nLab
Spin(9)

Contents

Contents

Idea

The spin group in dimension 9.

Properties

Relation to octonionic Hopf fibration

The octonionic Hopf fibration is equivariant with respect to the Spin(9)-action, the one on S 8=S( 9)S^8 = S(\mathbb{R}^9) induced from the canonical action of Spin(9)Spin(9) on 9\mathbb{R}^9, and on S 15=S( 16)S^{15} = S(\mathbb{R}^{16}) induced from the canonical inclusion Spin(9)Spin(16)Spin(9) \hookrightarrow Spin(16).

This equivariance is made fully manifest by realizing the octonionic Hopf fibration as a map of coset spaces as follows (Ornea-Parton-Piccinni-Vuletescu 12, p. 7):

S 7 fib(h 𝕆) S 15 h 𝕆 S 8 = = = Spin(8)Spin(7) Spin(9)Spin(7) Spin(9)Spin(8) \array{ S^7 &\overset{fib(h_{\mathbb{O}})}{\longrightarrow}& S^{15} &\overset{h_{\mathbb{O}}}{\longrightarrow}& S^8 \\ = && = && = \\ \frac{Spin(8)}{Spin(7)} &\longrightarrow& \frac{Spin(9)}{Spin(7)} &\longrightarrow& \frac{Spin(9)}{Spin(8)} }

Relation to standard model gauge group

The exact gauge group of the standard model of particle physics (see there) is isomorphic to the subgroup of the Jordan algebra automorphism group of the octonionic Albert algebra that “stabilizes a 4d sub-Minkowski spacetime” (see there for details).

More concretely, it is identified with the subgroup of Spin(9) which respects a splitting 3\mathbb{H} \oplus \mathbb{H} \simeq_{\mathbb{R}} \mathbb{C} \oplus \mathbb{C}^3 (Krasnov 19, see also at SO(10)-GUT)

rotation groups in low dimensions:

Dynkin labelsp. orth. groupspin grouppin groupsemi-spin group
SO(2)Spin(2)Pin(2)
B1SO(3)Spin(3)Pin(3)
D2SO(4)Spin(4)Pin(4)
B2SO(5)Spin(5)Pin(5)
D3SO(6)Spin(6)
B3SO(7)Spin(7)
D4SO(8)Spin(8)SO(8)
B4SO(9)Spin(9)
D5SO(10)Spin(10)
B5SO(11)Spin(11)
D6SO(12)Spin(12)
\vdots\vdots
D8SO(16)Spin(16)SemiSpin(16)
\vdots\vdots
D16SO(32)Spin(32)SemiSpin(32)

see also


References

Last revised on February 26, 2020 at 11:36:58. See the history of this page for a list of all contributions to it.