nLab Spin(5,1)

Idea

The Lorentzian spin group in dimension $5 + 1$

Properties

Exceptional isomorphism with $SL(2,\mathbb{H})$

The canonical subgroup inclusion of Spin(5) into $Spin(1,5)$ is eqjuivalent to the inclusion of the quaternionic unitary group Sp(2) $= U(2,\mathbb{H})$ into $SL(2,\mathbb{H})$:

See at Spin(5) the section Exceptional isomorphism and add SL(2,H), the sections Unitary subgroup and Relation to Spin(1,5).

exceptional spinors and real normed division algebras

Lorentzian spacetime dimension	$\phantom{AA}$spin group	normed division algebra	$\,\,$ brane scan entry
$3 = 2+1$	$Spin(2,1) \simeq SL(2,\mathbb{R})$	$\phantom{A}$ $\mathbb{R}$ the real numbers	super 1-brane in 3d
$4 = 3+1$	$Spin(3,1) \simeq SL(2, \mathbb{C})$	$\phantom{A}$ $\mathbb{C}$ the complex numbers	super 2-brane in 4d
$6 = 5+1$	$Spin(5,1) \simeq$ SL(2,H)	$\phantom{A}$ $\mathbb{H}$ the quaternions	little string
$10 = 9+1$	Spin(9,1) ${\simeq}$ “SL(2,O)”	$\phantom{A}$ $\mathbb{O}$ the octonions	heterotic/type II string

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

Via quaternion-calculus and SL(2,H):

• Joás Venâncio, Carlos Batista: Two-Component Spinorial Formalism using Quaternions for Six-dimensional Spacetimes, Adv. Appl. Clifford Algebras 31 71 (2021) [arXiv:2007.04296, doi:10.1007/s00006-021-01172-1]