Covering Lie group

A covering Lie group map is a smooth homomorphism of Lie groups

such that, as a map of topological spaces, $p$ is a covering map.

Basic properties

Assume $\widetilde G$ is connected.

1. The differential at the identity,

   $$dp_e:\mathrm{Lie}(\widetilde G)\to \mathrm{Lie}(G),$$

   is a linear isomorphism (so $p$ identifies the two Lie algebras ).

2. The kernel $\ker(p)$ is a discrete subgroup of $\widetilde G$.

3. In fact, $\ker(p)$ lies in the center $Z(\widetilde G)$: elements of the kernel act as deck transformations and must commute with the connected group generated by small neighborhoods of the identity.

Thus a covering homomorphism is a very rigid kind of quotient: it is (up to isomorphism) a quotient by a discrete central subgroup.

Universal covers

For every connected Lie group $G$, there exists a simply connected Lie group $\widetilde G$ and a covering homomorphism $\widetilde G\to G$; this is the existence of the universal covering group . The resulting universal covering group is unique up to unique isomorphism over $G$, and it is characterized by being simply connected with $\mathrm{Lie}(\widetilde G)\cong \mathrm{Lie}(G)$.

Context. Many global topological features (fundamental group, discrete central quotients) are invisible to the Lie algebra. Covering maps are the standard way to pass between Lie-algebraic data and different global forms of the “same” local Lie group.