A remark on Ado's Theorem for principal ideal domains

TL;DR

This paper provides a quantitative proof of Ado's Theorem over principal ideal domains, offering explicit bounds on the degree of Lie algebra representations and generalizing an embedding theorem for complex Lie algebras.

Contribution

It introduces a quantitative approach to Ado's Theorem over principal ideal domains, with explicit bounds and a generalized embedding theorem.

Findings

Explicit bounds on the degree of Lie algebra representations.

Embedding of Lie algebras into larger decomposable Lie algebras.

Generalization of an embedding theorem for complex Lie algebras.

Abstract

Ado's Theorem had been extended to principal ideal domains independently by Churkin and Weigel. They demonstrated that if $R$ is a principal ideal domain of characteristic zero and $\mathfrak{L}$ is a Lie algebra over $R$ which is also a free $R$-module of finite rank, then $\mathfrak{L}$ admits a finite faithful Lie algebra representation over $R$. We present a quantitative proof of this result, providing explicit bounds on the degree of the Lie algebra representations in terms of the rank of the free module. To achieve it, we generalise an established embedding theorem for complex Lie algebras: any Lie algebra as above embeds within a larger Lie algebra that decomposes as the direct sum of its nilpotent radical and another subalgebra.