Computing faithful representations for nilpotent Lie algebras

We describe various methods to determine a faithful representation of small dimension for a finite-dimensional nilpotent Lie algebra over an arbitrary field. We apply our methods to finding bounds for the smallest dimension μ(g) of a faithful g-module for some nilpotent Lie algebras g. In particular, we introduce an infinite family of filiform nilpotent Lie algebras fn of dimension n over Q and conjecture that μ(fn)>n+1 holds.