Contents

group theory

# 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 $S^8 = S(\mathbb{R}^9)$ induced from the canonical action of $Spin(9)$ on $\mathbb{R}^9$, and on $S^{15} = S(\mathbb{R}^{16})$ induced from the canonical inclusion $Spin(9) \hookrightarrow Spin(16)$.

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):

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

### 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 $\mathbb{H} \oplus \mathbb{H} \simeq_{\mathbb{R}} \mathbb{C} \oplus \mathbb{C}^3$ (Krasnov 19, see also at SO(10)-GUT)

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)

• Kirill Krasnov, $SO(9)$ characterisation of the Standard Model gauge group (arXiv:1912.11282)