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For a finite dimensional Lie algebra $\g$ over a field $\k$ of characteristiczero, the $\mu$-function (respectively $\mu_{nil}$-function) is defined to bethe minimal dimension of $V$ such that $\g$ admits a faithful representation(respectively a faithful nilrepresentation) on $V$. Let $\h_m$ be theHeisenberg Lie algebra of dimension $2m + 1$ and let $\mathfrak{a}_n$ be theabelian 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 $m,n \in \mathbb{N}$.

Minimal Faithful Representation of the Heisenberg Lie Algebra with Abelian Factor