Timeline for $SU(k)/SO(k)$ as a manifold, for each positive integer $k$ [closed]
Current License: CC BY-SA 4.0
21 events
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| Aug 12, 2021 at 1:03 | review | Reopen votes | |||
| Aug 14, 2021 at 8:46 | |||||
| Aug 12, 2021 at 0:39 | comment | added | Robert Bryant | In fact $SLag_k=\mathrm{SU}(k)/\mathrm{SO}(k)$ is not only homogeneous, it is an irreducible symmetric space on Cartan's list, where the defining involution $\sigma:\mathrm{SU}(k)\to \mathrm{SU}(k)$ is simply $\sigma(A)=\overline{A}$. In calibrated geometry, $SLag_k$ is known as the special Lagrangian Grassmannian. In addition to the cases you have looked at already, it may be worth noting that when $k=4$ the exceptional isomorphism $\mathrm{SU}(4)=\mathrm{Spin}(6)$ gives the identification $$SLag_4\simeq\mathrm{Gr}(3,6)\simeq\mathrm{SO}(6)/\bigl(\mathrm{SO}(3)\times \mathrm{SO}(3)\bigr)$$ | |
| Aug 12, 2021 at 0:32 | vote | accept | Марина Marina S | ||
| Aug 11, 2021 at 22:21 | history | closed |
abx mme Ben McKay Vivek Shende Andreas Blass |
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| Aug 11, 2021 at 22:12 | answer | added | Will Sawin | timeline score: 3 | |
| Aug 11, 2021 at 18:19 | comment | added | abx | @Ben McKay: For me (and some other authors, e.g. Bourbaki) a Lie subgroup is a subgroup which is a submanifold; this implies that it is closed. | |
| Aug 11, 2021 at 16:59 | comment | added | მამუკა ჯიბლაძე | @BenMcKay You probably need to impose some Hermitian structure or something, to get $SU_n$ rather than some bigger group, no? | |
| Aug 11, 2021 at 16:51 | comment | added | Ben McKay | This particular manifold is the collection of all choices of a totally real linear subspace of a complex vector space, as $SU_n$ acts transitively on such, while $SO_n$ is the subgroup fixing one such. | |
| Aug 11, 2021 at 16:50 | comment | added | Ben McKay | @МаринаMarinaS: the quotient $G/H$ of a Lie group by a closed subgroup is a smooth manifold, with a unique $G$-invariant smooth structure; see any textbook on Lie groups. | |
| Aug 11, 2021 at 16:49 | comment | added | Ben McKay | @abx: your mean ba a closed Lie subgroup; otherwise not. | |
| Aug 11, 2021 at 16:42 | comment | added | Марина Marina S | I edit the question to ask along the direction on which manifold is the SU(n)/SO(n). | |
| Aug 11, 2021 at 16:41 | history | edited | Марина Marina S | CC BY-SA 4.0 |
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| Aug 11, 2021 at 16:40 | review | Close votes | |||
| Aug 11, 2021 at 22:22 | |||||
| Aug 11, 2021 at 16:40 | comment | added | Марина Marina S | thanks David - so Lie group quotient space $G/H$ is only a topological manifold, but also a smooth manifold? | |
| Aug 11, 2021 at 16:34 | comment | added | David E Speyer | I would be interested in knowing simple descriptions of the manifold SU(n)/SO(n). Regarding that it is a manifold, see math.stackexchange.com/questions/2221222 | |
| Aug 11, 2021 at 16:29 | history | edited | Марина Marina S | CC BY-SA 4.0 |
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| Aug 11, 2021 at 16:23 | history | edited | Марина Marina S | CC BY-SA 4.0 |
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| Aug 11, 2021 at 16:23 | comment | added | abx | The quotient of a Lie group by a Lie subgroup is always a manifold (in a natural way). Please use Math.stackexchange for this kind of general questions. | |
| Aug 11, 2021 at 16:22 | history | edited | Марина Marina S | CC BY-SA 4.0 |
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| Aug 11, 2021 at 16:18 | review | First posts | |||
| Aug 11, 2021 at 17:22 | |||||
| Aug 11, 2021 at 16:17 | history | asked | Марина Marina S | CC BY-SA 4.0 |