nLab SL(2,H)

Definition

By $SL(2,\mathbb{H})$ one denotes the special linear group of $2 \times 2$ matrices with coefficients in the quaternions, where “special” refers to their Dieudonné determinant being unity:

$$SL(2,\mathbb{H}) \;:=\; \left\{ \left. \left[ \array{ a & b \\ c & d } \right] \;\right\vert\; \left\Vert a d - a c a^{-1} b \right\Vert \;=\; 1 \right\}$$

(here $\left\Vert q\right\Vert^2 \;:=\; q \bar q \in \mathbb{R}$ is the norm(-square) on quaternions).

Properties

Relation to $Sp(2)$

Every quaternionic unitary matrix (hence in Sp(2)) happens to have unit Dieudonné determinant (Cohen-De Leo 99, Cor. 6.4). Therefore we have a subgroup inclusion

$$Sp(2) \;=\; U(2,\mathbb{H}) \;\subset\; SL(2,\mathbb{H}) \,.$$

Relation to $Spin(5,1)$

Under the conjugation action on $2 \times 2$ Hermitian matrices with coefficients in the quaternions, $SL(2,\mathbb{H})$ is identified with Spin(5,1) and its canonical action on Minkowski spacetime $\mathbb{R}^{5,1}$.

For more on this see at spin representation, supersymmetry and division algebras and geometry of physics – supersymmetry.

Related concepts

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

References

• Joás Venâncio, Carlos Batista, Sections 2,3 of: 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)