Faithful representations of minimal dimension of current Heisenberg Lie algebras

Details Faithful representations of minimal dimension of current Heisenberg Lie algebras Faithful representations of minimal dimension of current Heisenberg Lie algebras

2008-03-07

arxiv.org/abs/0803.1076v1

Mathematics

Representation Theory

14 pages

Scientific paper

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$.

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