Is there a way to embed a Lie Group $G$ into a compact lie Group $H$, such that the inclusion is a Lie group homomorphism?
2026-04-23 04:52:56.1776919976
Compactification of Lie Group
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Let $f:G\to H$ be an injective Lie group morphism with $H$ compact. Since $H$ has a faithful finite dimensional representation, it suffices to compose with $f$ to obtain one for $G$.
This shows that any Lie group $G$ which does not have faithful finite dimensional representations provides an example of what you want.