Processing math: 100%
Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMar 25th 2019

    added pointer to

    diff, v4, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMar 25th 2019

    added this statement:


    Consider the canonical action of Spin(7) on the unit sphere in 8 (the 7-sphere),

    1. This action is is transitive;

    2. the stabilizer group of any point on S7 is G2;

    3. all G2-subgroups of Spin(7) arise this way, and are all conjugate to each other.

    Hence the coset space of Spin(7) by G2 is the 7-sphere

    Spin(7)/G2S7.

    will copy this also to G2 and 7-sphere

    diff, v4, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMar 25th 2019
    • (edited Mar 25th 2019)

    Hm, so it must be that we have this situation:

    S7BG2BSpin(7)(pb)S7BSpin(7)BSpin(8)
    • CommentRowNumber4.
    • CommentAuthorDavidRoberts
    • CommentTimeMar 25th 2019

    Intriguing!

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMar 25th 2019
    • (edited Mar 25th 2019)

    What further realizations of 7-spheres (or 4-spheres) as coset spaces do we have?

    So far I know that S7

    1. Spin(8)/Spin(7) (clear)

    2. Spin(7)/G2 (by what we just said)

    3. Sp(2)/(Sp(1)×Sp(1)) (the Gromoll-Meyer sphere)

    Anything else? I’d like to see more of the exceptional Lie groups show up – any chance?

    • CommentRowNumber6.
    • CommentAuthorDavid_Corfield
    • CommentTimeMar 26th 2019

    John Baez wrote a post years ago about 4 different ways to build the 7-sphere as a homogeneous space, including

    • Spin(6)/SU(3)

    • Spin(5)/SU(2)

    • CommentRowNumber7.
    • CommentAuthorDavidRoberts
    • CommentTimeMar 26th 2019

    So I guess that last one is Sp(2)/SU(2), which is pleasing.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeMar 26th 2019
    • (edited Mar 26th 2019)

    I only now realize that my reply to the above got sent to the wrong thread (here), where I said:

    Thanks! Have added these here

    diff, v20, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeMar 27th 2019

    So the diagram in #3 is just the last in a pasting array of similar diagrams, the de-homotopified version of which is a diagram of subgroup intersections in Spin(8) which I just typed out here

    (in xymatrix, so it doesn’t display here…)

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeMar 27th 2019
    • (edited Mar 27th 2019)

    [ removed ]