SU(2)

Introduction

From here.
The simplest compact Lie group is the circle $S^1\cong \text{SO}(2)$. Part of the reason it is so simple to understand is that Euler’s formula gives an extremely nice parameterization $e^{ix}=\cos x+i\sin x$ of its elements, showing that it can be understood either in terms of the group of elements of norm 1 in $\mathbb{C}$ (that is, the unitary group $\text{U}(1)$) or the imaginary subspace of $\mathbb{C}$.

In the case of $SU(2)$ we have a total analogy, with $\mathbb{C}$ replaced by the quaternions $\mathbb{H}$. First, $S^3\cong \text{SU}(2)$. And $\text{SU}(2)$ is isomorphic to the group of elements of norm 1 in $\mathbb{H}$, and there is an exponential map from the imaginary subspace of $\mathbb{H}$ to this group.

Composing with the double cover $\text{SU}(2)\to \text{SO}(3)$ lets us handle elements of $\text{SO}(3)$ almost as easily as we handle elements of $\text{SO}(2)$.

Definition

See also this.

It consists of 2x2 complex matrices of the form

with $a^2+b^2+c^2+d^2=1$. They are unitary ($U{U}^{†}=I$).
These matrices can be decomposed as

where ${\sigma }_x,{\sigma }_y,{\sigma }_z$ are the Pauli matrices. Since $i{\sigma }_x,i{\sigma }_y,i{\sigma }_z$ satisfy the same relation that the basis quaternions $i,j,j$, the group $SU(2)$ is in correspondence with unitary quaternions. The whole set of quaternions is not $U(2)$, but the set of matrices satisfying $M{M}^{†}\in \mathbb{R}$, i.e., a multiple of identity, and $det(M)\ge 0$. From here.

Topologically, $SU(2)$ is the 3-sphere.

Related: relation SO(3) and SU(2).

The Lie algebra