Timeline for The largest abelian subgroups of a Lie group
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
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| Jan 26, 2023 at 14:32 | comment | added | Callum | On a Lie algebra level, a compact Lie algebra $\mathfrak{g}$ has maximal dimension of a maximal abelian equal to $\mathrm{rank}(\mathfrak{g})$. For a non-compact simple Lie algebra you can often find the largest abelian subalgebra as the nilradical of one of the maximal parabolic subalgebras. I think this will work for $G$ complex or split for all the classical cases plus $E_6$ and $E_7$ but breaks down in some of the real forms and for $E_8$, $F_4$ and $G_2$. | |
| Jan 25, 2023 at 15:43 | comment | added | Yushi MuGiwara | @YCor I really appreciate your help, it was extremely helpful. | |
| Jan 25, 2023 at 7:47 | comment | added | YCor | If $G$ has a finite center, then $d(G)=c(G)$. The inequality $c(G)\le d(G)$ is trivial, and to get the reverse one, we can suppose $G$ has trivial center hence is algebraic, and pass to the Zariski closure. In general, if the center has $\mathbf{Q}$-rank $r$ one deduces $d(G)=c(G)+r$. So everything boils down to computing $c(G)$. | |
| Jan 24, 2023 at 19:42 | review | Close votes | |||
| Feb 4, 2023 at 3:02 | |||||
| Jan 24, 2023 at 19:41 | history | undeleted | Asaf Karagila♦ | ||
| Jan 24, 2023 at 18:28 | history | deleted | Yushi MuGiwara | via Vote | |
| Jan 24, 2023 at 18:28 | history | edited | Yushi MuGiwara | CC BY-SA 4.0 |
added 67 characters in body
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| Jan 24, 2023 at 18:25 | review | Close votes | |||
| Jan 24, 2023 at 18:34 | |||||
| Jan 24, 2023 at 18:21 | history | undeleted | Yushi MuGiwara | ||
| Jan 20, 2023 at 16:31 | history | deleted | Yushi MuGiwara | via Vote | |
| Jan 20, 2023 at 15:59 | comment | added | Yushi MuGiwara | I know that we have equality in the case of nilpotent Lie group ... to be specific I am looking for the case where G is semi-simple. | |
| Jan 20, 2023 at 15:51 | comment | added | YCor | This inequality is trivial. Aren't you asking whether equality holds? | |
| Jan 20, 2023 at 15:50 | history | edited | YCor | CC BY-SA 4.0 |
formatting, changed tag, clarified title
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| Jan 20, 2023 at 15:41 | history | asked | Yushi MuGiwara | CC BY-SA 4.0 |