers introduced, respectively, the concepts of c-ideality of subalgebras and the ideal index of maximal subalgebras of a Lie algebra and showed that these concepts play important roles in the study of Lie algebra theory (see also [6]). The first two authors in [5] presented the notion of θ -pair for maximal subalgebras and investigated its influence on the structure of some certain Lie algebras.…”