nLab SU(2)

Definition

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

Proposition

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

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

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

Properties

General

Proposition

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

Proposition

There are isomorphisms of Lie groups

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

   $$SU(2) \;\simeq\; Spin(3) \;\simeq\; Sp(1)$$

2. of the direct product group of $SU(2)$ with itself, to Spin(4)

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

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

See at spin group – Exceptional isomorphisms.

Proposition

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

$$\left( \array{ t & 0 \\ 0 & t^{-1} } \right) \;\; with t \in U(1) \hookrightarrow \mathbb{C} \,.$$

Lie algebra

Proposition

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

$$\left( \array{ i z & x + i y \\ - x + i y & - i z } \right) \;\;\; with \;\; x,y,z \in \mathbb{R} \,.$$

Definition

The standard basis elements of $\mathfrak{su}(2)$ given by the above presentation are

$$\sigma_1 \coloneqq \frac{1}{2} \left( \array{ 0 & 1 \\ -1 & 0 } \right)$$

$$\sigma_2 \coloneqq \frac{1}{2} \left( \array{ 0 & i \\ i & 0 } \right)$$

$$\sigma_3 \coloneqq \frac{1}{2} \left( \array{ i & 0 \\ 0 & -i } \right) \,.$$

These are called the Pauli matrices.

Proposition

The Pauli matrices satisfy the commutator relations

$$[\sigma_1, \sigma_2] = \sigma_3$$

$$[\sigma_2, \sigma_3] = \sigma_1$$

$$[\sigma_3, \sigma_1] = \sigma_2 \,.$$

Coadjoint orbits

Proposition

The coadjoint orbits of the coadjoint action of $SU(2)$ on $\mathfrak{su}(2)$ are equivalent to the subset of the above matrices with $x^2 + y^2 + z^2 = r^2$ for some $r \geq 0$.

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

Finite subgroups

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

$G$-Structure and exceptional geometry

Spin(8)-subgroups and reductions to exceptional geometry

reduction	from spin group	to maximal subgroup
Spin(7)-structure	Spin(8)	Spin(7)
G₂-structure	Spin(7)	G₂
CY3-structure	Spin(6)	SU(3)
SU(2)-structure	Spin(5)	SU(2)
generalized reduction	from Narain group	to direct product group
generalized Spin(7)-structure	$Spin(8,8)$	$Spin(7) \times Spin(7)$
generalized G₂-structure	$Spin(7,7)$	$G_2 \times G_2$
generalized CY3	$Spin(6,6)$	$SU(3) \times SU(3)$

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

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

Proof

Consider the canonical linear basis

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

by orthonormal quaternions

$$q_0, q_1, q_2, q_3 \;\in\; \mathbb{H}$$

$$\begin{aligned} & q_0 = 1\,, \\ & q_i q_i = -1 = - q_0 \;\; \text{for} i \in \{1,2,3\} \,, \\ & q_{\sigma(1)} q_{\sigma(3)} = \mathrm{sgn}(\sigma) q_{\sigma(3)} \,, \text{for} \sigma \in \mathrm{Sym}(3) \end{aligned}$$

On this, the action by Sp(1)

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

is by left quaternion multiplication.

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

$$\mathbf{4}\wedge \mathbf{4} \;\simeq\; \left\langle \left\{ \array{ a_1^+ := q_0 \wedge q_1 + q_2 \wedge q_3, \\ a_2^+ := q_0 \wedge q_2 + q_3 \wedge q_1, \\ a_3^+ := q_0 \wedge q_3 + q_1 \wedge q_2, } \array{ a_1^- := q_0 \wedge q_1 - q_2 \wedge q_3, \\ a_2^- := q_0 \wedge q_2 - q_3 \wedge q_1, \\ a_3^- := q_0 \wedge q_3 - q_1 \wedge q_2 } \right\} \right\rangle$$

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

$$q_i \cdot ( q_j \wedge q_k ) \;=\; (q_i q_j) \wedge q_k + q_j \wedge (q_i q_k) \,.$$

