Center of a Lie group

Elements commuting with all group elements; a closed normal subgroup.

Center of a Lie group

Let $G$ be a Lie group .

Definition. The center of $G$ is the subgroup

Z(G)=\{z\in G : zg=gz \text{ for all } g\in G\}.

Basic properties.

$Z(G)$ is a normal subgroup (indeed characteristic), hence a normal Lie subgroup whenever it is an embedded Lie subgroup.
$Z(G)$ is closed in $G$ , since it is the intersection of the closed sets $\{z:zg=gz\}$ over all $g\in G$ .

Relation to adjoint/conjugation. The conjugation action is trivial on $Z(G)$ . For connected $G$ , the kernel of the adjoint representation is exactly $Z(G)$ , so discreteness of $Z(G)$ is equivalent to discreteness of $\ker(\mathrm{Ad})$ ; see Ad has discrete kernel iff the center is discrete .

Context. The quotient $G/Z(G)$ is called the adjoint form in semisimple settings; modding out by the center removes precisely the elements invisible to conjugation.