# 圓群

$\mathbb T = \{ z \in \mathbb C : |z| = 1 \}.$

## 同構

$\mathbb T \cong \mbox{U}(1) \cong \mbox{SO}(2) \cong \mathbb R/\mathbb Z.\,$

$\theta \mapsto e^{i\theta} = \cos\theta + i\sin\theta$

$e^{i\theta_1}e^{i\theta_2} = e^{i(\theta_1+\theta_2)}$

$\mathbb T \cong \mathbb R/2\pi\mathbb Z$

$e^{i\theta} \leftrightarrow \exp\left(\theta\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}\right) = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \\ \end{bmatrix} = \cos{\theta}\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} +\sin{\theta}\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$

## 表示

$\phi_n(e^{i\theta}) = e^{in\theta},\qquad n\in\mathbb Z.$

$\phi_{-n} = \overline{\phi_n}.$

$\mathrm{Hom}(\mathbb T,\mathbb T) \cong \mathbb Z.$

$\rho_n(e^{i\theta}) = \begin{bmatrix} \cos n\theta & -\sin n\theta \\ \sin n\theta & \cos n\theta \\ \end{bmatrix},\quad n\in\mathbb Z^{+}.$

## 代數結構

$\mathbb T \cong \mathbb R \oplus (\mathbb Q / \mathbb Z).\,$

$\mathbb C^\times \cong \mathbb R \oplus (\mathbb Q / \mathbb Z)$