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nLab > Latest Changes: complex projective 3-space

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeAug 4th 2020

    for completeness and to satisfy links from Calabi-Penrose fibration

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeAug 5th 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

    ℂP 3≃Sp(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 ℍ 2≃ℂ 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,ℍ)/Sp(1)=?ℍ 2∖{0}≃ℂ 4∖{0}GL(2,\mathbb{H})/Sp(1) \stackrel{?}{=} \mathbb{H}^2 \setminus \{0\}\simeq \mathbb{C}^4 \setminus \{0\} and then ℂℙ 3=ℂ 4∖{0}/ℂ ×\mathbb{CP}^3 = \mathbb{C}^4 \setminus \{0\}/\mathbb{C}^\times …

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeAug 14th 2020

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

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeAug 14th 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 n−1⊕T)=PC 2n−1C_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
    • CommentTimeNov 21st 2020

    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
    • CommentTimeNov 23rd 2020

    added this pointer

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

    diff, v14, current

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeMar 31st 2021

    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.

    diff, v15, current

1 to 15 of 15

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