Groups Lecture 20

in which we meet the unitary group and start exploring Möbius maps as transformations of the Riemann sphere.

Having shown last time that every matrix in \mathrm{SO}_3 is a rotation around some axis, we finished off the proof today that any matrix in \mathrm{O}_3 is the product of at most three reflections. We really do need three for -I, which shows that we can’t quite think of \mathrm{O}_3 as “rotations and reflections”, at least not if we mean just reflections in planes.

The complex equivalent of orthogonal matrices are unitary matrices, satisfying A^\dagger A=I. They form the unitary group. The determinant of a unitary matrix is a complex number on the unit circle, and the kernel of the determinant homomorphism is the special unitary group. The equivalent to orthogonal matrices being isometries is that unitary matrices preserve complex dot product.

As our last chapter, we started looking at Möbius maps. A Möbius map is f\colon {\mathbb{C}_\infty \to \mathbb{C}_\infty} of the form f(z)=\frac{az+b}{cz+d} for a,b,c,d\in \mathbb{C} and ad-bc \neq 0. This condition is needed to make the map non-constant, and indeed injective on \mathbb{C}. To include f(-\frac{d}{c}), we added a point at infinity. So we think of \mathbb{C}_\infty as the Riemann Sphere via the Stereographic Projection: take the unit sphere, cut it in the equator with the complex plane. Then attach a rod at the North Pole.  If we move around the rod, keeping it fixed at the North Pole, we see it can meet any point z in the complex plane. Then it intersects one more point on the sphere, and this is where we map the number z. Then the North Pole corresponds to \infty. Doing it this way, we have the unit circle exactly at the equator of the sphere, and everything inside the unit disc is in the southern hemisphere, while everything outside the unit circle is in the northern hemisphere.

Equipped with this extra understanding, we saw that Möbius maps are bijections from the Riemann Sphere to itself, and form a group, the Möbius group M. We will finish the proof that it really is a group next time.

Understanding today’s lecture

For the Möbius maps, you might take a little time to get used to the point at infinity. We will see next time that, although we often have to write out special checks for that point, it is not really any different. If you are afraid that you will have to do a lot of fraction manipulation which will be tedious, don’t worry: we will get a short cut next time via matrices.

This area of the course is very well represented and explained in Beardon Algebra and Geometry, so do consider using that book as another source.

Preparing for Lecture 21

After finishing the proof that the Möbius maps really form a group, we will look at them using matrices. You might be able to guess how that will work already. Using a surjective homomorphism from \mathrm{GL}_2(\mathbb{C}), we will also find that the Möbius group is isomorphic to the projective general linear group (and also to the projective special linear group, which happens to be the same for \mathbb{C}). We will show that any Möbius map is the composition of dilation/rotations, translations and inversions. After that we will look at fixed points of Möbius maps, and their conjugacy classes.

Going a little deeper

Regarding our results of rotations and reflections in two and three dimensions: Beardon proves some more general results; he includes any isometry in the three-dimensional case (not just linear ones which keep the origin fixed), which means you actually need up to four reflections. He also touches on the general case of n dimensions. Do read up about it if you’re interested. It is in Chapter 11, mostly 11.2 Orthogonal Maps and a bit in 11.3 Isometries of Euclidian n-space.

I mentioned as an extra that the Riemann Sphere is also called the one-point compactification of \mathbb{C}. “One-point” is clear, as we are adding only one point, \infty. What does compact mean? You will learn it in Metric and Topological spaces. For subsets of \mathbb{R}^n, it means that the set is closed and bounded. Bounded means that the distances between points in the set can’t go to infinity. A bounded set is closed if “it has a sharp edge”. For example, the interval [0,1]\subset \mathbb{R} is closed, but (0,1) is not. The unit sphere, seen as a subset of \mathbb{R}^3, is compact = closed and bounded. The proper definition of compact is that every open cover has a finite subcover, so again it is something to do with finiteness.

We showed that Möbius maps are bijection from the Riemann sphere to itself. In fact, they are exactly the bijective conformal maps, which means they preserve angles and lengths. You could also say they are exactly the automorphisms of the Riemann sphere viewed as a complex manifold. Though that needs some more topology and complex analysis to understand properly.

 

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