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ON LOCALLY DEFINED FORMATIONS OF SOLUBLE LIE AND LEIBNIZ ALGEBRAS

Part of: Lie algebras and Lie superalgebras Representation theory of groups

Published online by Cambridge University Press:  02 February 2012

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

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It is well known that all saturated formations of finite soluble groups are locally defined and, except for the trivial formation, have many different local definitions. I show that for Lie and Leibniz algebras over a field of characteristic 0, the formations of all nilpotent algebras and of all soluble algebras are the only locally defined formations and the latter has many local definitions. Over a field of nonzero characteristic, a saturated formation of soluble Lie algebras has at most one local definition, but a locally defined saturated formation of soluble Leibniz algebras other than that of nilpotent algebras has more than one local definition.

Keywords

Lie algebrasLeibniz algebrassaturated formationslocal definition

MSC classification

Secondary: 17B30: Solvable, nilpotent (super)algebras 20D10: Solvable groups, theory of formations, Schunck classes, Fitting classes, $pi$-length, ranks

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 86 , Issue 2 , October 2012 , pp. 322 - 326
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

References

[1]Barnes, D. W., ‘Saturated formations of soluble Lie algebras in characteristic 0’, Arch. Math. (Basel) 30 (1978), 477–480.CrossRefGoogle Scholar
[2]Barnes, D. W., ‘Schunck classes of soluble Leibniz algebras’, arXiv:1101.3046 (2011).Google Scholar
[3]Barnes, D. W. and Gastineau-Hills, H. M., ‘On the theory of soluble Lie algebras’, Math. Z. 106 (1968), 343–354.CrossRefGoogle Scholar
[4]Doerk, K. and Hawkes, T., Finite Soluble Groups (De Gruyter, Berlin–New York, 1992).CrossRefGoogle Scholar
[5]Loday, J.-L. and Pirashvili, T., ‘Leibniz representations of Lie algebras’, J. Algebra 181 (1996), 414–425.CrossRefGoogle Scholar
[6]Jacobson, N., Lie Algebras (Interscience, New York, 1962).Google Scholar