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Abelian groups

For Abelian groups, the commutator is 0.

Circle group

The circle group consists of the rotations in a circle.

{$\mathrm{SO}(2)$} represents them as {$2 \times 2$} real matrices given in terms of sine and cosine of an angle theta.

\begin{pmatrix} \mathrm{cos\,\theta} & -\mathrm{sin\,\theta} \\ \mathrm{sin\,\theta} & \mathrm{cos\,\theta} \end{pmatrix}

Then the determinant is {$1$} regardless of angle. It gives the length of the radius of the unit circle.

Reflections are given by:

\begin{pmatrix} \mathrm{cos\,2\theta} & \mathrm{sin\,2\theta} \\ \mathrm{sin\,2\theta} & -\mathrm{cos\,2\theta} \end{pmatrix}

{$\mathrm{U}(1)$} represents rotations as {$1 \times 1$} matrices in a complex number {$e^{i\theta}$}. Then the determinant is {$1$} only when {$\theta = 0$}. Thus {$\mathrm{SU}(1)$} is the trivial group.

Complex conjugation is reflection about the {$x$} axis.

Readings

Wikipedia

• Rotations and reflections in two dimensions
• Special unitary group
• Rotation