nLab
SU(2)

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

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

Definition

The special unitary group SU(n)SU(n) for n=2n = 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.

Properties

General

Proposition

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

Proposition

There are isomorphisms of Lie groups

  1. of SU(2)SU(2) with the spin group in dimension 3 and with the quaternionic unitary group in one dimension

    SU(2)Spin(3)Sp(1) SU(2) \;\simeq\; Spin(3) \;\simeq\; Sp(1)
  2. of the direct product group of SU(2)SU(2) with itself, to Spin(4)

    SU(2)×SU(2)Spin(3)×Spin(3)Spin(4) SU(2) \times SU(2) \;\simeq\; Spin(3) \times Spin(3) \;\simeq\; Spin(4)

    with respect to which the canonical inclusion Spin(3)Spin(4)Spin(3) \hookrightarrow Spin(4) is given by the diagonal map.

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.

Finite subgroups

The finite subgroup of SU(2) have an ADE classification. See this theorem.

GG-Structure and exceptional geometry

Spin(8)-subgroups and reductions to exceptional geometry

reductionfrom spin groupto maximal subgroup
Spin(7)-structureSpin(8)Spin(7)
G2-structureSpin(7)G2
CY3-structureSpin(6)SU(3)
SU(2)-structureSpin(5)SU(2)
generalized reductionfrom Narain groupto direct product group
generalized Spin(7)-structureSpin(8,8)Spin(8,8)Spin(7)×Spin(7)Spin(7) \times Spin(7)
generalized G2-structureSpin(7,7)Spin(7,7)G 2×G 2G_2 \times G_2
generalized CY3Spin(6,6)Spin(6,6)SU(3)×SU(3)SU(3) \times SU(3)

see also: coset space structure on n-spheres

rotation groups in low dimensions:

sp. orth. groupspin grouppin groupsemi-spin group
SO(2)Spin(2)Pin(2)
SO(3)Spin(3)
SO(4)Spin(4)
SO(5)Spin(5)Pin(5)
Spin(6)
Spin(7)
SO(8)Spin(8)SO(8)
SO(9)Spin(9)
\vdots\vdots
SO(16)Spin(16)SemiSpin(16)
SO(32)Spin(32)SemiSpin(32)

see also

Last revised on March 30, 2019 at 10:07:29. See the history of this page for a list of all contributions to it.