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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 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} \,.

Lie algebra

Proposition

The Lie algebra 𝔰𝔲(2)\mathfrak{su}(2) (see also at su(2)) as a complex 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 \,.

Coadjoint orbits

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

Representation theory

To record some aspects of the linear representation theory of SU(2).

(…)

Lemma

We have

243113RO(Sp(1)) \wedge^2 \mathbf{4} \;\simeq\; 3 \cdot \mathbf{1} \;\oplus\; 1 \cdot \mathbf{3} \;\;\;\in\; \mathrm{RO}(\mathrm{Sp}(1))
Proof

Consider the canonical linear basis

(1)4{q 0,q 1,q 2,q 3} \mathbf{4} \;\simeq\; \big\langle \{q_0, q_1, q_2, q_3\}\big\rangle_{\mathbb{R}}

by orthonormal quaternions

q 0,q 1,q 2,q 3 q_0, q_1, q_2, q_3 \;\in\; \mathbb{H}
q 0=1, q iq i=1=q 0fori{1,2,3}, q σ(1)q σ(3)=sgn(σ)q σ(3),forσSym(3) \begin{aligned} & q_0 = 1\,, \\ & q_i q_i = -1 = - q_0 \;\; \text{for} i \in \{1,2,3\} \,, \\ & q_{\sigma(1)} q_{\sigma(3)} = \mathrm{sgn}(\sigma) q_{\sigma(3)} \,, \text{for} \sigma \in \mathrm{Sym}(3) \end{aligned}

On this, the action by Sp(1)

Sp(1){q|qq¯=1} \mathrm{Sp}(1) \;\simeq\; \{ q \in \mathbb{H} \;\vert\; q \bar q = 1 \}

is by left quaternion multiplication.

Consider then the following linear basis of 44\mathbf{4}\wedge \mathbf{4}:

44{a 1 +:=q 0q 1+q 2q 3, a 2 +:=q 0q 2+q 3q 1, a 3 +:=q 0q 3+q 1q 2,a 1 :=q 0q 1q 2q 3, a 2 :=q 0q 2q 3q 1, a 3 :=q 0q 3q 1q 2} \mathbf{4}\wedge \mathbf{4} \;\simeq\; \left\langle \left\{ \array{ a_1^+ := q_0 \wedge q_1 + q_2 \wedge q_3, \\ a_2^+ := q_0 \wedge q_2 + q_3 \wedge q_1, \\ a_3^+ := q_0 \wedge q_3 + q_1 \wedge q_2, } \array{ a_1^- := q_0 \wedge q_1 - q_2 \wedge q_3, \\ a_2^- := q_0 \wedge q_2 - q_3 \wedge q_1, \\ a_3^- := q_0 \wedge q_3 - q_1 \wedge q_2 } \right\} \right\rangle

with induced 𝔰𝔭(1)\mathfrak{sp}(1) Lie algebra action given by

q i(q jq k)=(q iq j)q k+q j(q iq k). q_i \cdot ( q_j \wedge q_k ) \;=\; (q_i q_j) \wedge q_k + q_j \wedge (q_i q_k) \,.

From this we find

q 1a 1 ± =q 1(q 0q 1±q 2q 3) =(q 1q 1=0+q 0(q 0)=0)±(q 3q 3=0+q 2(q 2)=0) =0 \begin{aligned} q_1 \cdot a_1^{\pm} & = \; q_1 \cdot \big( q_0 \wedge q_1 \pm q_2 \wedge q_3 \big) \\ & =\; \big( \underset{ = 0 }{ \underbrace{ q_1 \wedge q_1 } } + \underset{ = 0 }{ \underbrace{ q_0 \wedge (-q_0) } } \big) \pm \big( \underset{ = 0 }{ \underbrace{ q_3 \wedge q_3 } } + \underset{ = 0 }{ \underbrace{ q_2 \wedge (- q_2) } } \big) \\ & =\; 0 \end{aligned}

