nLab SL(2,H)

Definition

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:

$$SL(2,\mathbb{H}) \;\coloneqq\; \Big\{\left. A \in Mat_{2 \times 2}(\mathbb{H}) \;\right\vert\; det_D(A) = 1 \Big\} \mathrlap{\,,}$$

namely (cf. Venâncio & Batista 2021 (2.4))

(1)$$det_D\left( \array{ a & b \\ c & d } \right) \; = \; \left\{ \begin{array}{cl} 0 & \;\text{if}\; a = b = c = d = 0 \\ \left\vert a d - a c a^{-1} b \right\vert & \;\text{if}\; a \neq 0 \\ \left\vert d a - d b d^{-1} c \right\vert & \;\text{if}\; d \neq 0 \\ \left\vert b d b^{-1} a - b c \right\vert & \;\text{if}\; b \neq 0 \\ \left\vert c a c^{-1} d - c b \right\vert & \;\text{if}\; c \neq 0 \mathrlap{\,.} \end{array} \right.$$

Here

(2)$$\left\vert q \right\vert \;\coloneqq\; \textstyle{\sqrt{q q^\ast}} \;\in \mathbb{R}$$

is the standard norm on quaternions.

Remark

Since the norm (2) evidently satisfies

$$\left\vert q q'\right\vert \;=\; \left\vert q \right\vert \cdot \left\vert q' \right\vert \mathrlap{\,,}$$

the formulas (1) are furthermore equivalent to

(3)$$det_D\left( \array{ a & b \\ c & d } \right) \;=\; \left\{ \begin{array}{cl} 0 & \;\text{if}\; a = b = c = d = 0 \\ \left\vert a \right\vert \cdot \left\vert d - c a^{-1} b \right\vert & \;\text{if}\; a \neq 0 \\ \left\vert d \right\vert \cdot \left\vert a - b d^{-1} c \right\vert & \;\text{if}\; d \neq 0 \\ \left\vert b \right\vert \cdot \left\vert d b^{-1} a - c \right\vert & \;\text{if}\; b \neq 0 \\ \left\vert c \right\vert \cdot \left\vert a c^{-1} d - b \right\vert & \;\text{if}\; c \neq 0 \mathrlap{\,.} \end{array} \right.$$

In this form they appear in Cohen & De Leo 1999 p. 11.

Properties

Relation to $Sp(2)$

Lemma

The above $det_D$ (1) satisfies

$$\begin{aligned} det_D(A) & \in\; \mathbb{R}_{\geq 0} \\ det_D(I_2) & =\; 1 \\ det_D(A \cdot B) &=\; det_D(A) \cdot det_D(B) \\ det_D\big( A^\dagger \big) &=\; det_D(A) \mathrlap{\,.} \end{aligned}$$

(cf. Cohen & De Leo 1999 Thm. 5.1(5) & Cor. 6.4)

Proposition

Every quaternionic unitary matrix (hence in Sp(2)) happens to have unit Dieudonné determinant, whence we have a subgroup inclusion:

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

(Cohen-De Leo 99, Cor. 6.4)

Proof

By definition, $A \in Mat_{2 \times 2}(\mathbb{H})$ is in $Sp(2)$ iff $A \cdot A^\dagger = I_2$. From this, Lemma gives

$$\begin{aligned} 1 & = det_D\big(A \cdot A^\dagger \big) \\ & = det_D(A) \cdot det_D\big(A^\dagger\big) \\ & = \big( det_D(A) \big)^2 \,. \end{aligned}$$

But the only element $det_D(A) \in \mathbb{R}_{\geq 0}$ satisfying this equation is $det_D(A) = 1$.

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 $\mathbb{R}^{5,1}$.

(cf. Venâncio & Batista 2021)

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

This exceptional isomorphism is compatible with that between the subgroups Spin(5) and the quaternion unitary group Sp(2):

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

• Nir Cohen, Stefano De Leo, Cor. 6.4 in: The quaternionic determinant, Electronic Journal of Linear Algebra 7 (2000) 100-111 [arXiv:math-ph/9907015, doi:10.13001/1081-3810.1050, eudml:121484]

• 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]