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

Research output: Contribution to journalJournal article

<mark>Journal publication date</mark>2004
<mark>Journal</mark>Communications in Algebra
Issue number12
Number of pages19
Pages (from-to)4515-4533
<mark>Original language</mark>English


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