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 $\mu(\mathcal{L}_{r,2})= \left\lceil \sqrt{2r(r-1)} \right\rceil + 2$ for $r \geq 4$. It turns out that $\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 $\dim\mathcal{L}_{r,k}$.