# Lie algebras homomorphism induces Lie groups homomorphism

Given a homomorphism between two Lie algebras $\varphi:\mathfrak g\rightarrow \mathfrak h$. Let $G$ and $H$ be Lie groups with Lie algebras $\mathfrak g$ and $\mathfrak h$ resp. in which cases do we have a corresponding Lie group homomorphism $F:G\rightarrow H$ such that $F$ and $\varphi$ commute with exponential maps?

• $e^Ae^B \ne e^{A+B}$ in general... – W. Cadegan-Schlieper Nov 15 '17 at 19:48
• When $G$ is simply connected? – Angina Seng Nov 15 '17 at 19:49
• We need that $G$ and $H$ are connected, and $G$ is simply-connected: see here, Cor. 4.5.3. – Dietrich Burde Nov 15 '17 at 19:55

In general no, but if $G$ is simply connected yes. This is called Lie's Second Theorem.