Timeline for Double centralizer theorem for ($\ell$-adic) Lie algebras
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Dec 18, 2022 at 12:02 | vote | accept | Alireza Shavali | ||
| Dec 15, 2022 at 18:58 | answer | added | Yuri Zarhin | timeline score: 4 | |
| Dec 15, 2022 at 10:42 | comment | added | YCor | @AlirezaShavali Yes, because the normalizer of the Lie algebra in $\mathrm{SL}_2$ is Zariski-closed (since the action of the group on the Lie algebra is algebraic). So Zariski-density of the group, and the fact that the group normalizes the Lie algebra, implies that the Lie algebra is normalized by $\mathrm{SL}_2$, and hence is $\{0\}$ or $\mathfrak{sl}_2$. | |
| Dec 15, 2022 at 10:20 | comment | added | Alireza Shavali | @YCor Is this clear that Zariski density forces the Lie subalgebra to actually be a Lie ideal? | |
| Dec 12, 2022 at 11:22 | comment | added | YCor | Since this is Zariski-dense, the Lie algebra is an ideal in $\mathfrak{sl}_2$, hence $\{0\}$ or equal to $\mathfrak{sl}_2$. So if it's not $\mathfrak{sl}_2$, the image has to be finite (since it's compact), hence not Zariski-dense, contradiction. | |
| Dec 12, 2022 at 9:07 | history | edited | Alireza Shavali | CC BY-SA 4.0 |
corrected spelling
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| Dec 11, 2022 at 16:12 | comment | added | LSpice | Re, ah, sorry, I had not considered that gap. If it is easy to fix the gap, then I do not know how. I have retracted my close-as-duplicate vote. | |
| Dec 11, 2022 at 11:38 | comment | added | Alireza Shavali | @LSpice Thanks for the link. From what I understand, this only implies that the Zariski closure of the image contains $SL_2$. How does this imply that the $\ell$-adic Lie algebra (which might not be algebraic) contains $\mathfrak{sl}_2$? | |
| Dec 8, 2022 at 16:08 | review | Close votes | |||
| Dec 11, 2022 at 16:19 | |||||
| Dec 8, 2022 at 15:42 | comment | added | LSpice | I think this is a duplicate of Reductive subgroups of $\operatorname{GL}_2$ over an algebraically closed field of characteristic zero. | |
| Dec 8, 2022 at 15:30 | history | edited | YCor | CC BY-SA 4.0 |
formatting
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| S Dec 8, 2022 at 13:24 | review | First questions | |||
| Dec 8, 2022 at 20:11 | |||||
| S Dec 8, 2022 at 13:24 | history | asked | Alireza Shavali | CC BY-SA 4.0 |