$f_*$ isomorphism $\Rightarrow$ $f$ isomorphism?

Question

Consider a Lie group endomorphism (bijective homomorphism) $f: G \rightarrow G$ and its pushforward $f_*: \mathcal{A} \rightarrow \mathcal{A}$, where $\mathcal{A}$ is the Lie algebra of $G$. I have seen that if $f$ is an isomorphism (a diffeomorphism) then $f_*$ is an isomorphism.

Is the converse true? That is, given a group endomorphism $f: G \rightarrow G$ with pushforward $f_*: \mathcal{A} \rightarrow \mathcal{A}$ an algebra endomorphism, is it true that if $f_*$ is an isomorphism, the $f$ is an isomorphism too?

Answer 1

As Sebastian Schulz noted, the answer is obviously no if $G$ is not connected.

Furthermore, there are other Lie groups that have isomorphic Lie algebras, but which are not isomorphic since their global properties are different. A well-known example is given by $SU(2)$ and $SO(3)$. The Lie algebras of these groups are isomorphic, but the groups themselves are not: in fact, $SU(2)$ is the double cover of $SO(3)$.

Answer 2

OK, here is the minimal example that I could think of for a connected Lie group. Take $G = U(1) = \{ z \in \mathbb{C} | |z| = 1 \}$, it allows endomorphisms of the form $\phi_n : z \mapsto z^n$.

The Lie algebra of $U(1)$ is $\mathbb{R}$ and the derivative of $\phi_n$ is just multiplication by $n$ (which is an isomorphism unless $n=0$). However, unless $n= \pm 1$, the map is not injective on the level of Lie groups, so it is not an isomorphism.