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)} }

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 May 17, 2019 at 08:42:04. See the history of this page for a list of all contributions to it.