Lower semimodular Lie algebras

Abstract

This paper is concerned with the relationship between the properties of the subalgebra lattice ℒ(L) of a Lie algebra L and the structure of L. If the lattice ℒ(L) is lower semimodular, then the Lie algebra L is said to be lower semimodular. If a subalgebra S of L is a modular element in the lattice ℒ(L), then S is called a modular subalgebra of L. The easiest condition to ensure that L is lower semimodular is that dim A/B = 1 whenever B < A ≤ L and B is maximal in A (Lie algebras satisfying this condition are called sχ-algebras). Our aim is to characterize lower semimodular Lie algebras and sχ-algebras, over any field of characteristic greater than three. Also, we obtain results about the influence of two solvable modularmaximal subalgebras on the structure of the Lie algebra and some results on the structure of Lie algebras all of whose maximal subalgebras are modular.