Proceedings of the Edinburgh Mathematical Society (Series 2)

Research Article

ON UPPER-MODULAR SUBALGEBRAS OF A LIE ALGEBRA

Kevin Bowmana1, David A. Towersa2 and Vicente R. Vareaa3

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 (d.towers@lancaster.ac.uk)

a3 Department of Mathematics, University of Zaragoza, Zaragoza, 50009 Spain

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 $\mu$-algebra.

AMS 2000 Mathematics subject classification: Primary 17B05; 17B50; 17B30; 17B20

(Received January 30 2003)

Keywords

  • Lie algebra;
  • subalgebra lattice;
  • upper modular