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):

$\array{ S^7 &\overset{fib(h_{\mathbb{O}})}{\longrightarrow}& S^{15} &\overset{h_{\mathbb{O}}}{\longrightarrow}& S^8 \\ = && = && = \\ \frac{Spin(8)}{Spin(7)} &\longrightarrow& \frac{Spin(9)}{Spin(7)} &\longrightarrow& \frac{Spin(9)}{Spin(8)} }$
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 May 17, 2019 at 08:42:04. See the history of this page for a list of all contributions to it.