Lie Group Homomorphism - Imported Knowlpedia Content

A Lie group homomorphism is a map $\varphi:G\to H$ between Lie groups such that:

Equivalently, $\varphi$ is a group homomorphism that is smooth as a map of manifolds.

Differential at the identity

The differential at the identity,

(d\varphi)_e:T_eG\to T_eH,

is a Lie algebra homomorphism $(d\varphi)_e:\mathfrak{g}\to\mathfrak{h}$ , where $\mathfrak{g}=T_eG$ and $\mathfrak{h}=T_eH$ ; see Lie algebra of a Lie group.

Kernels, images, and coverings

$\ker(\varphi)$ is a closed subgroup of $G$ , hence an embedded Lie subgroup (by closed subgroup theorem).
The image $\varphi(G)$ is an immersed Lie subgroup of $H$ .
When $\ker(\varphi)$ is discrete and $\varphi$ is a local diffeomorphism, $\varphi$ is a covering Lie group map.

Lie group homomorphisms are the morphisms in the category underlying the Lie correspondence.