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special unitary group

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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

Dynkin diagramPlatonic solidfinite subgroup of SO(3)SO(3)finite subgroup of SU(2)SU(2)simple Lie group
A lA_lcyclic groupcyclic groupspecial unitary group
D lD_ldihedron/hosohedrondihedral groupbinary dihedral groupspecial orthogonal group
E 6E_6tetrahedrontetrahedral groupbinary tetrahedral groupE6
E 7E_7cube/octahedronoctahedral groupbinary octahedral groupE7
E 8E_8dodecahedron/icosahedronicosahedral groupbinary icosahedral groupE8

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 112(0 1 1 0) \sigma_1 \coloneqq \frac{1}{\sqrt{2}} \left( \array{ 0 & 1 \\ -1 & 0 } \right)
σ 212(0 i i 0) \sigma_2 \coloneqq \frac{1}{\sqrt{2}} \left( \array{ 0 & i \\ i & 0 } \right)
σ 312(i 0 0 i). \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

[σ 1,σ 2]=σ 3 [\sigma_1, \sigma_2] = \sigma_3
[σ 2,σ 3]=σ 1 [\sigma_2, \sigma_3] = \sigma_1
[σ 3,σ 1]=σ 2. [\sigma_3, \sigma_1] = \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(4)SU(4)

Proposition

There is an isomorphism of Lie groups

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

with the spin group in dimension 6.

See at spin group – Exceptional isomorphisms.

For SU(2)SU(2):

References

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

Revised on January 20, 2016 13:01:46 by Urs Schreiber (195.37.209.180)