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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeAug 14th 2015
   • (edited Aug 14th 2015)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added pointer to * Paul de Medeiros, [[José Figueroa-O'Farrill]], Sunil Gadhia, [[Elena Méndez-Escobar]], _Half-BPS quotients in M-theory: ADE with a twist_, JHEP 0910:038,2009 ([arXiv:0909.0163](http://arxiv.org/abs/0909.0163), [pdf slides](http://www.maths.ed.ac.uk/~jmf/CV/Seminars/YRM2010.pdf)) to the entries _[[7-sphere]]_, _[[ADE classification]]_, _[[Freund-Rubin compactification]]_. This article proves the neat result that the finite subgroups $\Gamma$ of $SO(8)$ such that $S^7/\Gamma$ is smooth and spin and has at least four Killing spinors has an ADE classification. The $\Gamma$s are the the "binary" versions of the symmetries of the Platonic solids.

   I have added pointer to

   • Paul de Medeiros, José Figueroa-O’Farrill, Sunil Gadhia, Elena Méndez-Escobar, Half-BPS quotients in M-theory: ADE with a twist, JHEP 0910:038,2009 (arXiv:0909.0163, pdf slides)

   to the entries 7-sphere, ADE classification, Freund-Rubin compactification.

   This article proves the neat result that the finite subgroups $\Gamma$ of $SO(8)$ such that $S^7/\Gamma$ is smooth and spin and has at least four Killing spinors has an ADE classification. The $\Gamma$s are the the “binary” versions of the symmetries of the Platonic solids.

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeAug 16th 2015
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   Author: Todd_Trimble
   Format: MarkdownItexI added some more material to [[7-sphere]], mostly on exotic smooth structures.

   I added some more material to 7-sphere, mostly on exotic smooth structures.

3. • CommentRowNumber3.
   • CommentAuthorDavidRoberts
   • CommentTimeAug 17th 2015
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexI mentioned that these exotic spheres are called _Brieskorn manifolds_ or _spheres_ and corrected a typo ($z^{42}$ -- !)

   I mentioned that these exotic spheres are called Brieskorn manifolds or spheres and corrected a typo ($z^{42}$ – !)

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeSep 8th 2016
   • PermaLink
   Author: Urs
   Format: MarkdownItexAdded statement of the canonical (nearly parallel) $G_2$-structure on the 7-sphere -- [here](https://ncatlab.org/nlab/show/7-sphere#G2Structure).

   Added statement of the canonical (nearly parallel) $G_2$-structure on the 7-sphere – here.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeMar 26th 2019
   • (edited Mar 26th 2019)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks! Have added these [here](https://ncatlab.org/nlab/show/7-sphere#CosetSpaceRealization) [this was in reply to the message [here](https://nforum.ncatlab.org/discussion/9744/spin7/?Focus=76866#Comment_76866), in another thread] <a href="https://ncatlab.org/nlab/revision/diff/7-sphere/20">diff</a>, <a href="https://ncatlab.org/nlab/revision/7-sphere/20">v20</a>, <a href="https://ncatlab.org/nlab/show/7-sphere">current</a>

   Thanks! Have added these here

   [this was in reply to the message here, in another thread]

   diff, v20, current

6. • CommentRowNumber6.
   • CommentAuthorDavid_Corfield
   • CommentTimeApr 19th 2019
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexAdded comment and reference to John Baez's account of the 4 coset realizations. <a href="https://ncatlab.org/nlab/revision/diff/7-sphere/22">diff</a>, <a href="https://ncatlab.org/nlab/revision/7-sphere/22">v22</a>, <a href="https://ncatlab.org/nlab/show/7-sphere">current</a>

   Added comment and reference to John Baez’s account of the 4 coset realizations.

   diff, v22, current

7. • CommentRowNumber7.
   • CommentAuthorStableHolonomy
   • CommentTimeOct 17th 2024
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   Author: StableHolonomy
   Format: TextIs there a Pontrjagin duality for SU(2) (unit ball in the quaternions), much like how S¹ is the unit ball in the complex numbers? There is a Haar integral for SU(2), and many of the same proofs appear to carry over. But its also possible that [[G,SU(2)],SU(2)] is better supposed to be isomorphic to (G ⋊ ℤ/2ℤ) ⋊ ℤ/2ℤ?

   Is there a Pontrjagin duality for SU(2) (unit ball in the quaternions), much like how S¹ is the unit ball in the complex numbers?
   
   There is a Haar integral for SU(2), and many of the same proofs appear to carry over. But its also possible that [[G,SU(2)],SU(2)] is better supposed to be isomorphic to (G ⋊ ℤ/2ℤ) ⋊ ℤ/2ℤ?