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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 of such that is smooth and spin and has at least four Killing spinors has an ADE classification. The 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
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   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 ( – !)

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) -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
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   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?