Minimal faithful representations of the free 2-step nilpotent Lie algebra of the rank r

Abstract

Given a finite dimensional Lie algebra g, let z(g) denote the center of g and let μ(g) be the minimal possible dimension for a faithful representation of g. In this paper we obtain μ(Lr,2), where Lr,k is the free k-step nilpotent Lie algebra of rank r. In particular we prove that μ(Lr,2)=⌈2r(r−1)⌉+2 for r≥4. It turns out that μ(Lr,2)∼μ(z(Lr,2))∼2dim⁡Lr,2 (as r→∞) and we present some evidence that this could be true for Lr,k for any k. This is considerably lower than the known bounds for μ(Lr,k), which are (for fixed k) polynomial in dim⁡Lr,k.