Universal covering group

Summary

The universal covering group of a connected Lie group $G$ is its universal cover $\tilde G$ with a Lie group structure. This structure makes the covering map $\tilde G \to G$ a homomorphism. It is simply connected, and $G$ is the quotient of $\tilde G$ by a discrete central subgroup isomorphic to the fundamental group of $G$. All connected Lie groups with a given Lie algebra are quotients of the same universal covering group.