1
$\begingroup$

I try to prove $\operatorname{SL}(2,\mathbb R)$ is not isomorphic to $S^1 \times \mathbb R^2 $ as a Lie group.

My idea is that since $\exp\colon \mathfrak{sl}(2,\mathbb{R}) \to \operatorname{SL}(2,\mathbb{R})$ is not surjective, it is enough to show that $\exp\colon \operatorname{Lie}(S^1 \times \mathbb{R^2}) \to S^1 \times \mathbb{R^2}$ is surjective. But I couldn't find how to do it. Are there other good ideas? Could you help me?

$\endgroup$
6
$\begingroup$

$S^1 \times \Bbb{R}^2$ is abelian while $\text{SL}_2(\Bbb{R})$ is not.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.