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special unitary group in nLab

Contents

Context

Group Theory

group theory

Classical groups

Finite groups

Group schemes

Topological groups

Lie groups

Super-Lie groups

Higher groups

Cohomology and Extensions

Related concepts

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 ℂ 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(ℂ n)SU(\mathbb{C}^n). By SU(p,q)SU(p,q), we mean SU(ℂ 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

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,ℂ)=SU(ℂ 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,ℂ)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,v∈ℂu,v \in \mathbb{C} are complex numbers and (−)¯\overline{(-)} denotes complex conjugation.

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∈ℝ. \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)witht∈U(1)↪ℂ. \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 r≥0r \geq 0.

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

SU(3)SU(3)

SU(4)SU(4)

See at spin group – Exceptional isomorphisms.

References

Last revised on August 24, 2024 at 11:44:33. See the history of this page for a list of all contributions to it.