A lower bound for faithful representations of nilpotent Lie algebras

Given a finite-dimensional nilpotent Lie algebra n, let mu(n) (resp. mu(nil) (n)) be the minimal dimension of V such that n admits a faithful representation (resp. nilrepresentation) on V. In this paper, we present a lower bound for mu(nil)(n) for a 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 mu(nil)(n) that involves the dimensions of n and its centre. This bound allows us to obtain mu(n) for some families of nilpotent Lie algebras.