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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeAug 4th 2020

for completeness and to satisfy links from Calabi-Penrose fibration

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeAug 4th 2020

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeAug 13th 2020

I have added a reference (Zandi 88) with a clear statement (in section 7) of the coset space realization

$\mathbb{C}P^3 \simeq Sp(2)/(Sp(1) \times \mathrm{U}(1)) \,.$

But I am still looking for a more canonical citation for this fact.

• CommentRowNumber4.
• CommentAuthorDavidRoberts
• CommentTimeAug 13th 2020
• (edited Aug 13th 2020)

Couldn’t one explain this via $S^7 = Sp(2)/Sp(1)$ (unit sphere in $\mathbb{H}^2\simeq \mathbb{C}^4$) and then $\mathbb{CP}^3 = S^7/U(1)$, using (I assume) the fact $U(1)$ is central? Or do you want the explicit statement as written? One could do a non-compact version, with $GL(2,\mathbb{H})/Sp(1) \stackrel{?}{=} \mathbb{H}^2 \setminus \{0\}\simeq \mathbb{C}^4 \setminus \{0\}$ and then $\mathbb{CP}^3 = \mathbb{C}^4 \setminus \{0\}/\mathbb{C}^\times$

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeAug 13th 2020

Just to clarify that I am not worried about the explanation of this fact but about citing it properly.

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeAug 13th 2020

• Kouyemon Iriye, Manifolds which have two projective space bundle structures from the homotopical point of view, J. Math. Soc. Japan Volume 42, Number 4 (1990), 639-658 (euclid:jmsj/1227108441)

where it’s equation (3)

• CommentRowNumber7.
• CommentAuthorDavidRoberts
• CommentTimeAug 14th 2020
• (edited Aug 14th 2020)

Try Таблица 1 of

• . L. Onishchik, “On compact Lie groups transitive on certain manifolds”, Dokl. Akad. Nauk SSSR, 135:3 (1960), 531–534, Math-Net.ru

where we have, essentially, $C_n/(C_{n-1}\oplus T)= \mathbf{PC}^{2n-1}$, where $C_n=Sp(n)$ and $T=U(1)$.

I’m sure its much older than this, though.

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeAug 14th 2020

Thanks! I have added that pointer to the entry.

But it’s not necessarily for about earliest reference, though that may be interesting, too. Best would be a modern textbook account.

• CommentRowNumber9.
• CommentAuthorDavidRoberts
• CommentTimeAug 14th 2020
• (edited Aug 14th 2020)

Oh, well, then this: https://hal.archives-ouvertes.fr/hal-00121742 (See section 3), which appeared as a chapter in the Handbook of Pseudo-Riemannian Geometry and Supersymmetry would be good. More specifically: the discussion before Prop 3.2 proves what you want, I think.

• Onishchik A.L. (1993) Compact Homogeneous Spaces. In: Onishchik A.L. (eds) Lie Groups and Lie Algebras I. Encyclopaedia of Mathematical Sciences, vol 20. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-57999-8_11

in particular table 3 (which reproduces the result from the 1960s)

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeAug 14th 2020
• (edited Aug 14th 2020)

Thanks again!

The pointer to Butruille I already had in the entry, in its arXiv version (but this is one of those reference that just takes it all for granted).

The Onishchik Encyclopedia is good. Now I remember that we we had been citing this table elsewhere already (such as at SU(3)). Thanks for reminding me! Have added it here now, too.

• CommentRowNumber11.
• CommentAuthorDavidRoberts
• CommentTimeAug 14th 2020

Added more publication data for the Butruille chapter

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeNov 6th 2020

• Michael Atiyah, E. Rees, Vector bundles on projective 3-space, Invent. Math.35, 131–153 (1976)
• CommentRowNumber13.
• CommentAuthorUrs
• CommentTime2 days ago

• Mark Hughes, Symmetries of Homotopy Complex Projective Three Spaces, Transactions of the American Mathematical Society Vol. 337, No. 1 (May, 1993), pp. 291-304 (doi:10.2307/2154323)
• CommentRowNumber14.
• CommentAuthorUrs
• CommentTime16 hours ago

• Dagan Karp, Dhruv Ranganathan, Paul Riggins, Ursula Whitcher, Toric symmetry of $\mathbb{C}P^3$, Advances in Theoretical and Mathematical Physics, Vol. 16, No. 4, 2012 (arXiv:1109.5157)