Computing faithful representations for nilpotent Lie algebras

TL;DR

This paper introduces three methods to find minimal faithful representations of nilpotent Lie algebras and explores bounds for their smallest faithful module dimensions, with conjectures and experimental evidence for specific families.

Contribution

It presents new algorithms for constructing faithful representations and provides bounds and conjectures for the minimal faithful module dimensions of certain nilpotent Lie algebras.

Findings

Three methods for faithful representation construction.

Bounds for the minimal dimension of faithful modules.

Conjecture that _n > n+1 for filiform nilpotent Lie algebras.

Abstract

We describe three methods to determine a faithful representation of small dimension for a finite-dimensional nilpotent Lie algebra over an arbitrary field. We apply our methods in finding bounds for the smallest dimension $\mu(\Lg)$ of a faithful $\Lg$-module for some nilpotent Lie algebras $\Lg$. In particular, we describe an infinite family of filiform nilpotent Lie algebras $\Lf_n$ of dimension $n$ over $\Q$ and conjecture that $\mu(\Lf_n) > n+1$. Experiments with our algorithms suggest that $\mu(\Lf_n)$ is polynomial in $n$.