# nLab SL(2,H)

Contents

group theory

## Spin geometry

spin geometry

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)

string geometry

# Contents

## 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.

exceptional spinors and real normed division algebras

Lorentzian
spacetime
dimension
$\phantom{AA}$spin groupnormed division algebra$\,\,$ brane scan entry
$3 = 2+1$$Spin(2,1) \simeq SL(2,\mathbb{R})$$\phantom{A}$ $\mathbb{R}$ the real numberssuper 1-brane in 3d
$4 = 3+1$$Spin(3,1) \simeq SL(2, \mathbb{C})$$\phantom{A}$ $\mathbb{C}$ the complex numberssuper 2-brane in 4d
$6 = 5+1$$Spin(5,1) \simeq$ SL(2,H)$\phantom{A}$ $\mathbb{H}$ the quaternionslittle string
$10 = 9+1$Spin(9,1) ${\simeq}$SL(2,O)$\phantom{A}$ $\mathbb{O}$ the octonionsheterotic/type II string