Are all $\mathfrak{su}(n)$ irreducible representations faithful?

Question

For a result I want to obtain I need to know whether there exist non-faithful irreducible representations of $\mathfrak{su}(n)$ Lie-algebras. Sadly my expertise in Lie algebra theory is insufficient to answer this question and quick research has revealed a nontrivial connection between irreducibility and faithfulness that is somewhat over my head.

So, do such representations exist and if so can we name an example?

Answer 1

Since $\mathfrak{su}(n)$ is simple and since the kernel of a representation is an ideal of $\mathfrak{su}(n)$, the only irreducible non-faithful representation of that Lie algebra is the trivial representation on a $1$-dimensional vector space.