Compact Lie group with non discrete center?

Question

Could someone please give me an example of a compact Lie group with non discrete center which is not just a product of a group with a torus?

Answer 1

The group $U_2({\mathbb C})$ is compact with non-discrete center $Z(U_2({\mathbb C}))=\left.\left\{\tiny\begin{pmatrix} \lambda & 0 \\ 0 & \lambda\end{pmatrix}\ \right|\ \lambda\in {\mathbb S}^1\right\}$, but it does not split as the product of two nontrivial Lie groups: for if it did, in the hypothetical splitting $U_2({\mathbb C})=T\times K$ the subgroups $T,K$ would necessarily be connected, hence the decomposition would descend to the unique nontrivial Lie algebra decomposition of ${\mathfrak u}_2({\mathbb C})$, namely $${\mathfrak u}_2({\mathbb C})={\mathfrak z}\oplus{\mathfrak s}{\mathfrak u}_2({\mathbb C}),\quad {\mathfrak z}:=\left.\left\{\scriptsize\begin{pmatrix}2\pi i t & 0 \\ 0 & 2\pi i t\end{pmatrix}\ \right|\ t\in{\mathbb R}\right\}.$$ However, $\text{exp}({\mathfrak z})$ and $\text{exp}({\mathfrak s}{\mathfrak u}_2({\mathbb C}))$ intersect in $\scriptsize\begin{pmatrix} -1 & 0 \\ 0 & -1\end{pmatrix}$, hence $T\cap K\neq\{e\}$ - contradiction.

A general compact Lie group only admits a covering of the form $Z\times K$ with $Z$ a torus and $K$ semisimple.

The book Structure and Geometry of Lie groups by Hilgert and Neeb is very well written and a great source in these questions.