Representing Lie algebras using approximations with nilpotent ideals

TL;DR

This paper refines Ado's theorem by introducing a method to approximate Lie algebras with nilpotent ideals, leading to explicit bounds on the minimal degree of faithful representations.

Contribution

It defines approximation of Lie algebras with nilpotent ideals and constructs faithful representations with explicit bounds, improving understanding of Lie algebra representations.

Findings

Explicit upper bounds for minimal faithful representation degree.

A new approach using nilpotent ideal approximations.

Application of universal enveloping algebra techniques.

Abstract

We prove a refinement of Ado's theorem for Lie algebras over an algebraically-closed field of characteristic zero. We first define what it means for a Lie algebra $L$ to be approximated with a nilpotent ideal, and we then use such an approximation to construct a faithful representation of $L$. The better the approximation, the smaller the degree of the representation will be. We obtain, in particular, explicit and combinatorial upper bounds for the minimal degree of a faithful $L$-representation. The proofs use the universal enveloping algebra of Poincar\'e-Birkhoff-Witt and the almost-algebraic hulls of Auslander and Brezin.