Lie Group Homomorphism

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

• $\varphi(gh)=\varphi(g)\varphi(h)$ for all $g,h\in G$, and
• $\varphi$ is a smooth map .

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

Differential at the identity

The differential at the identity,

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 .