Faithful representations of minimal dimension of current Heisenberg Lie algebras

Abstract

Given a Lie algebra $\mathfrak{g}$ over a field of characteristic zero $k$, let $\mu(\mathfrak{g})=\min\{\dim \pi: \pi\text{is a faithful representation of}\mathfrak{g}\}$. Let $\mathfrak{h}_{m}$ be the Heisenberg Lie algebra of dimension $2m+1$ over $k$ and let $k[t]$ be the polynomial algebra in one variable. Given $m\in\mathbb{N}$ and $p\in k[t]$, let $\mathfrak{h}_{m,p}=\mathfrak{h}_m\otimes k[t]/(p)$ be the current Lie algebra associated to $\mathfrak{h}_m$ and $k[t]/(p)$, where $(p)$ is the principal ideal in $k[t]$ generated by $p$. In this paper we prove that $ mu(\mathfrak{h}_{m,p}) = m \deg p + \left \lceil 2\sqrt{\deg p} \right\rceil$.