and

q 1a 2 ± =q 1(q 0q 2±q 3q 1) =(q 1q 2=q 1q 2+q 0q 3=q 0q 3)±((q 2)q 1=q 1q 2+q 3(q 0)=q 0q 3) ={2a 3 + 0 \begin{aligned} q_1 \cdot a_2^{\pm} & =\; q_1 \cdot \big( q_0 \wedge q_2 \pm q_3 \wedge q_1 \big) \\ & = \; \big( \underset{ = q_1 \wedge q_2 }{ \underbrace{ q_1 \wedge q_2 } } + \underset{ = q_0 \wedge q_3 }{ \underbrace{ q_0 \wedge q_3 } } \big) \pm \big( \underset{ = q_1 \wedge q_2 }{ \underbrace{ (- q_2) \wedge q_1 } } + \underset{ = q_0 \wedge q_3 }{ \underbrace{ q_3 \wedge (- q_0) } } \big) \\ & =\; \left\{ \begin{array}{l} 2 a_3^+ \\ 0 \end{array} \right. \end{aligned}

and

q 1a 3 ± =q 1(q 0q 3±q 1q 2) =(q 1q 3=q 3q 1+q 0(q 2)=q 0q 2)±((q 0)q 2=q 0q 2+q 1q 3=q 3q 1) ={2a 2 + 0. \begin{aligned} q_1 \cdot a_3^{\pm} & =\; q_1 \cdot \big( q_0 \wedge q_3 \pm q_1 \wedge q_2 \big) \\ & = \; \big( \underset{ = - q_3 \wedge q_1 }{ \underbrace{ q_1 \wedge q_3 } } + \underset{ = - q_0 \wedge q_2 }{ \underbrace{ q_0 \wedge (-q_2) } } \big) \pm \big( \underset{ = - q_0 \wedge q_2 }{ \underbrace{ (- q_0) \wedge q_2 } } + \underset{ = - q_3 \wedge q_1 }{ \underbrace{ q_1 \wedge q_3 } } \big) \\ & =\; \left\{ \begin{array}{l} -2 a_2^+ \\ 0 \end{array} \right. \end{aligned} \,.

Since everything here is invariant under cyclic permutation of the three non-zero indices it follows generally that

(12q i)a j +=kϵ ijka k +,(12q i)a j =0for alli,j{1,2,3} (\tfrac{1}{2} q_i) \cdot a_j^+ \;=\; \underset{k}{\sum} \epsilon_{i j k} a_k^+ \,, \;\; (\tfrac{1}{2} q_i) \cdot a_j^- \;=\; 0 \;\; \text{for all} i,j \in \{1,2,3\}

But this means that

{a 1 +,a 2 +,a 3 +}3,aa{a i }1RO(Sp(1)). \big\langle \{a^+_1, a^+_2, a^+_3\} \big\rangle \;\simeq\; \mathbf{3} \,, \phantom{aa} \big\langle \{a^-_i\} \big\rangle \;\simeq\; \mathbf{1} \;\;\;\in \mathrm{RO}(\mathrm{Sp}(1)) \,.


Lemma

We have

344RO(Sp(1)). \wedge^3 \mathbf{4} \;\simeq\; \mathbf{4} \;\; \in RO(Sp(1)) \,.
Proof

Consider the following linear basis

34 {b 0:=+q 1q 2q 3 b 1:=q 0q 2q 3 b 2:=+q 0q 1q 3 b 3:=q 0q 1q 2}. \wedge^3 \mathbf{4} \;\simeq_{\mathbb{R}}\; \left\langle \left\{ \begin{array}{l} b_0 := + q_1 \wedge q_2 \wedge q_3 \\ b_1 := - q_0 \wedge q_2 \wedge q_3 \\ b_2 := + q_0 \wedge q_1 \wedge q_3 \\ b_3 := - q_0 \wedge q_1 \wedge q_2 \end{array} \right\} \right\rangle \,.

