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CHIEF FACTORS COVERED BY PROJECTORS OF SOLUBLE LEIBNIZ ALGEBRAS

Part of: General nonassociative rings Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  30 March 2012

DONALD W. BARNES
DONALD W. BARNES*
Affiliation:
1 Little Wonga Rd., Cremorne NSW 2090, Australia (email: donwb@iprimus.com.au)
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Abstract

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Let 𝔉 be a saturated formation of soluble Leibniz algebras. Let K be an 𝔉-projector and A/B a chief factor of the soluble Leibniz algebra L. It is well known that if A/B is 𝔉-central, then K covers A/B. I prove the converse: if K covers A/B, then A/B is 𝔉-central.

Keywords

Leibniz algebrasLie algebrassaturated formationsprojectors

MSC classification

Secondary: 17A32: Leibniz algebras 17B30: Solvable, nilpotent (super)algebras
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 86 , Issue 3 , December 2012 , pp. 353 - 355
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

References

[1]Barnes, D. W., ‘On Engel’s theorem for Leibniz algebras’, arXiv:1012.0608v2 (2011).CrossRefGoogle Scholar
[2]Barnes, D. W., ‘Schunck classes of soluble Leibniz algebras’, arXiv:1101.3046 (2011).Google Scholar
[3]Barnes, D. W., ‘On 𝔉-hypercentral modules for Lie algebras’, Arch. Math. 30 (1978), 1–7.CrossRefGoogle Scholar
[4]Barnes, D. W., ‘On 𝔉-hyperexcentric modules for Lie algebras’, J. Aust. Math. Soc. 74 (2003), 235–238.CrossRefGoogle Scholar
[5]Loday, J.-L. and Pirashvili, T., ‘Leibniz representations of Lie algebras’, J. Algebra 181 (1996), 414–425.CrossRefGoogle Scholar