Minimal Faithful Representation of the Heisenberg Lie Algebra with Abelian Factor

For a finite dimensional Lie algebra $\text{g}$ over a field $\text{k}$ of characteristic zero, the $\mu$-function (respectively ${\mu }_{nil}$-function) is defined to be the minimal dimension of $V$ such that $\text{g}$ admits a faithful representation (respectively a faithful nilrepresentation) on $V$. Let ${\text{h}}_m$ be the Heisenberg Lie algebra of dimension $2m+1$ and let ${\mathfrak{a}}_n$ be the abelian Lie algebra of dimension $n$. The aim of this paper is to compute $\mu ({\text{h}}_m\oplus {\mathfrak{a}}_n)$ and ${\mu }_{nil}({\text{h}}_m\oplus {\mathfrak{a}}_n)$ for all $m,n\in \mathbb{N}$.