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 $mu({\mathfrak{h}}_{m,p})=m°p+\lceil 2\sqrt{°p}\rceil$.

14 pages

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