Irreducible representations of Heisenberg group

Question

Lately, I've been struggling with the following problem. Let $H$ be the 3 dimensional Heisenberg group and let $\rho:H\to\text{GL}(n,\mathbb{C})$ be a irreducible representation. Show that $n=1$.

I do not have any concrete idea where to begin, so any hint would be appreciated.

Answer 1

An irreducible representation $H\rightarrow GL(V)$ induces an irreducible representation on the Lie algebra level. By Lie's theorem, every irreducible representation of a complex solvable Lie algebra is $1$-dimensional. The Heisenberg Lie algebra $\mathfrak{h}$ is nilpotent, hence solvable.