A lower bound for faithful representations of nilpotent Lie algebras

Abstract

In this paper we present a lower bound for the minimal dimension µ(n) of a faithful representation of a finite dimensional p-step nilpotent Lie algebra n over a field of characteristic zero. Our bound is given as the minimum of a quadratically constrained linear optimization problem, it works for arbitrary p and takes into account a given filtration of n. We present some estimates of this minimum which leads to a very explicit lower bound for µ(n) that involves the dimensions of n and its center. This bound allows us to obtain µ(n) for some families of nilpotent Lie algebras.