Timeline for Do all closed connected subgroups of $SO(2n+1)$ embed into $SO(2n)$?
Current License: CC BY-SA 4.0
12 events
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May 2, 2018 at 13:35 | comment | added | Robert Bryant | @IgorBelegradek: YCor answered the question you actually asked, and that answer was first, so you made the correct choice. | |
May 2, 2018 at 13:07 | comment | added | Igor Belegradek | I love both answers but can only accept one. | |
May 2, 2018 at 11:45 | comment | added | Igor Belegradek | (cont.) except possibly if $n=3$ where one has to deal with $G_2$ structure group. | |
May 2, 2018 at 11:30 | comment | added | Igor Belegradek | Thank you, in the weaker question I am also asking about Lie algebras of compact Lie groups. In my application they appear as structure groups of $SO(2n+1)$ principal bundles. I wanted to know what can be said about the bundle if the structure group is a closed proper (!) subgroup of $SO(2m+1)$. From the answer to the weaker question it follows that for bundles with smaller structure group the top rational Pontryagin class is a polynomial in lower-dimensional Pontryagin classes and the Euler class, which (potentially) restricts the bundle. | |
May 2, 2018 at 11:17 | comment | added | YCor | Thanks (sorry I didn't read "of compact type") | |
May 2, 2018 at 11:12 | comment | added | Robert Bryant | @YCor: Well, I would imagine that "simple Lie algebra over $\mathbb{R}$ of compact type" does imply "real form". Since the original question was about closed subgroups of $\mathrm{SO}(2n{+}1)$ which are compact, I feel that it is justified to restrict to simple Lie algebra over $\mathbb{R}$ of compact type, and the 'weaker' question seems to be about simple subalgebras of ${\mathfrak{so}}_{2n+1}$, which are also of this type. Hence my rephrasing of the question in the form that I did. | |
May 2, 2018 at 11:05 | comment | added | YCor | @RobertBryant you mean, all real forms of these types? | |
May 2, 2018 at 11:02 | comment | added | Robert Bryant | @YCor: When one phrases the question as "Which simple Lie algebras over $\mathbb{R}$ of compact type have the property that their nontrivial representation(s) of lowest dimension have odd dimension?", then, yes, the answer is only ${\mathfrak{so}}_{2n+1}$ and ${\mathfrak{g}}_{2}$. | |
May 2, 2018 at 8:31 | comment | added | YCor | It might me the only exception? I guess the answer is yes for large $n$. | |
May 2, 2018 at 0:50 | history | edited | Robert Bryant | CC BY-SA 4.0 |
Fixed a typo, added a reference
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May 1, 2018 at 23:05 | comment | added | paul garrett | Haha! A crazy fact! :) | |
May 1, 2018 at 22:36 | history | answered | Robert Bryant | CC BY-SA 3.0 |