Universal covering group

Definition

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 that is also a Lie group homomorphism,
• and $p$ is universal among covering Lie groups of $G$ in the sense that any covering Lie group $q:H\to G$ factors uniquely through $p$ by a Lie group homomorphism $H\to \widetilde G$ commuting with the projections to $G$.

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 $\widetilde G$ (see discrete subgroups) and in fact lies in the center of $\\widetilde G$. Topologically, $\ker(p)$ is naturally isomorphic to the fundamental group $\pi_1(G)$ once basepoints are chosen. Consequently,

as a quotient Lie group.

Lie algebra

The differential $dp_e:\mathrm{Lie}(\widetilde G)\to \mathrm{Lie}(G)$ is an isomorphism of Lie algebras (compare differentials of Lie group homomorphisms). Thus covering changes global topology but not the infinitesimal structure.