Editing Covering group (section)

== Lie groups ==
{{See also|Group extension#Central extension}}

The above definitions and constructions all apply to the special case of [[Lie group]]s. In particular, every covering of a [[manifold]] is a manifold, and the covering homomorphism becomes a [[smooth map]]. Likewise, given any discrete normal subgroup of a Lie group the quotient group is a Lie group and the quotient map is a covering homomorphism.

Two Lie groups are locally isomorphic if and only if their [[Lie algebras]] are isomorphic. This implies that a homomorphism {{nowrap|''φ'' : ''G'' → ''H''}} of Lie groups is a covering homomorphism if and only if the induced map on Lie algebras
: <math>\phi_* : \mathfrak g \to \mathfrak h</math>
is an isomorphism.

Since for every Lie algebra <math>\mathfrak g</math> there is a unique simply connected Lie group ''G'' with Lie algebra {{tmath|1= \mathfrak g }}, from this follows that the universal covering group of a connected Lie group ''H'' is the (unique) simply connected Lie group ''G'' having the same Lie algebra as ''H''.