Differential of a Lie Group Homomorphism

Let $G$ and $H$ be Lie groups with identity elements $e_G$ and $e_H$, and let $\varphi:G\to H$ be a Lie group homomorphism. Since $\varphi$ is a smooth map, it has a differential (pushforward) at every point; in particular at the identity:

Using the standard identification of the Lie algebra of a Lie group with the tangent space at the identity, we view this as a linear map

Theorem (Lie algebra homomorphism induced by $\varphi$). Let $\varphi:G\to H$ be a Lie group homomorphism. Then the differential at the identity $d\varphi_{e_G}:\mathfrak g\to\mathfrak h$ is a homomorphism of Lie algebras, meaning it preserves the Lie bracket:

Moreover, it intertwines the exponential maps:

It is also functorial: if $\psi:H\to K$ is another Lie group homomorphism, then

Examples

1. Inclusion of a Lie subgroup. If $i:H\hookrightarrow G$ is the inclusion of a Lie subgroup, then $di_{e_H}:\mathfrak h\to\mathfrak g$ is the natural inclusion of tangent spaces at the identity. Concretely, it identifies $\mathfrak h$ as a Lie subalgebra of $\mathfrak g$.

1. Determinant on $GL(n,\mathbb R)$. The determinant is a Lie group homomorphism $\det:GL(n,\mathbb R)\to \mathbb R^\times$. Identifying $\mathfrak{gl}(n,\mathbb R)\cong M_n(\mathbb R)$, the induced map on Lie algebras is
   $$d(\det)_{I}(A)=\operatorname{tr}(A),$$
   and more generally $d(\det)_B(V)=\det(B)\operatorname{tr}(B^{-1}V)$.

1. Covering map $\mathbb R\to S^1$. The map $\varphi:\mathbb R\to S^1\subset\mathbb C$, $\varphi(t)=e^{it}$, is a Lie group homomorphism (additive to multiplicative). Then $d\varphi_0:\mathbb R\to T_1S^1$ sends $a\mapsto ia$ (since $\frac{d}{dt}e^{it}\big|_{t=0}=i$). Under the common identification $T_1S^1\cong \mathbb R$ via $a\mapsto ia$, this differential is the identity.