Groups Unfolding

The Möbius Group

Consider the complex plane ${{\mathbb C}}$ extended with the point at infinity, ${\hat{\mathbb C}={\mathbb C}\cup\{\infty\}}$. The linear fractional transformations of ${\hat{\mathbb C}}$, that is

$\displaystyle m\colon\hat{\mathbb C}\rightarrow\hat{\mathbb C},\ z\mapsto m(z):=\frac{az+b}{cz+d},\ \text{with}\ a,b,c,d\in{\mathbb C}: ad-bc\neq 0, \ \ \ \ \ (1)$

form a group ${M}$ called the Möbius group. There are four basic types of such transformations:

1. Translations: ${m_1(z)=z+z_0}$ (${z_0\in{\mathbb C}}$);
2. Dilations: ${m_2(z)=cz}$ (${c>0}$);
3. Rotations: ${m_3(z)=e^{\text{i}\varphi}z}$ (${-\pi<\varphi\leq \pi}$);
4. Inversions1: ${m_4(z)=r^2/\bar z}$ (${r>0}$).

These can be combined to reproduce any transformations of the form (1). This fact shows that the Möbius group consist of conformal (= angle preserving) transformations of the complex plane. Möbius transformations can be visualized by placing a sphere on the plane, and using stereographic projection to identify a point ${(\xi,\eta,\zeta)}$ on the surface of the sphere with a point ${z=x+\mathrm{i}y}$ of the extended plane as follows:

$\displaystyle \xi=\frac{2x}{|z^2|+1},\quad \eta=\frac{2y}{|z^2|+1},\quad \zeta=\frac{|z^2|-1}{|z^2|+1},\quad\text{and}\quad z=x+\mathrm{i}y=\frac{\xi+\mathrm{i}\eta}{1-\zeta}. \ \ \ \ \ (2)$

Then any motion of the sphere induces a Möbius transformation of ${\hat{\mathbb C}}$. This identification is usually referred to as the Riemann sphere. Here is a nice video [1] demonstrating this:

Looking at (1) it’s clear that any ${m\in M}$ can be encoded in a ${2\times 2}$ invertible complex matrix, providing a natural link (homomorphism) between ${\text{GL}(2,{\mathbb C})}$ and ${M}$

$\displaystyle \text{GL}(2,{\mathbb C})\ni m:=\begin{pmatrix}a&b\\c&d\end{pmatrix} \mapsto\frac{az+b}{cz+d}=m(z)\in M. \ \ \ \ \ (3)$

Another trivial observation is that multiplying the components ${a}$, ${b}$, ${c}$, ${d}$ by an overall non-zero factor gives the same transformation. On `matrix language’ this means an equivalence relation on ${\text{GL}(2,{\mathbb C})}$ given by

$\displaystyle m\sim m',\ m,m'\quad\Leftrightarrow\quad \exists\lambda\in{\mathbb C}\setminus\{0\}: m'=\lambda m, \ \ \ \ \ (4)$

and one has to consider the quotient group ${\text{GL}(2,{\mathbb C})/\sim}$ which is called the projective linear group ${\text{PGL}(2,{\mathbb C})}$ and isomorphic to the Möbius group

$\displaystyle M\cong\text{PGL}(2,{\mathbb C}). \ \ \ \ \ (5)$

This shows that ${M}$ is connected, i.e. there always exists a path (of transformations) between any two elements. One might ask whether it’s simply connected, which is a stronger property, meaning the any loop can be (continuously) deformed to a point. To answer this question it’s useful to choose such a ${\lambda}$ in (4) that

$\displaystyle \det(\lambda m)=\lambda^2(ad-bc)=+1, \ \ \ \ \ (6)$

and by that restricting the homomorphism, given in (3), to ${\text{SL}(2,{\mathbb C})}$. Then it’s apparent that any path connecting ${I=\text{diag}(1,1)}$ and ${-I}$ in ${\text{SL}(2,{\mathbb C})}$ is a loop around the identity in ${M}$ and therefore clearly cannot be shrunk to a point. Thus the Möbius group is not simply connected.

