Today I Learned: On De-Klein-ing Lie's - Part 1

Update: Part 0 Part 2. Also, a crazy thought comes to mind - has anyone ever done linear modelling (of the statistical variety) with projective geometry and other geometries? It might be worthwhile doing this in situations with physical interpretations that admit conformal inversion-type maps on variables.

Okay okay okay, ugh...it's too early in the morning for anything other than maths: let's have another look at Lie's line-sphere correspondence.

Today's menus: projectivisation (again, I probably should say projective completion, but whatevs man!).

To begin, draw a circle. Now draw a bigger one. Now draw a much much much bigger circle. You'll notice that arcs on a circles become straighter as you increase the radius of the circle. So naturally, if you draw a really...perhaps...infinitely big circle, you should probably wind up with just a straight line. This idea that lines are pretty much circles (and planes are pretty much spheres and so forth) is formalised in projective geometry.

The basic idea is to extend spaces like $\mathbb{R}^n$ and $\mathbb{C}^n$ to these projective spaces called $\mathrm{RP}^n$ and $\mathbb{CP}^n$ which are a little bit bigger - but they're only larger in the sense that they have points "at infinity".

As an illustrative example, let's consider $\mathbb{R}^1=\mathbb{R}$ - the real numbers.

Let's regard a real number, like $2\in\mathbb{R}$, as a ratio $a:b$, where $b=2a$. This may seem rather silly, but when you think about it, there's something entirely natural and primitive about this - I mean, it's a tiny bit like how we build rational numbers, which is the foundation upon which we construct the real numbers.

In this way, any real number $r\in\mathbb{R}$ may be written as $1:r$, which we naturally expect to be equal to $a:ar$. So we see the emergence of the following space: $$\begin{align*}\mathbb{RP}^1:=\{\text{pairs of the form }[a:b]\text{ where }[a:b]=[ax:bx]\text{ if }x\neq0\\\text{and }a,b\in\mathbb{R}\text{ aren't both }0\}.\end{align*}$$ So we've got these funny square-bracketed pairs of numbers $[a:b]$, so notated as to resemble a ratio, and an embedding map $$\begin{align*}\iota:\mathbb{R}\rightarrow\mathbb{RP}^1,\, r\mapsto[1:r]\end{align*}$$ that covers almost all of $\mathbb{RP}^1$. In fact, the only point that isn't covered is the one point where the first coordinate is $0$ - the point $[0:1]$. Now, it shouldn't be too hard to believe that this is $\infty$ because if $[a:b]=\frac{b}{a}$, then we should expect that $[0:1]=\frac{1}{0}=\infty$.

Let's generalise to a general projective $n$-space: $$\begin{align*}\mathbb{RP}^n:=\{[x_0:x_1:\ldots:x_n]\,|\,[x_0:x_1:\ldots:x_n]=[rx_0:rx_1:\ldots:rx_n]\text{ if }r\neq0,\\\text{ and the }x_i\in\mathbb{R}\text{ aren't all }0,\}\end{align*}$$ naturally, the complex projective $n$-space $\mathbb{CP}^n$ is defined identically - just with $\mathbb{R}$ replaced with $\mathbb{C}$.

In these higher dimensional projective spaces, the points at infinity are given by points of the form $[0:x_1:x_2:\ldots:x_n]$. Which, if you think about it, is a projective space of $1$ lower dimension. That is (as sets and also, I guess, as a CW-complex): $$\begin{align*}\mathbb{RP}^n=\mathbb{RP}^{n-1}\cup\mathbb{RP}^{n-2}\cup\ldots\cup\mathbb{RP}^{0},\end{align*}$$
and similarly with $\mathbb{CP}^n$ of course. I should probably point out at this point that this decomposition is nice enough for calculating the homology of these spaces. And now, time for something completely different.

Notice that each point $[x_0:\ldots:x_n] \in\mathbb{RP}^n$ is secretly a whole line of points in $\mathbb{R}^{n+1}$ given by $$\begin{align*}\{t(x_0,\ldots,x_1)\in\mathbb{R}^{n+1}\}. \end{align*}$$ Therefore, we can think of $\mathbb{RP}^n$ as the space of lines going through the origin in $\mathbb{R}^{n+1}$. In fact, since each line is orthogonal to precisely one $n$-dimensional subspace (or hyperplane), we could actually think of $\mathbb{RP}^n$ as the space of hyperplanes in $\mathbb{R}^{n+1}$ if we so wished. And this duality between lines and hyperplanes is a powerful technique in projective geometry, and I won't say too much more about it because I really ought to have breakfast.

But one important thing that I ought to say is that since invertible $(n+1)\times(n+1)$-linear transformations take lines to lines in $\mathbb{R}^{n+1}$, this gives us a natural action of the matrices $\mathrm{GL}_{n+1}(\mathbb{R})$ on $\mathbb{RP}^n$. Maybe I'll write a bit more about that, after I've had something to eat.

Or maybe not? Who knows.