The Geometry of $\Ug(1)$

We saw in §2.4 that complex multiplication can be interpreted geometrically as a rescaling and a rotation. A pure rotation is therefore obtained by multiplying by a unit complex number. In other words, if $|w|=1$, then $|wz|=|z|$, that is, the length of $z$ is preserved under multiplication by $w$. What do unit-normed elements $w\in\CC$ look like? Since $|r e^{i\theta}|=r$, we have \begin{equation} w = e^{i\theta} \end{equation} for some $\theta$. But these are precisely the complex numbers with norm $1$! Thus, \begin{equation} \Ug(1) = \{ w\in\CC : \bar{w}w = 1 \} = \{ e^{i\theta}: \theta\in[0,2\pi) \} \\ \end{equation} which can also be written as \begin{equation} \Ug(1) = \{ w\in\CC : |w|=1 \} \end{equation} Equivalently, we can describe $\Ug(1)$ as the group of transformations \begin{equation} \Ug(1) = \{ z \longmapsto wz : w,z\in\CC, |w|=1 \} \end{equation} As already noted, a special case is when $\theta=\frac\pi2$, in which case $w=i$.

But we already have a name for the rotations in the plane, namely $\SO(2)$, which we studied in §2.4. Thus, $\SO(2)$ and $\Ug(1)$ are the same group, which we write as \begin{equation} \Ug(1) \cong \SO(2) \end{equation} where the symbol “$\cong$” is read as “is isomorphic to”. This isomorphism is just the first of several we will encounter relating different descriptions of the same group.

What about reflections?

A reflection about the $y$-axis is easy; that's just complex conjugation, namely the map \begin{equation} z \longmapsto \bar{z} \end{equation} A reflection about any other line through the origin can be obtained by a combination of rotations and conjugation. For instance, reflection about the line $y=x$ is given by \begin{equation} z \longmapsto e^{i\theta/4}(\bar{e^{-i\theta/4}z}) = i\bar{z} \end{equation} and reflection about the $x$-axis is given by \begin{equation} z \longmapsto i(\bar{-iz}) = -\bar{z} \end{equation}

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