Bulletin of the Australian Mathematical Society

Research Article

ON LOCALLY DEFINED FORMATIONS OF SOLUBLE LIE AND LEIBNIZ ALGEBRAS

DONALD W. BARNESa1

a1 1 Little Wonga Road, Cremorne, NSW 2090, Australia (email: donwb@iprimus.com.au)

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.

(Received October 27 2011)

2010 Mathematics subject classification

  • primary 17B30; secondary 17A32;
  • 20D10

Keywords and phrases

  • Lie algebras;
  • Leibniz algebras;
  • saturated formations;
  • local definition