April 12, 2021

Continued Fractions, Attempt 3, Part 1: Degenerate Projective Transformation

1. Complex Projective Line

Let $V$ be a 2-dimensional complex coordinate space. We consider $V$ consists of column vectors, that is, \[ V=\left\{\V{x}{y}\bigg|\,x,y\in\C\right\}. \] Define $W$ by \[ W=V\setminus\left\{\V{0}{0}\right\} \] and a binary relation $\sim$ over $W$ by \[ \V{x}{y}\sim\V{x'}{y'}\Longleftrightarrow \exists\lambda\in\C^\times:\V{x}{y}=\lambda\V{x'}{y'}, \] where $\C^\times=\C\setminus\{0\}$. $\sim$ is an equivalence relation. The quotient $W/\!\sim$ is called the complex projective line, and denoted by $\CP^1$. We denote the class of $\CP^1$ to which $\V{x}{y}\in W$ belongs by $\left[\V{x}{y}\right]$, or simply $\B{x}{y}$.

Define $U$ by \[ U=\CP^1\setminus\left\{\B{1}{0}\right\} \] and $\phy:U\to\C$ by \[ \phy:\B{x}{y}\mapsto\frac{x}{y}. \] $\phy$ is a (well-defined)bijection between $U$ and $\C$. If one want to map $\B{1}{0}$ also by $\phy$, the image should be $1/0=\infty$. Therefore, we consider $\CP^1$ and the extended complex plane $\whC=\C\cup\left\{\infty\right\}$ are same in the view of $\phy$.

2. Projective Transformation

For any $M=\M{a}{b}{c}{d}\in\GL(2,\C)$, we associate a map $\pi_M:\CP^1\to\CP^1$ (well-)defined by \[ \pi_M:\B{x}{y}\mapsto\left[\M{a}{b}{c}{d}\V{x}{y}\right]. \] This is called a projective transformation. The set of all the projective transformations of $\CP^1$ is denoted by $\PGL(1,\C)$, which forms a group.

Let $\pi$ be a map $\GL(2,\C)\to\PGL(1,\C)$ defined by \[ \pi:M\mapsto\pi_M. \] $\pi$ is a group homomorphism. We have \[ \ker\pi=\left\{\lambda\M{1}{0}{0}{1}\bigg|\,\lambda\in\C^\times\right\}, \] and therefore \[ \PGL(1,\C)\cong\GL(2,\C)/\C^\times. \]

Remark. Confusingly, $\PGL(1,\C)$ is often denoted as $\PGL(2,\C)$. This is because of the above fact [U].

In the view of $\phy$, $\pi_M$ is represented in $\whC$ as \[ z\mapsto\frac{az+b}{cz+d}, \] where $z=x/y$. This is called a linear fractional transformation, or a Möbius transformation.

3. Degenerate Projective Transformation

Let \[ N=\M{a}{b}{c}{d}\in\Mat(2,\C)\setminus\GL(2,\C). \] Also for $N$, we want to associate a map $\pi_N:\CP^1\to\CP^1$, which would be called a degenerate projective transformation.

Assume that $c\ne0$. For $\V{x}{y}\in W$, we have \[ ax+by=\frac{acx+bcy}{c}=\frac{a(cx+dy)}{c} \] and therefore \[ \M{a}{b}{c}{d}\V{x}{y}=\V{0}{0} \Longleftrightarrow cx+dy=0 \Longleftrightarrow \B{x}{y}=\B{d}{-c}. \] For any $\B{x}{y}\in\CP^1\setminus\left\{\B{d}{-c}\right\}$, $\pi_N$ maps $\B{x}{y}$ to $\B{a}{c}$. Therefore, we dare to define $\pi_N$ as a constant function $\B{a}{c}$ entirely on $\CP^1$.

Remark. If $\pi_N$ had mapped $\B{d}{-c}$ to an element in $\CP^1$ other than $\B{a}{c}$, then we should have accepted the discontinuity at that point.

With similar arguments, we conclude that $\pi_N$ is a constant function $\B{a}{c}$ when $\V{a}{c}\in W$, and $\B{b}{d}$ when $\V{b}{d}\in W$. Of course, these two values coincide when $\V{a}{c}\in W$ and $\V{b}{d}\in W$ because $ad-bc=0$.

Remark. There seems no reasonable definition of $\pi_N$ when $N=\M{0}{0}{0}{0}$.


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