Restricted Lie algebras having a distributive lattice of restricted subalgebras

Abstract

Let L be a restricted Lie algebra over a field of characteristic p>0. We investigate the structure of L when its lattice of restricted subalgebras satisfies some prescribed properties. In particular, we establish when is distributive. The special form that this result takes when is either the field with p elements or an algebraically closed field is also discussed. Furthermore, we establish when is Boolean or the two-elements lattice.