nLab special unitary group

Contents

Context

Group Theory

Lie theory

∞-Lie theory (higher geometry)

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Related topics

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

Idea

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

For nn a natural number, the special unitary group SU(n)SU(n) is the group of isometries of the nn-dimensional complex Hilbert space n\mathbb{C}^n which preserve the volume form on this space. It is the subgroup of the unitary group U(n)U(n) consisting of the n×nn \times n unitary matrices with determinant 11.

More generally, for VV any complex vector space equipped with a nondegenerate Hermitian form QQ, SU(V,Q)SU(V,Q) is the group of isometries of VV which preserve the volume form derived from QQ. One may write SU(V)SU(V) if QQ is obvious, so that SU(n)SU(n) is the same as SU( n)SU(\mathbb{C}^n). By SU(p,q)SU(p,q), we mean SU( p+q,Q)SU(\mathbb{C}^{p+q},Q), where QQ has pp positive eigenvalues and qq 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 n1A_{n \geq 1}cyclic group
n+1\mathbb{Z}_{n+1}
cyclic group
n+1\mathbb{Z}_{n+1}
special unitary group
SU(n+1)SU(n+1)
A1cyclic group of order 2
2\mathbb{Z}_2
cyclic group of order 2
2\mathbb{Z}_2
SU(2)
A2cyclic group of order 3
3\mathbb{Z}_3
cyclic group of order 3
3\mathbb{Z}_3
SU(3)
A3
=
D3
cyclic group of order 4
4\mathbb{Z}_4
cyclic group of order 4
2D 2 42 D_2 \simeq \mathbb{Z}_4
SU(4)
\simeq
Spin(6)
D4dihedron on
bigon
Klein four-group
D 4 2× 2D_4 \simeq \mathbb{Z}_2 \times \mathbb{Z}_2
quaternion group
2D 42 D_4 \simeq Q8
SO(8), Spin(8)
D5dihedron on
triangle
dihedral group of order 6
D 6D_6
binary dihedral group of order 12
2D 62 D_6
SO(10), Spin(10)
D6dihedron on
square
dihedral group of order 8
D 8D_8
binary dihedral group of order 16
2D 82 D_{8}
SO(12), Spin(12)
D n4D_{n \geq 4}dihedron,
hosohedron
dihedral group
D 2(n2)D_{2(n-2)}
binary dihedral group
2D 2(n2)2 D_{2(n-2)}
special orthogonal group, spin group
SO(2n)SO(2n), Spin(2n)Spin(2n)
E 6E_6tetrahedrontetrahedral group
TT
binary tetrahedral group
2T2T
E6
E 7E_7cube,
octahedron
octahedral group
OO
binary octahedral group
2O2O
E7
E 8E_8dodecahedron,
icosahedron
icosahedral group
II
binary icosahedral group
2I2I
E8

Representation theory

See at representation theory of the special unitary group.

Examples

SU(2)SU(2)

We discuss aspects of SU(2), hence

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

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

(u v v¯ u¯)with|u| 2+|v| 2=1, \left( \array{ u & v \\ - \overline{v} & \overline{u} } \right) \;\;\; with \;\; {\vert u\vert}^2 + {\vert v\vert}^2 = 1 \,,

where u,vu,v \in \mathbb{C} are complex numbers and ()¯\overline{(-)} denotes complex conjugation.

Proposition

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

Proposition

There is an isomorphism of Lie groups

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

with the spin group in dimension 3.

See at spin group – Exceptional isomorphisms.

Proposition

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

(iz x+iy x+iy iz)withx,y,z. \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 𝔰𝔲(2)\mathfrak{su}(2) given by the above presentation are

σ 1(0 1 1 0) \sigma_1 \coloneqq \left( \array{ 0 & 1 \\ -1 & 0 } \right)
σ 2(0 i i 0) \sigma_2 \coloneqq \left( \array{ 0 & i \\ i & 0 } \right)
σ 3(i 0 0 i). \sigma_3 \coloneqq \left( \array{ i & 0 \\ 0 & -i } \right) \,.

These are called the Pauli matrices.

Proposition

The Pauli matrices satisfy the commutator relations

[σ 1,σ 2]=2σ 3 [\sigma_1, \sigma_2] = 2\sigma_3
[σ 2,σ 3]=2σ 1 [\sigma_2, \sigma_3] = 2\sigma_1
[σ 3,σ 1]=2σ 2. [\sigma_3, \sigma_1] = 2\sigma_2 \,.
Proposition

The maximal torus of SU(2)SU(2) is the circle group U(1)U(1). In the above matrix group presentation this is naturally identified with the subgroup of matrices of the form

(t 0 0 t 1)withtU(1). \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)SU(2) on 𝔰𝔲(2)\mathfrak{su}(2) are equivalent to the subset of the above matrices with x 2+y 2+z 2=r 2x^2 + y^2 + z^2 = r^2 for some r0r \geq 0.

These are regular coadjoint orbits for r>0r \gt 0.

SU(3)SU(3)

SU(4)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.

References

Last revised on September 7, 2023 at 12:37:08. See the history of this page for a list of all contributions to it.