Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
added pointer to
added this statement:
Consider the canonical action of Spin(7) on the unit sphere in ℝ8 (the 7-sphere),
This action is is transitive;
the stabilizer group of any point on S7 is G2;
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)/G2≃S7.Hm, so it must be that we have this situation:
S7⟶BG2⟶BSpin(7)≃↓↓(pb)↓S7⟶BSpin(7)⟶BSpin(8)Intriguing!
What further realizations of 7-spheres (or 4-spheres) as coset spaces do we have?
So far I know that S7≃⋯
⋯Spin(8)/Spin(7) (clear)
⋯Spin(7)/G2 (by what we just said)
⋯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?
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)
So I guess that last one is Sp(2)/SU(2), which is pleasing.
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…)
[ removed ]
1 to 10 of 10