I seem to be the only one around who thinks this is hilarious, or even gets it. I am the biggest geek in the department of mathematics.
There is a special kind of function between rings, just like we have in groups. Given rings and , a function is called a homomorphism if it preserves all the ring structure.
The sort of odd thing here is that we’ve got two different kinds of rings to consider: those with and without identities. If we’re considering rings in general, we require that
but if we’re restricting ourselves to rings with identities, we also require that
where the on the left is the identity of , and the one on the right is the identity for . If we have two rings with identities but we consider them as general rings there will be more homomorphisms than if we consider them as rings with identity. It becomes important to pay a bit of attention to what kind of rings we’re really concerned with.
As an exercise, consider an arbitrary ring and see what ring homomorphisms exist from to . If has an identity, which of these homomorphisms preserve the identity?
Oh, and I probably should mention this: all the terminology from groups comes along for the ride. An injective (one-to-one) ring homomorphism is a monomorphism. A surjective (onto) ring homomorphism is an epimorphism. One that’s both is an isomorphism. A homomorphism from a ring to itself is an endomorphism, and an isomorphism from a ring to itself is an automorphism.
[EDIT: cleaned up LaTeX error and added comments at the end about terminology.]
I just got home from a long discussion with Dr. Zuckerman about this whole business. I’m not quite ready to say exactly what’s going on, but I want to correct a couple errors that I’ve made. Let it not be said that I don’t admit when I’m wrong.
Firstly, in my little added remarks about the Monster group in my “Why We Care” post, I was oversimplifying. First of all, the lattice is not the Leech lattice. The Leech lattice lives in 24-dimensional space for one thing (doh). Basically, you put together three copies of the lattice and then tweak it a bit.
Putting them together I can explain. The simplest lattice is just the integers sitting inside the real line. If you move to the plane, the points with integer coordinates sit at the corners of the squares in a checkerboard tiling of the plane. This is “adding two copies of the integer lattice”. For three copies of , we want 24-tuples of numbers so the first eight, second eight, and third eight are each the coordinates of a point in the lattice.
When you do this, it turns out there’s just enough room to squeeze in some more points to get a new lattice. That’s the Leech lattice. The Monster also isn’t quite just a group of symmetries of this lattice, so there’s still a few more steps to go, but it’s definitely related. So the connection isn’t quite as close as I’d implied, but it’s there.
The other thing is about real forms. I’d forgotten that not every choice of “realification” of the Killing form gives a Lie group, and further that not every choice that does work gives a unique Lie group.
What is true is that to every real form of a complex Lie group , there’s a largest compact subgroup . This means that its ends curve back in on themselves like the circle or the torus, and don’t run off to infinity like the line or the cylinder. Then we can “complexify” this group to get another complex group that’s really interesting to us. This group is a subgroup of , which will be important. In particular, if we take the compact real form of , its maximal compact subgroup is just itself, so its complexification is just back again.