Timeline for Is there a unique "natural" action of $\mathsf{SL}_{n+1}$ on $\mathbb{R}^n$?
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Jun 24, 2021 at 2:59 | vote | accept | Giovanni Moreno | ||
| Jun 13, 2021 at 13:36 | answer | added | Robert Bryant | timeline score: 10 | |
| Jun 12, 2021 at 15:19 | history | edited | Giovanni Moreno | CC BY-SA 4.0 |
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| Jun 11, 2021 at 6:41 | history | edited | YCor | CC BY-SA 4.0 |
removed irrelevant question to help the reader
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| Jun 11, 2021 at 4:31 | comment | added | Giovanni Moreno | I think that VladimirDotsenko in his second remark had in mind the parabolic sub-algebra $\mathfrak{p}=\mathfrak{gl}_n\oplus\mathbb{R}^n$ of index $n$ mentioned in the comment of @Kapil . If I got him right, if there is a point $x\in \mathbb{R}^n$, such that the kernel of $i_x:\mathfrak{sl}_{n+1}\to T_x\mathbb{R}^n$ is isomorphic to $\mathfrak{p}$, then there is hope of "rectifying" the embedding $i$ into $i_{\textrm{nat}}$. This looks like a good necessary condition; actually constructing the "rectifying diffeomorfism" is another story. | |
| Jun 11, 2021 at 3:26 | history | edited | Giovanni Moreno | CC BY-SA 4.0 |
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| Jun 10, 2021 at 11:11 | comment | added | Kapil | Perhaps one can look at the "isotropy" of a point. In this case, it is a parabolic sub-algebra associated with a point of $\mathbb{P}^n$. It is possible that there are other sub-algebras of index $n$ in the lie algebra $\mathfrak{sl}_{n+1}$. | |
| Jun 10, 2021 at 10:13 | comment | added | Giovanni Moreno | About your confusion about question 1, my bad: by linear I meant linear or constant (I.e., of degree less or equal to one). In any case, there is only 2+4=6 of them, so that pure dimensional considerations are enough. The true question is the second one. | |
| Jun 10, 2021 at 9:55 | comment | added | Vladimir Dotsenko | For question 2, a good start would probably be to ask if $gl_n\subset sl_{n+1}$ can be embedded in vector fields on $\mathbb{R}^n$ in an essentially unique way. | |
| Jun 10, 2021 at 9:54 | comment | added | Vladimir Dotsenko | I am confused about your question 1. How do you embed a vector space of dimension $8$ into a vector space of dimension $4$? | |
| Jun 10, 2021 at 9:39 | history | asked | Giovanni Moreno | CC BY-SA 4.0 |