We claim that in terms of these basis elements and those of (1) the isomorphism is given by b 0q 0b_0 \mapsto q_0, b iq ib_i \mapsto q_i. This follows by direct inspection. For instance for the Lie algebra action of q 1q_1 we find:

q 1b 0 =q 1(q 1q 2q 3) =(q 0)q 2q 3=b 1+q 1q 3q 3=0+q 1q 2(q 2)=0 =b 1 \begin{aligned} q_1 \cdot b_0 & \; = \; q_1 \cdot ( q_1 \wedge q_2 \wedge q_3 ) \\ & \; = \; \underset{ = b_1 }{ \underbrace{ (- q_0) \wedge q_2 \wedge q_3 } } \;+\; \underset{ = 0 }{ \underbrace{ q_1 \wedge q_3 \wedge q_3 } } \;+\; \underset{ = 0 }{ \underbrace{ q_1 \wedge q_2 \wedge (- q_2) } } \\ & \;=\; b_1 \end{aligned}
q 1b 1 =q 1(q 0q 2q 3) =q 1q 2q 3b 0q 0q 3q 3=0q 0q 2(q 2)=0 =b 0 \begin{aligned} q_1 \cdot b_1 & \;=\; q_1 \cdot ( - q_0 \wedge q_2 \wedge q_3 ) \\ & \;=\; - \underset{ b_0 }{ \underbrace{ q_1 \wedge q_2 \wedge q_3 } } - \underset{ = 0 }{ \underbrace{ q_0 \wedge q_3 \wedge q_3 } } - \underset{ = 0 }{ \underbrace{ q_0 \wedge q_2 \wedge (-q_2) } } \\ & \;=\; - b_0 \end{aligned}
q 1b 2 =q 1(q 0q 1q 3) =q 1q 1(q 2)=0+q 0(q 0)q 3=0+q 0q 1(q 2)=b 3 =b 3 \begin{aligned} q_1 \cdot b_2 & \;=\; q_1 \cdot (q_0 \wedge q_1 \wedge q_3) \\ & \;=\; \underset{ = 0 }{ \underbrace{ q_1 \wedge q_1 \wedge (- q_2) } } + \underset{ = 0 }{ \underbrace{ q_0 \wedge (- q_0) \wedge q_3 } } + \underset{ = b_3 }{ \underbrace{ q_0 \wedge q_1 \wedge (- q_2) } } \\ & \;=\; b_3 \end{aligned}
q 1b 3 =q 1(q 0q 1q 2) =q 1q 1q 2=0q 0(q 0)q 2=0q 0q 1q 3=b 2 =b 2 \begin{aligned} q_1 \cdot b_3 & \;=\; q_1 \cdot (- q_0 \wedge q_1 \wedge q_2) \\ & \;=\; - \underset{ = 0 }{ \underbrace{ q_1 \wedge q_1 \wedge q_2 } } - \underset{ = 0 }{ \underbrace{ q_0 \wedge (- q_0) \wedge q_2 } } - \underset{ = b_2 }{ \underbrace{ q_0 \wedge q_1 \wedge q_3 } } \\ & \;=\; - b_2 \end{aligned}

rotation groups in low dimensions:

Dynkin labelsp. orth. groupspin grouppin groupsemi-spin group
SO(2)Spin(2)Pin(2)
B1SO(3)Spin(3)Pin(3)
D2SO(4)Spin(4)Pin(4)
B2SO(5)Spin(5)Pin(5)
D3SO(6)Spin(6)
B3SO(7)Spin(7)
D4SO(8)Spin(8)SO(8)
B4SO(9)Spin(9)
D5SO(10)Spin(10)
B5SO(11)Spin(11)
D6SO(12)Spin(12)
\vdots\vdots
D8SO(16)Spin(16)SemiSpin(16)
\vdots\vdots
D16SO(32)Spin(32)SemiSpin(32)

see also

References

See also

Last revised on August 16, 2020 at 00:38:14. See the history of this page for a list of all contributions to it.