= Solution
The <Lie group–Lie algebra correspondence> separates local differential structure from global topology. For a <Lie group> $G$, define $LG=T_eG$ as a <real vector space>. Extend each $X\in T_eG$ to the <left-invariant vector field>
$$
X^L(g)=d_eL_g(X).
$$
The <commutator> of two such <vector fields> is again left-invariant. Set
$$
\boxed{[X,Y]=[X^L,Y^L](e).}
$$
The vector-field <commutator> is bilinear, antisymmetric and satisfies the <Jacobi identity>, so this makes $T_eG$ a <Lie algebra>. For a matrix <group> this bracket is $XY-YX$.
For a smooth <Lie group homomorphism> $\phi:G_1\to G_2$, its differential at the identity is the <linear map>
$$
L\phi=d_{e_1}\phi:\mathfrak g_1\longrightarrow\mathfrak g_2.
$$
The identity $\phi\circ L_g=L_{\phi(g)}\circ\phi$ shows that the corresponding left-invariant fields are $\phi$-related. Brackets of related <vector fields> are related, hence
$$
\boxed{L\phi([X,Y])=[L\phi(X),L\phi(Y)].}
$$
The chain rule also gives $L(\psi\circ\phi)=L\psi\circ L\phi$ and $L(\operatorname{id})=\operatorname{id}$. Thus differentiation is a functor from <Lie groups> and smooth <homomorphisms> to real <Lie algebras> and bracket-preserving <linear maps>.
To see what information the differential retains, apply $\phi$ to a <one-parameter subgroup>. Its initial tangent is $L\phi(X)$, so uniqueness yields
$$
\boxed{\phi(\exp X)=\exp(L\phi(X)).}
$$
If $G_1$ is connected, a small exponential neighbourhood generates it. Every $g\in G_1$ is a finite product $\exp X_1\cdots\exp X_k$, and therefore
$$
\phi(g)=\exp(L\phi(X_1))\cdots\exp(L\phi(X_k)).
$$
For an already existing <homomorphism> this expression is independent of the chosen product because it equals $\phi(g)$. Two <homomorphisms> with the same differential agree on that neighbourhood and hence everywhere. This proves that a <Lie group homomorphism determined by its differential> requires a connected source.
On a disconnected source only the restriction to $G_1^\circ$ is determined. For instance a nontrivial finite discrete <group> has zero <Lie algebra>, so its identity and trivial <endomorphisms> have the same zero differential but are different maps. The discussion of all components must therefore include additional global data. The remaining parts explain the corresponding existence and topology conditions.
Back to article page
