Real and complex groups
Okay, another thing to make clear is that there’s not just one group we could mean by . There’s one complex group, and a bunch of “real forms” of the group.
The difference between a real group and a complex group is pretty simply stated: implicitly what I’ve been talking about are real groups. Complex Lie groups are group structures on complex manifolds. That is, they “locally look like” complex -dimensional space. You may remember that the complex numbers look like a plane with the real numbers sitting inside on a line. A complex -manifold looks like a real -manifold, but there’s some extra structure floating around I’ll try to ignore. Basically it deals with how we can “scale” shapes in the manifold by imaginary amounts — how to “multiply by ” — but that’s really horribly oversimplifying.
If we’ve got the complex plane, how do we find the real numbers? You might think we can just read off which points have zero imaginary part, but this actually sort of begs the question: it assumes you already know what the real line in the complex plane is.
What we can do is think of the complex plane as a -dimensional complex manifold. Now there’s a “reflection” of the plane to itself that plays nice with the complex structure: complex conjugation, . The points that are their own conjugates make up the real line. But there’s another reflection that plays nice: . The fixed points here are the circle of radius one!
Now we can see the nonzero complex numbers as a group with multiplication as its operation. Similarly we can see the nonzero real numbers with multiplication and the circle with addition of angles as groups. These are all one-dimensional Lie groups. Each of the latter two is a real form of the first one, and together they make up all the simple real and complex commutative Lie groups.
In general, real forms work something like this. There’s a “reflection” in the complex -manifold whose fixed points form a real -manifold. The technical details of how to find these things are more than I want to go into right now, but this is the visual geometric intuition I use.
As another more interesting example, consider the group . This consists of all matrices with complex entries:
with the property that . This is a complex Lie group of dimension . It has two real forms. One you might be able to guess is , where all the entries in the matrix are real. The other is , which is a subgroup of satisfying the requirement
Both and are -dimensional real Lie groups.
Another interesting thing about them is looking for the biggest subgroup of either that can be made from the two -dimensional real groups above. You can only fit one copy of the nonzero real numbers into and no copies of the circle. On the other hand, you can fit one copy of the circle into and no copies of the nonzero reals. At the complex level, we see this in the fact that you can only fit one copy of the nonzero complex numbers into . Since these are the biggest commutative Lie groups we can find inside these groups, we say in each case that the group has “rank “. In fact, is the group . The subscript tells the rank of the group — the biggest product of copies of the nonzero complex numbers you can fit inside.
Okay, so what about ? We see that it has rank , so there’s a product of eight copies of the nonzero complex numbers sitting inside. When we break down to a real form, each of these will collapse either into a circle or a copy of the nonzero complex numbers. If each one becomes a circle, the whole real form is called “compact” and things are actually pretty fantastically well-behaved. If we collapse each to a copy of the nonzero real numbers we get the “split” real form of , and things are actually pretty fantastically evil. That’s the real Lie group that Adams’ team was working on.
[EDIT: Okay, as I’ve found I have to say, I’ve pretty drastically oversimplified things. More info in the link]