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ON LOCALLY DEFINED FORMATIONS OF SOLUBLE LIE AND LEIBNIZ ALGEBRAS

  • DONALD W. BARNES (a1)
Abstract
Abstract

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.

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[1]D. W. Barnes , ‘Saturated formations of soluble Lie algebras in characteristic 0’, Arch. Math. (Basel) 30 (1978), 477480.

[3]D. W. Barnes and H. M. Gastineau-Hills , ‘On the theory of soluble Lie algebras’, Math. Z. 106 (1968), 343354.

[4]K. Doerk and T. Hawkes , Finite Soluble Groups (De Gruyter, Berlin–New York, 1992).

[5]J.-L. Loday and T. Pirashvili , ‘Leibniz representations of Lie algebras’, J. Algebra 181 (1996), 414425.

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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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