Faithful Lie algebra modules and quotients of the universal enveloping algebra

TL;DR

This paper introduces a new method for finding small faithful representations of finite dimensional nilpotent Lie algebras, providing applications including improved bounds for certain counterexamples to Milnor's conjecture.

Contribution

The paper presents a novel approach to determine minimal faithful modules for nilpotent Lie algebras and applies it to improve bounds related to Milnor's conjecture.

Findings

New upper bound on minimal faithful module dimension for specific Lie algebras

Application of the method to counterexamples of Milnor's conjecture

Enhanced understanding of faithful representations in nilpotent Lie algebras

Abstract

We describe a new method to determine faithful representations of small dimension for a finite dimensional nilpotent Lie algebra. We give various applications of this method. In particular we find a new upper bound on the minimal dimension of a faithful module for the Lie algebras being counter examples to a well known conjecture of J. Milnor.