Universal covering group

Let $G$ be a connected Lie group. A universal covering group of $G$ is a pair $(\widetilde G,p)$ where:

$\widetilde G$ is a simply connected Lie group;
$p:\widetilde G\to G$ is a smooth covering map and a Lie group homomorphism;
$p$ is universal among connected covering Lie groups of $G$ : for every such covering $q:H\to G$ , there is a unique Lie group homomorphism $r:\widetilde G\to H$ such that $q\circ r=p$ .

The existence of such a pair is guaranteed by the existence theorem for universal covering groups.

Kernel and fundamental group

The kernel $\ker(p)$ is a discrete normal subgroup of $\widetilde G$ and lies in the center of $\widetilde G$ . After choosing basepoints, $\ker(p)$ is naturally isomorphic to $\pi_1(G)$ . Consequently,

G \cong \widetilde G / \ker(p)

as a quotient Lie group.

Lie algebra

The differential $dp_e:\operatorname{Lie}(\widetilde G)\to \operatorname{Lie}(G)$ is an isomorphism of Lie algebras. Thus passage to the universal cover changes global topology but not the infinitesimal structure.