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ON UPPER-MODULAR SUBALGEBRAS OF A LIE ALGEBRA

Published online by Cambridge University Press:  01 July 2004

Kevin Bowman ,
David A. Towers  and
Vicente R. Varea
Kevin Bowman
Affiliation:
Department of Physics, Astronomy and Mathematics, University of Central Lancashire, Preston PR1 2HE, UK
David A. Towers
Affiliation:
Department of Mathematics, Lancaster University, Lancaster LA1 4YF, UK (d.towers@lancaster.ac.uk)
Vicente R. Varea
Affiliation:
Department of Mathematics, University of Zaragoza, Zaragoza, 50009 Spain
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Abstract

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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

Keywords

Lie algebrasubalgebra latticeupper modular
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 47 , Issue 2 , June 2004 , pp. 325 - 337
Copyright
Copyright © Edinburgh Mathematical Society 2004