special unitary group

- group, ∞-group
- group object, group object in an (∞,1)-category
- abelian group, spectrum
- group action, ∞-action
- representation, ∞-representation
- progroup
- homogeneous space

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.

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

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

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.

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

There is an isomorphism of Lie groups

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

with the spin group in dimension 3.

See at *spin group – Exceptional isomorphisms*.

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

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

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*.

The Pauli matrices satisfy the commutator relations

$[\sigma_1, \sigma_2] = \sigma_3$

$[\sigma_2, \sigma_3] = \sigma_1$

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

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

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

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 $r \gt 0$.

There is an isomorphism of Lie groups

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

with the spin group in dimension 6.

See at *spin group – Exceptional isomorphisms*.

For $SU(2)$:

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

- Alexandre Kirillov,
*Lectures on the Orbit Method*, Graduate Studies in Mathematics, 64, American Mathematical Society, (2004)

Revised on August 19, 2015 14:59:19
by Noam Zeilberger
(176.189.43.179)