A general result of Lie Theory is that every Lie group homomorphism $\Phi: G\rightarrow H$ induces a Lie algebra homomorphism $\phi: \frak{g} \rightarrow \frak{h}$.
Which Lie algebra homomorphism is induced by left (or right)-translations:
$L_g h = gh$ for $h,g \in G$, which is a map $G \rightarrow G$ ?
A first idea would be looking at the corresponding pushforward: $L_{ g \star} $ is a map between tangent vectors at $g$ and $h$ respectively. For $ X \in T_g G$ the pushforward is
$L_{ g \star} X = X' \in T_{gh} G$ and therefore this is a map between different tangent spaces which does not help me, because A Lie algebra homomorphism has to be a map from $T_e G$ to $T_e G$.
Any help or ideas finding the corresponding Lie algebra homomorphism for left translations would be much appreciated.
Best Answer
Left and right translations are not Lie group homomorphisms; they don't even preserve the identity, and the induced map on Lie algebras is obtained by looking at derivatives at the identity. However, conjugation by a fixed element $g \in G$ is, and the induced map on $\mathfrak{g}$ gives a representation $G \to \text{Aut}(\mathfrak{g})$ called the adjoint representation. This is itself a Lie group homomorphism, and differentiating it gives the adjoint representation
$$\mathfrak{g} \ni x \mapsto (y \mapsto [x, y]) \in \text{Der}(\mathfrak{g})$$
of $\mathfrak{g}$ (and this is one way to define the Lie bracket).