Lie Groups
See also: Group extension#Central_extensionThe above definitions and constructions all apply to the special case of Lie groups. 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 the their Lie algebras are isomorphic. This implies that a homomorphism φ : G → H of Lie groups is a covering homomorphism if and only if the induced map on Lie algebras
is an isomorphism.
Since for every Lie algebra there is a unique simply connected Lie group G with Lie algebra, from this follows that the universal convering group of a connected Lie group H is the (unique) simply connected Lie group G having the same Lie algebra as H.
Read more about this topic: Covering Group
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