nLab special unitary group

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

ADE classification and McKay correspondence

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 $SU(n+1)$
A1		cyclic group of order 2 $\mathbb{Z}_2$	cyclic group of order 2 $\mathbb{Z}_2$	SU(2)
A2		cyclic 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)
D4	dihedron 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)
D5	dihedron on triangle	dihedral group of order 6 $D_6$	binary dihedral group of order 12 $2 D_6$	SO(10), Spin(10)
D6	dihedron 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 $SO(2n)$, $Spin(2n)$
$E_6$	tetrahedron	tetrahedral 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

Representation theory

See at representation theory of the special unitary group.

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.

See at spin group – Exceptional isomorphisms.

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

$$\sigma_1 \coloneqq \left( \array{ 0 & 1 \\ -1 & 0 } \right)$$

$$\sigma_2 \coloneqq \left( \array{ 0 & i \\ i & 0 } \right)$$

$$\sigma_3 \coloneqq \left( \array{ i & 0 \\ 0 & -i } \right) \,.$$

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 $r \gt 0$.

$SU(3)$

• SU(3)

$SU(4)$

Proposition

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.

Related concepts

• For SU(2):

  • quaternion

  • geometric quantization of the 2-sphere

• Euler angles

• Pauli group

References

• Howard Georgi, §13 in: Lie Algebras In Particle Physics, Westview Press (1999), CRC Press (2019) [doi:10.1201/9780429499210]

  with an eye towards application to (the standard model of) particle physics