nLab su(2)

Idea

The Lie algebra $\mathfrak{su}(2)$ is the special case of special unitary Lie algebras $\mathfrak{su}(n)$ for $n = 2$, underlying the Lie group SU(2) (the special unitary group $SU(n)$ for $n =2$).

Idea

The Lie algebra $\mathfrak{su}(2)$ is equivalently given as follows:

1. the Lie algebra on 3 generators $\{\sigma_1, \sigma_2,\sigma_3\}$ subject to the following relations on their Lie bracket:

   $$[e_i, e_j] \;=\; \textstyle{\underset{k}{\sum}} \epsilon_{i j k} e_k$$

2. the Lie algebra spanned by ($i$ times) the three Pauli matrices with Lie bracket their commutator in their matrix algebra.

Properties

Pauli matrix presentation

Proposition

The Lie algebra $\mathfrak{su}(2)$ 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

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

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

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

Another common basis in use is the Cartan-Weyl basis

$$\begin{aligned} \sigma^\pm & \coloneqq \sigma_2 \pm i \sigma_3 \\ \sigma^0 & \coloneqq \sigma_1 \end{aligned}$$

Complexification

The complexification of $\mathfrak{su}(2)$ is the special linear Lie algebra $\mathfrak{sl}(2, \mathbb{C})$ (see at sl(2)) (…)

Related concepts

• fuzzy 2-sphere

  fuzzy funnel, Dp-D(p+2)-brane bound state

• SU(2)

• Pauli quantum gate

References

Textbook accounts:

• Walter Pfeifer, The Lie algebra $\mathfrak{su}(2)$, In: The Lie Algebras $\mathfrak{su}(N)$, Birkhäuser, Basel (2003) (doi:10.1007/978-3-0348-8097-8_3, pdf)

• 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

See also

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

• Wikipedia, Representation theory of SU(2)