Computing faithful representations for nilpotent Lie algebras

Burde, Dietrich; Eick, Bettina; de Graaf, Willem

Computing faithful representations for nilpotent Lie algebras

Authors

Burde, Dietrich

Eick, Bettina

de Graaf, Willem

Description

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 $μ(\Lg)$ 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 $μ(\Lf_n) > n+1$. Experiments with our algorithms suggest that $μ(\Lf_n)$ is polynomial in $n$.
14 pages