a1 Department of Physics, Astronomy and Mathematics, University of Central Lancashire, Preston PR1 2HE, UK
a2 Department of Mathematics, Lancaster University, Lancaster LA1 4YF, UK (email@example.com)
a3 Department of Mathematics, University of Zaragoza, Zaragoza, 50009 Spain
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 $\mu$-algebra.
AMS 2000 Mathematics subject classification: Primary 17B05; 17B50; 17B30; 17B20
(Received January 30 2003)