Contents

group theory

# Contents

## Definition

The special unitary group $SU(n)$ for $n = 2$.

###### Proposition

As a matrix group $SU(2)$ is equivalent to the subgroup of the general linear group $GL(2, \mathbb{C})$ on those of the form

$\left( \array{ u & v \\ - \overline{v} & \overline{u} } \right) \;\;\; with \;\; {\vert u\vert}^2 + {\vert v\vert}^2 = 1 \,,$

where $u,v \in \mathbb{C}$ are complex numbers and $\overline{(-)}$ denotes complex conjugation.

## Properties

### General

###### Proposition

The underlying manifold of $SU(2)$ is diffeomorphic to the 3-sphere $S^3$.

###### Proposition

There are isomorphisms of Lie groups

1. of $SU(2)$ with the spin group in dimension 3 and with the quaternionic unitary group in one dimension

$SU(2) \;\simeq\; Spin(3) \;\simeq\; Sp(1)$
2. of the direct product group of $SU(2)$ with itself, to Spin(4)

$SU(2) \times SU(2) \;\simeq\; Spin(3) \times Spin(3) \;\simeq\; Spin(4)$

with respect to which the canonical inclusion $Spin(3) \hookrightarrow Spin(4)$ is given by the diagonal map.

###### Proposition

The Lie algebra $\mathfrak{su}(2)$ as a matrix Lie algebra is the sub Lie algebra on those matrices of the form

$\left( \array{ i z & x + i y \\ - x + i y & - i z } \right) \;\;\; with \;\; x,y,z \in \mathbb{R} \,.$
###### Definition

The standard basis elements of $\mathfrak{su}(2)$ given by the above presentation are

$\sigma_1 \coloneqq \frac{1}{\sqrt{2}} \left( \array{ 0 & 1 \\ -1 & 0 } \right)$
$\sigma_2 \coloneqq \frac{1}{\sqrt{2}} \left( \array{ 0 & i \\ i & 0 } \right)$
$\sigma_3 \coloneqq \frac{1}{\sqrt{2}} \left( \array{ i & 0 \\ 0 & -i } \right) \,.$

These are called the Pauli matrices.

###### Proposition

The Pauli matrices satisfy the commutator relations

$[\sigma_1, \sigma_2] = \sigma_3$
$[\sigma_2, \sigma_3] = \sigma_1$
$[\sigma_3, \sigma_1] = \sigma_2 \,.$
###### Proposition

The maximal torus of $SU(2)$ is the circle group $U(1)$. In the above matrix group presentation this is naturally identified with the subgroup of matrices of the form

$\left( \array{ t & 0 \\ 0 & t^{-1} } \right) \;\; with t \in U(1) \hookrightarrow \mathbb{C} \,.$
###### Proposition

The coadjoint orbits of the coadjoint action of $SU(2)$ on $\mathfrak{su}(2)$ are equivalent to the subset of the above matrices with $x^2 + y^2 + z^2 = r^2$ for some $r \geq 0$.

These are regular coadjoint orbits for $r \gt 0$.

### Finite subgroups

The finite subgroup of SU(2) have an ADE classification. See this theorem.

### $G$-Structure and exceptional geometry

Spin(8)-subgroups and reductions to exceptional geometry

reductionfrom spin groupto maximal subgroup
Spin(7)-structureSpin(8)Spin(7)
G2-structureSpin(7)G2
CY3-structureSpin(6)SU(3)
SU(2)-structureSpin(5)SU(2)
generalized reductionfrom Narain groupto direct product group
generalized Spin(7)-structure$Spin(8,8)$$Spin(7) \times Spin(7)$
generalized G2-structure$Spin(7,7)$$G_2 \times G_2$
generalized CY3$Spin(6,6)$$SU(3) \times SU(3)$

sp. orth. groupspin grouppin groupsemi-spin group
SO(2)Spin(2)Pin(2)
SO(3)Spin(3)
SO(4)Spin(4)
SO(5)Spin(5)Pin(5)
Spin(6)
Spin(7)
SO(8)Spin(8)SO(8)
SO(9)Spin(9)
$\vdots$$\vdots$
SO(16)Spin(16)SemiSpin(16)
SO(32)Spin(32)SemiSpin(32)