Timeline for Homomorphism from noncompact semisimple Lie group to compact Lie group
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Oct 9, 2020 at 12:58 | comment | added | YCor | ... with algebraic homomorphisms. (Which is automatic among Lie group homomorphisms between semisimple algebraic groups, which is not hard but strictly harder this whole argument). | |
| Oct 8, 2020 at 10:35 | comment | added | Mikhail Borovoi | @YCor: I agree. I had in mind real semisimple algebraic groups. | |
| Oct 8, 2020 at 10:30 | history | edited | Mikhail Borovoi | CC BY-SA 4.0 |
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| Oct 8, 2020 at 10:11 | history | edited | Mikhail Borovoi | CC BY-SA 4.0 |
Edited taking into account a comment of YCor.
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| Oct 8, 2020 at 9:02 | comment | added | YCor | I don't think this is a proof. It's a trivial reduction to showing that there's no injective homomorphism from a [nontrivial!] semisimple group with no compact factor into a compact Lie group. The remainder of the proof (which I mentioned in a comment, or alternatively doing the case when $G$ is locally isomorphic to $\mathrm{SL}_2(\mathbf{R})$) is fairly standard (and probably quite a duplicate here), but contains all the substance of the proof, which is not apparent here. Just to keep in mind, there's an injective hom from the noncompact Lie group $\mathbf{R}$ into some compact Lie group. | |
| Oct 8, 2020 at 7:18 | history | edited | Mikhail Borovoi | CC BY-SA 4.0 |
A new short proof is written.
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| Oct 8, 2020 at 7:06 | history | edited | Mikhail Borovoi | CC BY-SA 4.0 |
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| Oct 8, 2020 at 6:54 | history | answered | Mikhail Borovoi | CC BY-SA 4.0 |