Timeline for Annihilate a simple Lie algebra using two commutators
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Feb 25, 2022 at 18:14 | comment | added | Salvatore Siciliano | OK LSpice, thank you. | |
| Feb 25, 2022 at 15:54 | comment | added | LSpice |
TeX note: $\operatorname{ad} x$ \operatorname{ad} x spaces better than $\mathrm{ad} x$ \mathrm{ad} x. I have edited accordingly.
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| Feb 25, 2022 at 15:53 | history | edited | LSpice | CC BY-SA 4.0 |
`\operatorname` and article link
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| Feb 24, 2022 at 13:06 | comment | added | D. Dona | @YCor Yes, I see. That's why I kind of want to hold off on making this the accepted answer (no offence to Salvatore). It's being helpful nonetheless, and it's a connection to the classification problem that I did not expect, not being familiar with the area myself. | |
| Feb 23, 2022 at 16:35 | comment | added | YCor | @D.Dona but this doesn't answer the question in, say, the complex case, which remains the "main" case of the first question. | |
| Feb 23, 2022 at 11:46 | comment | added | Salvatore Siciliano | @D. Dona: I would expect that there exist also examples with $x$ and $y$ non-proportional. | |
| Feb 23, 2022 at 11:41 | comment | added | Salvatore Siciliano | @YCor: Sure, I agree: thanks for the remark. | |
| Feb 23, 2022 at 8:59 | comment | added | D. Dona | Huh, so it is indeed possible, thanks! I'll go through the related literature, whose existence I ignored. About my second question, it is natural for me to ask then: is this the only case in which it happens, or are there examples when $x\neq y$ (or rather $x\notin Ky$)? | |
| Feb 22, 2022 at 23:21 | comment | added | YCor | Just for completeness: in char. zero the result of non-existence of sandwich element follows from the Jacobson-Morozov theorem (which ensures that in a semisimple Lie algebra for every $x$ such that $\mathrm{ad}(x)$ is nilpotent and nonzero, there exists $y$ such that $[x,[x,y]]=x$, so that $\mathrm{ad}(x)^2$ is nonzero as well). | |
| Feb 22, 2022 at 14:58 | history | edited | YCor | CC BY-SA 4.0 |
fixed italics instead of crooked math mode
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| Feb 22, 2022 at 12:54 | history | answered | Salvatore Siciliano | CC BY-SA 4.0 |