Contents

# Contents

## 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:

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

$\sigma_1 \coloneqq \frac{1}{\sqrt{2}} \left( \array{ 0 & 1 \\ -1 & 0 } \right)$
$\sigma_2 \coloneqq \frac{1}{\sqrt{2}} \left( \array{ 0 & i \\ i & 0 } \right)$
$\sigma_3 \coloneqq \frac{1}{\sqrt{2}} \left( \array{ i & 0 \\ 0 & -i } \right) \,.$

These are called the Pauli matrices.

###### Proposition

The Pauli matrices satisfy the commutator relations

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

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)) (…)

## 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)