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

Abstract

Given a finite dimensional Lie algebra $\mathfrak{g}$, let $\mathfrak{z}(\mathfrak{g})$ denote the center of $\mathfrak{g}$ and let $\mu (\mathfrak{g})$ be the minimal possible dimension for a faithful representation of $\mathfrak{g}$. In this paper we obtain $\mu ({\mathcal{L}}_{r,2})$, where ${\mathcal{L}}_{r,k}$ is the free $k$-step nilpotent Lie algebra of rank $r$. In particular we prove that $\mu ({\mathcal{L}}_{r,2})=\lceil \sqrt{2r(r-1)}\rceil +2$ for $r\ge 4$. It turns out that $\mu ({\mathcal{L}}_{r,2})\sim \mu (\mathfrak{z}({\mathcal{L}}_{r,2}))\sim 2\sqrt{\dim {\mathcal{L}}_{r,2}}$ (as $r\to \infty$) and we present some evidence that this could be true for ${\mathcal{L}}_{r,k}$ for any $k$, this is considerably lower than the known bounds for $\mu ({\mathcal{L}}_{r,k})$, which are (for fixed $k$) polynomial in $\dim {\mathcal{L}}_{r,k}$.