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for completeness and to satisfy links from Calabi-Penrose fibration
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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 $\mathbb{CP}^3 = \mathbb{C}^4 \setminus \{0\}/\mathbb{C}^\times$ …
Just to clarify that I am not worried about the explanation of this fact but about citing it properly.
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where it’s equation (3)
Try Таблица 1 of
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.
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.
Added Or else see
in particular table 3 (which reproduces the result from the 1960s)
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.
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I have added the following pointer to the references:
Complex projective 3-space is conceived in the guise as the twistor space of 4d Minkowski spacetime in
Over on Twitter with Sam Walters (here) we are wondering whether the twistor fibration is made explicit by Penrose, or when/where it is. In the above reference it is certainly implicit, but I don’t see it stated explicitly.
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