Intersections of maximal subalgebras in Lie algebras

Abstract

It is well known that imposing relatively weak conditions on the subalgebra lattice of a (finite-dimensional) Lie algebra may have strong consequences for its structure. This is illustrated by the condition of duality: a Lie algebra is said to have a dual if its subalgebra lattice is anti-isomorphic to that one of another Lie algebra. Towers [3] proved that solvable Lie algebras with a dual are abelian or almost abelian, and that every Lie algebra with a dual is of this kind if the base field is algebraically closed of characteristic zero. But the general case remained open for arbitrary base fields. We depart from the apparently much weaker condition that every element of the lattice should be the inlimum of maximal elements. It turns out that for arbitrary base fields of characteristic zero every Lie algebra with this property is already self-dual, and either solvable or three-dimensional non-split simple. This gives a positive answer to the question raised at the end of [3].