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.

Examples

$SU(2)$

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

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

$SU(4)$

Proposition

There is an isomorphism of Lie groups

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

with the spin group in dimension 6.

For $SU(2)$:

References

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

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