J. Aust. Math. Soc.  78 (2005), 407-421
Ado-Iwasawa extras

Donald W. Barnes
  1 Little Wonga Road
  Cremorne NSW 2090
  Australia
  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 $\rho$ of $L$. Let $S$ be a subnormal subalgebra of $L$, let $\mathfrak{F}$ be a saturated formation of soluble Lie algebras and suppose that $S \in \mathfrak{F}$. I show that there exists a module $V$ with the extra property that it is $\mathfrak{F}$-hypercentral as $S$-module. Further, there exists a module $V$ which has this extra property simultaneously for every such $S$ and $\mathfrak{F}$, along with the Hochschild extra that $\rho(x)$ is nilpotent for every $x \in L$ with $\operatorname{ad}(x)$ nilpotent. In particular, if $L$ is supersoluble, then it has a faithful representation by upper triangular matrices.
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