Journal of the Australian Mathematical Society

Research Article

Ado-Iwasawa extras

Donald W. Barnesa1

a1 1 Little Wonga Road Cremorne NSW 2090 Australia e-mail: donwb@iprimus.com.au

Abstract

Let L be a finite-dimensional Lie algebra over the field F. The Ado-Iwasawa Theorem asserts the existence of a finite-dimensional L-module which gives a faithful representation ρ of L. Let S be a subnormal subalgebra of L, let S1446788700008600_inline1 be a saturated formation of soluble Lie algebras and suppose that S xs2208 S1446788700008600_inline1. I show that there exists a module V with the extra property that it is S1446788700008600_inline1-hypercentral as S-module. Further, there exists a module V which has this extra property simultaneously for every such S and S1446788700008600_inline1, along with the Hochschild extra that ρ(x) is nilpotent for every x xs2208 L with ad(x) nilpotent. In particular, if L is supersoluble, then it has a faithful representation by upper triangular matrices.

(Received November 22 2003)

(Revised March 17 2004)

2000 Mathematics subject classification

  • primary 17B30;
  • 17B56;
  • secondary 17B50;
  • 17B55

Keywords and phrases

  • Lie algebras;
  • faithful representations;
  • saturated formations