From this we find

$$\begin{aligned} q_1 \cdot a_1^{\pm} & = \; q_1 \cdot \big( q_0 \wedge q_1 \pm q_2 \wedge q_3 \big) \\ & =\; \big( \underset{ = 0 }{ \underbrace{ q_1 \wedge q_1 } } + \underset{ = 0 }{ \underbrace{ q_0 \wedge (-q_0) } } \big) \pm \big( \underset{ = 0 }{ \underbrace{ q_3 \wedge q_3 } } + \underset{ = 0 }{ \underbrace{ q_2 \wedge (- q_2) } } \big) \\ & =\; 0 \end{aligned}$$

and

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

and

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

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

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

But this means that

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

Lemma

We have

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

Proof

Consider the following linear basis

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

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

$$\begin{aligned} q_1 \cdot b_0 & \; = \; q_1 \cdot ( q_1 \wedge q_2 \wedge q_3 ) \\ & \; = \; \underset{ = b_1 }{ \underbrace{ (- q_0) \wedge q_2 \wedge q_3 } } \;+\; \underset{ = 0 }{ \underbrace{ q_1 \wedge q_3 \wedge q_3 } } \;+\; \underset{ = 0 }{ \underbrace{ q_1 \wedge q_2 \wedge (- q_2) } } \\ & \;=\; b_1 \end{aligned}$$

$$\begin{aligned} q_1 \cdot b_1 & \;=\; q_1 \cdot ( - q_0 \wedge q_2 \wedge q_3 ) \\ & \;=\; - \underset{ b_0 }{ \underbrace{ q_1 \wedge q_2 \wedge q_3 } } - \underset{ = 0 }{ \underbrace{ q_0 \wedge q_3 \wedge q_3 } } - \underset{ = 0 }{ \underbrace{ q_0 \wedge q_2 \wedge (-q_2) } } \\ & \;=\; - b_0 \end{aligned}$$

$$\begin{aligned} q_1 \cdot b_2 & \;=\; q_1 \cdot (q_0 \wedge q_1 \wedge q_3) \\ & \;=\; \underset{ = 0 }{ \underbrace{ q_1 \wedge q_1 \wedge (- q_2) } } + \underset{ = 0 }{ \underbrace{ q_0 \wedge (- q_0) \wedge q_3 } } + \underset{ = b_3 }{ \underbrace{ q_0 \wedge q_1 \wedge (- q_2) } } \\ & \;=\; b_3 \end{aligned}$$

$$\begin{aligned} q_1 \cdot b_3 & \;=\; q_1 \cdot (- q_0 \wedge q_1 \wedge q_2) \\ & \;=\; - \underset{ = 0 }{ \underbrace{ q_1 \wedge q_1 \wedge q_2 } } - \underset{ = 0 }{ \underbrace{ q_0 \wedge (- q_0) \wedge q_2 } } - \underset{ = b_2 }{ \underbrace{ q_0 \wedge q_1 \wedge q_3 } } \\ & \;=\; - b_2 \end{aligned}$$

Related concepts

• su(2)

• complex Hopf fibration

• Euler angles

• SU(3), SU(4), SU(5)

• Pauli quantum gate

• Solovay-Kitaev theorem

rotation groups in low dimensions:

Dynkin label	sp. orth. group	spin group	pin group	semi-spin group
	SO(2)	Spin(2)	Pin(2)
B1	SO(3)	Spin(3)	Pin(3)
D2	SO(4)	Spin(4)	Pin(4)
B2	SO(5)	Spin(5)	Pin(5)
D3	SO(6)	Spin(6)
B3	SO(7)	Spin(7)
D4	SO(8)	Spin(8)		SO(8)
B4	SO(9)	Spin(9)
D5	SO(10)	Spin(10)
B5	SO(11)	Spin(11)
D6	SO(12)	Spin(12)
	$\vdots$	$\vdots$
D8	SO(16)	Spin(16)		SemiSpin(16)
	$\vdots$	$\vdots$
D16	SO(32)	Spin(32)		SemiSpin(32)

see also

• Spin(5).Spin(3)

• finite rotation groups

• ADE-classification

References

• Howard Georgi, §3 in: Lie Algebras In Particle Physics, Westview Press (1999), CRC Press (2019) [doi:10.1201/9780429499210]

  with an eye towards application to (the standard model of) particle physics

On the coadjoint orbits of $SU(2)$:

• Alexandre Kirillov, p. 183 in: Lectures on the Orbit Method, Graduate Studies in Mathematics, 64, American Mathematical Society, (2004)

See also

• Qiaochu Yuan, The representation theory of SU(2)

• Wikipedia, Representation theory of SU(2)