# 圓群

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

## 同構

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

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

${\displaystyle e^{i\theta _{1}}e^{i\theta _{2}}=e^{i(\theta _{1}+\theta _{2})}}$

${\displaystyle \mathbb {T} \cong \mathbb {R} /2\pi \mathbb {Z} }$

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

## 表示

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

${\displaystyle \phi _{-n}={\overline {\phi _{n}}}.}$

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

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

## 代數結構

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

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