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

Bowman, Kevin and Towers, David A. and Varea, Vicente R. (2004) On Lie algebras all of whose minimal subalgebras are lower modular. Communications in Algebra, 32 (12). pp. 4515-4533. ISSN 0092-7872

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    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.

    Item Type: Article
    Journal or Publication Title: Communications in Algebra
    Additional Information: The final, definitive version of this article has been published in the Journal, Communications in Algebra, 32 (12), 2004, © Informa Plc
    Uncontrolled Keywords: Lie algebras ; Lattice of subalgebras ; Modular subalgebra
    Subjects: Q Science > QA Mathematics
    Departments: Faculty of Science and Technology > Mathematics and Statistics
    ID Code: 1660
    Deposited By: Dr David A. Towers
    Deposited On: 15 Feb 2008 14:47
    Refereed?: Yes
    Published?: Published
    Last Modified: 09 Oct 2013 12:43
    Identification Number:
    URI: http://eprints.lancs.ac.uk/id/eprint/1660

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