nLab
su(2)

Contents

Context

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

Idea

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

Idea

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

  1. the Lie algebra on 3 generators {σ 1,σ 2,σ 3}\{\sigma_1, \sigma_2,\sigma_3\} subject to the following relations on their Lie bracket:

    [e i,e j]=kϵ ijke k [e_i, e_j] = \underset{k}{\sum} \epsilon_{i j k} e_k
  2. the Lie algebra spanned by (ii times) the three Pauli matrices with Lie bracket their commutator in their matrix algebra.

Properties

Pauli matrix presentation

Proposition

The Lie algebra 𝔰𝔲(2)\mathfrak{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 \,.

Complexification

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

References

Textbook accounts:

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

See also

Last revised on February 1, 2020 at 07:06:08. See the history of this page for a list of all contributions to it.