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 $\mathrm{SU}\left(n\right)$ is the group of isometries of the $n$-dimensional complex Hilbert space ${ℂ}^{n}$ which preserve the volume form on this space. It is the subgroup of the unitary group $U\left(n\right)$ consisting of the $n×n$ unitary matrices with determinant $1$.

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

## Examples

### $\mathrm{SU}\left(2\right)$

We discuss aspects of the special unitary group for $n=2$, hence

$\mathrm{SU}\left(2\right)≔\mathrm{SU}\left(2,ℂ\right)=\mathrm{SU}\left({ℂ}^{2}\right)\phantom{\rule{thinmathspace}{0ex}}.$SU(2) \coloneqq SU(2,\mathbb{C}) = SU(\mathbb{C}^2) \,.
###### Proposition

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

$\left(\begin{array}{cc}u& v\\ -\overline{v}& \overline{u}\end{array}\right)\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\mathrm{with}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}{\mid u\mid }^{2}+{\mid v\mid }^{2}=1\phantom{\rule{thinmathspace}{0ex}},$\left( \array{ u & v \\ - \overline{v} & \overline{u} } \right) \;\;\; with \;\; {\vert u\vert}^2 + {\vert v\vert}^2 = 1 \,,

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

###### Proposition

The underlying manifold of $\mathrm{SU}\left(2\right)$ is diffeomorphic to the 3-sphere ${S}^{3}$.

###### Proposition

There is an isomorphism of Lie groups

$\mathrm{SU}\left(2\right)\simeq \mathrm{Spin}\left(3\right)$SU(2) \simeq Spin(3)

with the spin group in dimension 3.

###### Proposition

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

$\left(\begin{array}{cc}iz& x+iy\\ -x+iy& -iz\end{array}\right)\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\mathrm{with}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}x,y,z\in ℝ\phantom{\rule{thinmathspace}{0ex}}.$\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 $\mathrm{𝔰𝔲}\left(2\right)$ given by the above presentation are

${\sigma }_{1}≔\frac{1}{\sqrt{2}}\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$\sigma_1 \coloneqq \frac{1}{\sqrt{2}} \left( \array{ 0 & 1 \\ -1 & 0 } \right)
${\sigma }_{2}≔\frac{1}{\sqrt{2}}\left(\begin{array}{cc}0& i\\ i& 0\end{array}\right)$\sigma_2 \coloneqq \frac{1}{\sqrt{2}} \left( \array{ 0 & i \\ i & 0 } \right)
${\sigma }_{3}≔\frac{1}{\sqrt{2}}\left(\begin{array}{cc}i& 0\\ 0& -i\end{array}\right)\phantom{\rule{thinmathspace}{0ex}}.$\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

$\left[{\sigma }_{1},{\sigma }_{2}\right]={\sigma }_{3}$[\sigma_1, \sigma_2] = \sigma_3
$\left[{\sigma }_{2},{\sigma }_{3}\right]={\sigma }_{1}$[\sigma_2, \sigma_3] = \sigma_1
$\left[{\sigma }_{3},{\sigma }_{1}\right]={\sigma }_{2}\phantom{\rule{thinmathspace}{0ex}}.$[\sigma_3, \sigma_1] = \sigma_2 \,.
###### Proposition

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

$\left(\begin{array}{cc}t& 0\\ 0& {t}^{-1}\end{array}\right)\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\mathrm{with}t\in U\left(1\right)↪ℂ\phantom{\rule{thinmathspace}{0ex}}.$\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 $\mathrm{SU}\left(2\right)$ on $\mathrm{𝔰𝔲}\left(2\right)$ are equivalent to the subset of the above matrices with ${x}^{2}+{y}^{2}+{z}^{2}={r}^{2}$ for some $r\ge 0$.

These are regular coadjoint orbits for $r>0$.

### $\mathrm{SU}\left(4\right)$

###### Proposition

There is an isomorphism of Lie groups

$\mathrm{SU}\left(4\right)\simeq \mathrm{Spin}\left(6\right)$SU(4) \simeq Spin(6)

with the spin group in dimension 6.

For $\mathrm{SU}\left(2\right)$:

## References

The coadjoint orbitsof $\mathrm{SU}\left(2\right)$ are discussed around p. 183 of

Revised on August 5, 2013 21:51:07 by Urs Schreiber (89.204.135.189)