Category of Lie group representations equivalent to the category of representations of their Lie algebra

Solution 1:

Let $V$ and $W$ be two representations of $G$ and $f : V \rightarrow W$ be invariant by $\frak{g}$, i.e. for all $X \in \frak{g}$ and $v \in V$: $$f(X.u) = X.f(u).$$ Let $X \in \frak{g}$ and $g=\exp(X)$. Define $$ \phi : \mathbb{R} \longrightarrow \mathrm{End}(V,W), \quad t \mapsto f \circ \exp(tX).$$ and $$ \psi : \mathbb{R} \longrightarrow \mathrm{End}(V,W), \quad t \mapsto \exp(tX) \circ f.$$ Then $$\frac{d}{dt} \phi(t) = f \circ X \circ \exp(tX) = X \circ f \circ \exp(tX) = X.\phi(t).$$ and $$\frac{d}{dt} \psi(t) = X \circ \exp(tX) \circ f = X.\psi(t).$$ Hence $\phi$ and $\psi$ both satisfy the differential equation. $$\begin{cases} \frac{d}{dt} y(t) &= X.y(t) \\ y(0) & = f \end{cases} $$ So $\phi=\psi$ by Cauchy-Lipschitz, and $f.g = g.f$.