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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?

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  • $\begingroup$ $e^Ae^B \ne e^{A+B}$ in general... $\endgroup$ – W. Cadegan-Schlieper Nov 15 '17 at 19:48
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    $\begingroup$ When $G$ is simply connected? $\endgroup$ – Angina Seng Nov 15 '17 at 19:49
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    $\begingroup$ We need that $G$ and $H$ are connected, and $G$ is simply-connected: see here, Cor. 4.5.3. $\endgroup$ – Dietrich Burde Nov 15 '17 at 19:55
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In general no, but if $G$ is simply connected yes. This is called Lie's Second Theorem.

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