nLab Spin(9)

Idea

The spin group in dimension 9.

Properties

Relation to octonionic Hopf fibration

The octonionic Hopf fibration is equivariant with respect to a 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):

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

Octonionic construction of representation

The group $Spin(9)$ has a 16-dimensional faithful irreducible complex representation called its Dirac spinor representation. (Bryant 20) constructs this using octonions as follows.

For $(r,x) \in \mathbb{R} \oplus \mathbb{O}$ define a linear map $m_{r,x} \colon \mathbb{C} \otimes \mathbb{O}^2 \to \mathbb{C} \otimes \mathbb{O}^2$ as follows:

$$m_{r,x} = i \left( \begin{array}{cc} r & C R_x \\ -C L_x & -r \end{array} \right)$$

where $r$ here denotes multiplication by the real number $r$ and $L_x, R_x, C \colon \mathbb{O} \to \mathbb{O}$ are left multiplication by the octonion $x$, right multiplication by $x$ and octonionic conjugation, respectively. Since $m_{r,x}^2 = - r^2 -{|x|}^2$ this map induces an action of the Clifford algebra $Cliff(\mathbb{R} \oplus \mathbb{O})$ on $\mathbb{C} \otimes \mathbb{O}^2$. (To be precise, this is Clifford algebra generated by 9 anticommuting square roots of minus 1). One can show that this action gives an isomorphism of complex associative algebras

$$Cliff(\mathbb{R} \oplus \mathbb{O}) \cong End(\mathbb{C} \otimes \mathbb{O}^2)$$

where $End(\mathbb{C} \otimes \mathbb{O}^2)$, the algebra of all complex-linear transformations of $\mathbb{C} \otimes \mathbb{O}^2$, is isomorphic to the algebra of $16 \times 16$ complex matrices.

The group $Spin(9)$ is isomorphic to the group of linear transformations generated by products $m_{r,x} m_{s,y}$ where $r,s \in \mathbb{R}$ have ${|r|} = {|s|} = 1$ and $x, y \in \mathbb{O}$ have ${|x|} = {|y|} = 1$. This group is also generated by elements

$$\left( \begin{array}{cc} r & C R_x \\ -C L_x & r \end{array} \right)$$

where $r \in \mathbb{R}$ and $x \in \mathbb{O}$ have $r^2 + {|x|}^2 = 1$.

This description of $Spin(9)$ makes manifest its 16-dimensional Dirac spinor representation, which is a faithful complex representation.

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)

Homotopy groups

$\pi_1$	$\pi_2$	$\pi_3$	$\pi_4$	$\pi_5$	$\pi_6$	$\pi_7$	$\pi_8$	$\pi_9$	$\pi_10$	$\pi_11$	$\pi_12$	$\pi_13$	$\pi_14$	$\pi_15$	$\pi_16$	$\pi_17$
$0$	$0$	$\mathbb{Z}$	$0$	$0$	$0$	$\mathbb{Z}$	$\mathbb{Z}_2^2$	$\mathbb{Z}_2^2$	$\mathbb{Z}_8$	$\mathbb{Z}\oplus\mathbb{Z}_2$	$0$	$\mathbb{Z}_2$	$\mathbb{Z}_8\oplus\mathbb{Z}_2$	$\mathbb{Z}\oplus\mathbb{Z}_2^3$	$\mathbb{Z}_2^6$	$\mathbb{Z}_8\oplus\mathbb{Z}_2^2$

$\pi_18$	$\pi_19$	$\pi_20$	$\pi_21$	$\pi_22$
$\mathbb{Z}_2835\oplus\mathbb{Z}_16\oplus\mathbb{Z}_8\oplus\mathbb{Z}_2$	$\mathbb{Z}_2$	$\mathbb{Z}_2$	$\mathbb{Z}_12$	$\mathbb{Z}_{1247400}\oplus\mathbb{Z}_8\oplus\mathbb{Z}_2^2$

(Mimura 67)

Related concepts

rotation groups in low dimensions:

Dynkin label	sp. orth. group	spin group	pin group	semi-spin group
	SO(2)	Spin(2)	Pin(2)
B1	SO(3)	Spin(3)	Pin(3)
D2	SO(4)	Spin(4)	Pin(4)
B2	SO(5)	Spin(5)	Pin(5)
D3	SO(6)	Spin(6)
B3	SO(7)	Spin(7)
D4	SO(8)	Spin(8)		SO(8)
B4	SO(9)	Spin(9)
D5	SO(10)	Spin(10)
B5	SO(11)	Spin(11)
D6	SO(12)	Spin(12)
	$\vdots$	$\vdots$
D8	SO(16)	Spin(16)		SemiSpin(16)
	$\vdots$	$\vdots$
D16	SO(32)	Spin(32)		SemiSpin(32)

see also

• Spin(5).Spin(3)

• finite rotation groups

• ADE-classification

References

• Robert Bryant, Notes on spinors in low dimension. (arXiv:2011.05568)

• Mamoru Mimura, The homotopy groups of Lie groups of low rank, Math. Kyoto Univ. Volume 6, Number 2 (1967), 131-176. (Euclid)

• Thomas Friedrich, Weak Spin(9)-Structures on 16-dimensional Riemannian Manifolds (1999), (arXiv:math/9912112)

• Maurizio Parton, Paolo Piccinni, Spin(9) and almost complex structures on 16-dimensional manifolds (2011), (arXiv:1105.5318)

• Liviu Ornea, Maurizio Parton, Paolo Piccinni, Victor Vuletescu, Spin(9) geometry of the octonionic Hopf fibration, (arXiv:1208.0899, doi:10.1007/s00031-013-9233-x)

• Maurizio Parton, Paolo Piccinni, The Role of Spin(9) in Octonionic Geometry, (arXiv:1810.06288)

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