Differential of a Lie group homomorphism

Let $\Phi:G\to H$ be a Lie group homomorphism between Lie groups, and let $\mathfrak g=\mathrm{Lie}(G)$ and $\mathfrak h=\mathrm{Lie}(H)$ (see Lie algebra of a Lie group).

The differential at the identity,

d\Phi_e:\mathfrak g \longrightarrow \mathfrak h,

is a Lie algebra homomorphism; i.e.

d\Phi_e([X,Y]) = [\,d\Phi_e(X),\, d\Phi_e(Y)\,] \qquad\text{for all }X,Y\in\mathfrak g.

Moreover, $\Phi$ intertwines exponential maps:

\Phi(\exp_G X) \;=\; \exp_H\!\bigl(d\Phi_e(X)\bigr) \quad\text{for all }X\in\mathfrak g,

where $\exp_G$ and $\exp_H$ are the exponential maps of $G$ and $H$ .

Idea of proof

Identify $\mathfrak g$ and $\mathfrak h$ with left-invariant vector fields using the left-invariant fields Lie algebra lemma. The pushforward $\Phi_*$ carries left-invariant vector fields on $G$ to left-invariant vector fields on $H$ , and pushforwards preserve Lie brackets of vector fields. Evaluating at $e$ yields bracket preservation for $d\Phi_e$ .

Context. This is the functorial bridge from global group maps to infinitesimal algebra maps; it is the starting point for studying representations of Lie groups via their differentiated Lie algebra representations.