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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeAug 4th 2020

    for completeness and to satisfy links from Calabi-Penrose fibration

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeAug 4th 2020

    added pointer to:

    diff, v2, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeAug 13th 2020

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

    P 3Sp(2)/(Sp(1)×U(1)). \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.

    diff, v5, current

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

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

    added pointer to

    • 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)

    diff, v6, current

    • 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 n1T)=PC 2n1C_n/(C_{n-1}\oplus T)= \mathbf{PC}^{2n-1}, where C n=Sp(n)C_n=Sp(n) and T=U(1)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.

    diff, v7, current

    • 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.

    Added Or else see

    • 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.

    diff, v8, current

    • CommentRowNumber11.
    • CommentAuthorDavidRoberts
    • CommentTimeAug 14th 2020

    Added more publication data for the Butruille chapter

    diff, v9, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeNov 6th 2020

    added pointer to:

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

    diff, v11, current

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTime2 days ago

    added pointer to:

    • 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)

    diff, v13, current

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTime16 hours ago

    added this pointer

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

    diff, v14, current

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