A remark on the Ado-Iwasawa theorem
The Ado-Iwasawa theorem states that every finite dimensional Lie algebra over a field \(K\) has a finite dimensional faithful representation. The paper under review shows that this theorem holds for Lie algebras over certain classes of rings instead of fields. Thus let \(L\) be Lie algebra over the commutative ring \(R\), \(1\in R\), and suppose \(L\) is finitely generated as an \(R\)-module. If \(R\) is either a principal ideal domain of characteristic 0, or a Noetherian ring with 1 of prime characteristic \(p\) then there exists a finitely generated free \(R\)-module \(V\) and a faithful representation of \(L\) in \(V\) (Theorem A). One refinement of this theorem is Theorem B which deals with the case \(R\) residually positive (i.e. R is a PID and for every nontrivial prime ideal \(I\) of \(R\), the characteristic of \(R/I\) is positive). Then there exists a faithful representation of \(L\) in some finitely generated \(R\)-module \(V\). As a corollary the author gives an application to pro-\(p\)-groups, namely Theorem C. Let \(\hat{G}\) be torsion-free finitely generated powerful pro-\(p\)-group. Then there exists a finitely generated \(Z_p\) module \(W\) and an injective morphism (of topological groups) \(\phi\colon \hat{G}\to PL(W)\). Here \(PL(W)=\{1_W+\text{\textbf{p}}\alpha\mid \alpha\in \hom_{Z_p}(W)\}\subseteq \hom_{Z_p}(W)\), \(\text{\textbf{p}}=p\) if \(p\) is odd, and 4 if \(p=2\).
Faithful representations of Lie algebras
scientific article; zbMATH DE number 1271823