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

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 $r \geq 4$\mathrm{.}Itturnsoutthat$\mu($\mathcal{L}${r,2}) \sim\mu\big(\mathfrak{z}($\mathcal{L}${r,2})\big) \sim 2\sqrt{\dim$\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 $\dim {\mathcal{L}}_{r,k}$.