Contents

group theory

# Contents

## Idea

The special unitary group is the subgroup of the unitary group on the elements with determinant equal to 1.

For $n$ a natural number, the special unitary group $SU(n)$ is the group of isometries of the $n$-dimensional complex Hilbert space $\mathbb{C}^n$ which preserve the volume form on this space. It is the subgroup of the unitary group $U(n)$ consisting of the $n \times n$ unitary matrices with determinant $1$.

More generally, for $V$ any complex vector space equipped with a nondegenerate Hermitian form $Q$, $SU(V,Q)$ is the group of isometries of $V$ which preserve the volume form derived from $Q$. One may write $SU(V)$ if $Q$ is obvious, so that $SU(n)$ is the same as $SU(\mathbb{C}^n)$. By $SU(p,q)$, we mean $SU(\mathbb{C}^{p+q},Q)$, where $Q$ has $p$ positive eigenvalues and $q$ negative ones.

## Properties

### As part of the ADE pattern

Dynkin diagram/
Dynkin quiver
dihedron,
Platonic solid
finite subgroups of SO(3)finite subgroups of SU(2)simple Lie group
$A_{n \geq 1}$cyclic group
$\mathbb{Z}_{n+1}$
cyclic group
$\mathbb{Z}_{n+1}$
special unitary group
$SU(n+1)$
A1cyclic group of order 2
$\mathbb{Z}_2$
cyclic group of order 2
$\mathbb{Z}_2$
SU(2)
A2cyclic group of order 3
$\mathbb{Z}_3$
cyclic group of order 3
$\mathbb{Z}_3$
SU(3)
A3
=
D3
cyclic group of order 4
$\mathbb{Z}_4$
cyclic group of order 4
$2 D_2 \simeq \mathbb{Z}_4$
SU(4)
$\simeq$
Spin(6)
D4dihedron on
bigon
Klein four-group
$D_4 \simeq \mathbb{Z}_2 \times \mathbb{Z}_2$
quaternion group
$2 D_4 \simeq$ Q8
SO(8), Spin(8)
D5dihedron on
triangle
dihedral group of order 6
$D_6$
binary dihedral group of order 12
$2 D_6$
SO(10), Spin(10)
D6dihedron on
square
dihedral group of order 8
$D_8$
binary dihedral group of order 16
$2 D_{8}$
SO(12), Spin(12)
$D_{n \geq 4}$dihedron,
hosohedron
dihedral group
$D_{2(n-2)}$
binary dihedral group
$2 D_{2(n-2)}$
special orthogonal group, spin group
$SO(2n)$, $Spin(2n)$
$E_6$tetrahedrontetrahedral group
$T$
binary tetrahedral group
$2T$
E6
$E_7$cube,
octahedron
octahedral group
$O$
binary octahedral group
$2O$
E7
$E_8$dodecahedron,
icosahedron
icosahedral group
$I$
binary icosahedral group
$2I$
E8

## Examples

### $SU(2)$

We discuss aspects of SU(2), hence

$SU(2) \coloneqq SU(2,\mathbb{C}) = SU(\mathbb{C}^2) \,.$
###### 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.

###### Proposition

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

###### Proposition

There is an isomorphism of Lie groups

$SU(2) \simeq Spin(3)$

with the spin group in dimension 3.

###### 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

$[\sigma_1, \sigma_2] = 2\sigma_3$
$[\sigma_2, \sigma_3] = 2\sigma_1$
$[\sigma_3, \sigma_1] = 2\sigma_2 \,.$
###### 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 $r \gt 0$.

### $SU(4)$

###### Proposition

There is an isomorphism of Lie groups

SU(4)$\simeq$ Spin(6)

with the spin group in dimension 6.

The coadjoint orbits of $SU(2)$ are discussed around p. 183 of