12,000

We have over 12,000 students, from over 100 countries, within one of the safest campuses in the UK

93%

93% of Lancaster students go into work or further study within six months of graduating

Home > Research > Publications & Outputs > On Lie algebras all of whose minimal subalgebra...
View graph of relations

« Back

On Lie algebras all of whose minimal subalgebras are lower modular.

Research output: Contribution to journalJournal article

Published

Journal publication date2004
JournalCommunications in Algebra
Journal number12
Volume32
Number of pages19
Pages4515-4533
Original languageEnglish

Abstract

The main purpose of this paper is to study Lie algebras L such that if a subalgebra U of L has a maximal subalgebra of dimension one then every maximal subalgebra of U has dimension one. Such an L is called lm(0)-algebra. This class of Lie algebras emerges when it is imposed on the lattice of subalgebras of a Lie algebra the condition that every atom is lower modular. We see that the effect of that condition is highly sensitive to the ground field F. If F is algebraically closed, then every Lie algebra is lm(0). By contrast, for every algebraically non-closed field there exist simple Lie algebras which are not lm(0). For the real field, the semisimple lm(0)-algebras are just the Lie algebras whose Killing form is negative-definite. Also, we study when the simple Lie algebras having a maximal subalgebra of codimension one are lm(0), provided that the characteristic of F is different from 2. Moreover, lm(0)-algebras lead us to consider certain other classes of Lie algebras and the largest ideal of an arbitrary Lie algebra L on which the action of every element of L is split, which might have some interest by themselves.

Bibliographic note

The final, definitive version of this article has been published in the Journal, Communications in Algebra, 32 (12), 2004, © Informa Plc