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This might be a dumb question, but are all continuous maps between Lie groups also homomorphisms? I can only seem to think of examples in which they are (i.e., $GL(n,\mathbb{R}) \to \mathbb{R}$ via the determinant, the covering space map from $\mathbb{R} \to S^1$,...). Conversely, a Lie group homomorphism is defined as a homomorphism that is smooth, so what are some examples of homomorphisms between Lie groups that are not smooth?

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Re: first question. So... all continuous maps $\mathbb{R} \to \mathbb{R}$ are of the form $x \mapsto ax$? Re: last question: you can show that measurable homomorphisms between Lie groups are smooth, so you need to seek non-measurable examples. One way would be to take a non-measurable solution to the Cauchy functional equation. – t.b. Mar 24 '12 at 19:13
$S^1$ is a Lie group and there are many continuous maps $S^1 \rightarrow S^1$ which are continuous but no Lie group homomorphisms. Are you sure your formulation expresses your question correctly? – user20266 Mar 24 '12 at 19:15
Take $G$ and $H$ any lie groups and $e \ne h_0 \in H$. Then $G \to H, g \mapsto h_0$ is always continuous but never a homomorphism. – Eric O. Korman Mar 24 '12 at 19:32
@ThomasAndrews: Continuous homomorphisms are automatically smooth. – t.b. Mar 24 '12 at 20:04
@ThomasAndrews: It's even worse in some cases. For compact Lie groups with finite centers, homomorphisms are automatically continuous, hence automatically smooth. – Jason DeVito Mar 24 '12 at 20:06

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