Another interesting thing, easily seen from (6), is that each Möbius transformation is covered by two complex matrices of determinant one, viz. ${\lambda m}$ and ${-\lambda m}$, meaning that ${\text{SL}(2,{\mathbb C})}$ is a ${2}$-fold covering group of ${M}$. Moreover, ${\text{SL}(2,{\mathbb C})}$ is simply connected, thus it’s the universal covering group of ${M}$.

The Rotation Group

Now, let’s just consider those Möbius transformations which correspond to rotations of the Riemann sphere. The rotations about third coordinate axis is obviously coming from a rotation ${m_3(z)=e^{\text{i}\varphi}z}$ (${-\pi<\varphi\leq \pi}$); the corresponding matrices in ${\text{SL}(2,{\mathbb C})}$ is ${\pm\mathrm{diag}(e^{\mathrm{i}\varphi/2},e^{-\mathrm{i}\varphi/2})}$. The homomorphism

$\displaystyle \pm U_\zeta(\varphi):=\pm\begin{pmatrix} e^{\mathrm{i}\varphi/2}&0\\&e^{-\mathrm{i}\varphi/2} \end{pmatrix}\mapsto \begin{pmatrix} \cos\varphi&-\sin\varphi&0\\ \sin\varphi&\cos\varphi&0\\ 0&0&1 \end{pmatrix} \ \ \ \ \ (7)$

then implies, by the same argument as the one below (6), that ${\text{SO}(3)}$ is not simply connected. The rotations around the other two axes are given by the matrices

$\displaystyle \pm U_\xi(\varphi):=\pm\begin{pmatrix} \cos\varphi/2&\mathrm{i}\sin\varphi/2\\ \mathrm{i}\sin\varphi/2&\cos\varphi/2 \end{pmatrix} \quad\text{and}\quad \pm U_\eta(\varphi):=\pm\begin{pmatrix} \cos\varphi/2&-\sin\varphi/2\\ \sin\varphi/2&\cos\varphi/2. \end{pmatrix} \ \ \ \ \ (8)$

The three (actually six) ${2\times 2}$ matrices given in (7) and (8) are unitary with determinant one. This means that we obtained a homomorphism between the groups ${\text{SU}(2)}$ and ${\text{SO}(3)}$. Again, it’s a ${2}$-fold cover, and since ${\text{SU}(2)}$ is simply connected, it’s the universal covering of ${\text{SO}(3)}$. As a side note, we mention that the ${2\times 2}$ unitary matrices in (7) and (8) can be written as

$\displaystyle U_\xi(\alpha)=e^{\mathrm{i}(\alpha/2)\sigma_1},\quad U_\eta(\beta)=e^{-\mathrm{i}(\beta/2)\sigma_2},\quad U_\zeta(\gamma)=e^{\mathrm{i}(\gamma/2)\sigma_3}, \ \ \ \ \ (9)$

where ${\sigma_1}$, ${\sigma_2}$, ${\sigma_3}$ are the famous Pauli matrices

$\displaystyle \sigma_1=\begin{pmatrix}0&1\\1&0\end{pmatrix},\quad \sigma_2=\begin{pmatrix}0&-\mathrm{i}\\\mathrm{i}&0\end{pmatrix},\quad \sigma_3=\begin{pmatrix}1&0\\&-1\end{pmatrix}. \ \ \ \ \ (10)$

A useful application that a rotation about an axis, given by the unit vector ${\vec n\in{\mathbb R}^3}$, through an angle ${\varphi}$ can be obtained from the unitary matrix

$\displaystyle U_{\hat n}(\varphi)=e^{-\mathrm{i}(\varphi/2)\vec n\cdot\vec\sigma}, \ \ \ \ \ (11)$

where ${\vec n\cdot\vec\sigma=n_1+\sigma_1+n_2\sigma_2+n_3\sigma_3}$.

Footnote 1. Or rather combined with a reflection to the real axis. ^

References

1. D.N. Arnold and J. Rogness, Möbius Transformations Revealed, video, June 2007.
2. D.H. Sattinger and O.L. Weaver, Lie Groups and Algebras with Applications to Physics, Geometry, and Mechanics. Applied Mathematical Sciences 61, Springer, 1986.