On Upper-Modular Subalgebras of a Lie Algebra

Abstract

This paper is a further contribution to the extensive study by a number of authors of the subalgebra lattice of a Lie algebra. We give some necessary and some sufficient conditions for a subalgebra to be upper modular. For algebraically closed fields of any characteristic these enable us to determine the structure of Lie algebras having abelian upper-modular subalgebras which are not ideals. We then study the structure of solvable Lie algebras having an abelian upper-modular subalgebra which is not an ideal and which has trivial intersection with the derived algebra; in particular, the structure is determined for algebras over the real field. Next we classify non-solvable Lie algebras over fields of characteristic zero having an upper-modular atom which is not an ideal. Finally, it is shown that every Lie algebra over a field of characteristic different from two and three in which every atom is upper modular is either quasi-abelian or a µ-algebra.