Minimal Faithful Representation of the Heisenberg Lie Algebra with Abelian Factor

For a finite dimensional Lie algebra $\g$ over a field $\k$ of characteristic zero, the $\mu$-function (respectively $\mu_{nil}$-function) is defined to be the minimal dimension of $V$ such that $\g$ admits a faithful representation (respectively a faithful nilrepresentation) on $V$. Let $\h_m$ 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 $\mu(\h_m \oplus \mathfrak{a}_n)$ and $\mu_{nil}(\h_m \oplus \mathfrak{a}_n)$ for all $m,n \in \mathbb{N}$.