Are Lie algebras $u_n$ and $su_n$ simple?

Question

I think, that $u_n$ isn't simple, because, for example, any matrix $\begin{pmatrix} ia & 0 \\ 0 & ia \end{pmatrix} \in Z(u_n)$, and hence $u_n$ has non-trivial ideal. But i don't know anything about $su_n$.

Answer 1

The Lie algebra $\mathfrak{su}(n)$ is indeed simple, see http://en.wikipedia.org/wiki/Special_unitary_group. And yes, the unitary Lie algebra $\mathfrak{u}(n)$ has a $1$-dimensional abelian center, so that it is not even semisimple.