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Home > Research > Publications & Outputs > On upper modular subalgebras of a Lie algebra.
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On upper modular subalgebras of a Lie algebra.

Research output: Contribution to journalJournal article

Published

Journal publication date1/06/2004
JournalProceedings of the Edinburgh Mathematical Society
Journal number2
Volume47
Number of pages13
Pages325-337
Original languageEnglish

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 μ-algebra.

Bibliographic note

RAE_import_type : Journal article RAE_uoa_type : Pure Mathematics http://journals.cambridge.org/action/displayJournal?jid=UHY The final, definitive version of this article has been published in the Journal, Proceedings of the Edinburgh Mathematical Society, 47 (2), pp 325-337 2004, © 2004 Cambridge University Press.