Covering Group - Lie Groups

Lie Groups

The 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.