The Unapologetic Mathematician

Real and complex groups

Okay, another thing to make clear is that there’s not just one group we could mean by $E_8$. 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 $n$-dimensional space. You may remember that the complex numbers look like a plane with the real numbers sitting inside on a line. A complex $n$-manifold looks like a real $2n$-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 $i$” — 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 $1$-dimensional complex manifold. Now there’s a “reflection” of the plane to itself that plays nice with the complex structure: complex conjugation, $z\mapsto \bar{z}$. The points that are their own conjugates make up the real line. But there’s another reflection that plays nice: $z\mapsto 1/\bar{z}$. 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 $n$-manifold whose fixed points form a real $n$-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 $SL(2,\mathbb{C})$. This consists of all $2\times 2$ matrices with complex entries:
$\left(\begin{array}{cc}a&b\\c&d\end{array}\right)$
with the property that $ad-bc=1$. This is a complex Lie group of dimension $3$. It has two real forms. One you might be able to guess is $SL(2,\mathbb{R})$, where all the entries in the matrix are real. The other is $SU(2)$, which is a subgroup of $SL(2,\mathbb{C})$ satisfying the requirement
$\left(\begin{array}{cc}a&b\\c&d\end{array}\right)\left(\begin{array}{cc}\bar{a}&\bar{c}\\\bar{b}&\bar{d}\end{array}\right)=\left(\begin{array}{cc}1& 0\\ 0&1\end{array}\right)$
Both $SL(2,\mathbb{R})$ and $SU(2)$ are $3$-dimensional real Lie groups.

Another interesting thing about them is looking for the biggest subgroup of either that can be made from the two $1$-dimensional real groups above. You can only fit one copy of the nonzero real numbers into $SL(2,\mathbb{R})$ and no copies of the circle. On the other hand, you can fit one copy of the circle into $SU(2)$ 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 $SL(2,\mathbb{C})$. Since these are the biggest commutative Lie groups we can find inside these groups, we say in each case that the group has “rank $1$“. In fact, $SL(2,\mathbb{C})$ is the group $A_1$. 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 $E_8$? We see that it has rank $8$, so there’s a product of eight copies of the nonzero complex numbers sitting inside. When we break $E_8$ 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 $E_8$, 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]

March 22, 2007 Posted by | Atlas of Lie Groups | 2 Comments

A rough overview

I’ve had a flood of incoming people in the past couple days, and have even been linked from the article in The New York Times (or at least in their list of blogs commenting on the news). As I said before, their coverage is pretty superficial, and I’ve counted half a dozen errors in their picture captions alone.

One of the main reasons I write this weblog is because I believe anyone can follow the basic ideas of even the most bleeding-edge mathematics. Few mathematicians write towards the generally interested lay audience (“GILA”) the way physicists tend to do, and when mathematics does make it into the popular press the journalists don’t even make the effort they do in physics to get what they do say right.

My uncle, no mathematician he but definitely a GILA member, emailed me to mention he’d read that mathematicians had “solved E8″, but had no idea what it meant. Mostly he was asking if I knew Adams (I do), but I responded with a high-level overview of what they were doing and why. I’m going to post here what I told him. It’s designed to be pretty self-contained, and has been refined from a few days of explaining the ideas to other nonmathematicians.

Oh, and I’m not above link-baiting. If you find this coherent and illuminating, please pass the link to this post around. If there’s something that I’ve horribly screwed up in here, please let me know and I’ll try to smooth it over while keeping it accessible. I’m also trying to explain the ideas at a somewhat higher level (though not in full technicality) within the category “Atlas of Lie Groups”. If you want to know more, please keep watching there.

[UPDATE: I now also have another post trying to answer the "what's it good for?" question. That response starts at the fourth paragraph: "I also want to...".]

I understand not knowing what the news reports mean, because most of them are pretty horrible. It’s possible to give a stripped-down explanation, but the popular press doesn’t seem to want to bother.

A group is a collection of symmetries. A nice one is all the transformations of a square. You can flip it over left-to-right, flip it up-to-down, or rotate it by quarter turns. This group isn’t “simple” because there are smaller groups sitting inside it [yes, it's a bit more than that as readers here should know. --ed] — you could forget the flips and just consider the group of rotations. All groups can be built up from simple groups that have no smaller ones sitting inside them, so those are the ones we really want to understand. Think of it sort of like breaking a number into its prime factors.

The kinds of groups this project is concerned with are called Lie groups (pronounced “lee”) after the Norwegian mathematician Sophus Lie. They’re made up of continuous transformations like rotations of an object in 3-dimensional space. Again, the Lie groups we’re really interested in are the simple ones that can’t be broken down into smaller ones.

A hundred years ago, Élie Cartan and others came up with a classification of all these simple Lie groups. There are four infinite families like rotations in spaces of various dimensions or square matrices of various sizes with determinant 1 (if you remember any matrix algebra). These are called $A_n$, $B_n$, $C_n$, and $D_n$. There are also five extras that don’t fit into those four families, called $G_2$, $F_4$, $E_6$, $E_7$, and $E_8$. That last one is the biggest. It takes three numbers to describe a rotation in 3-D space, but 248 numbers to describe an element of $E_8$.

Classifying the groups is all well and good, but they’re still hard to work with. We want to know how these groups can act as symmetries of various objects. In particular, we want to find ways of assigning a matrix to each element of a group so that if you take two transformations in the group and do them one after the other, the matrix corresponding to that combination is the product of the matrices corresponding to the two transformations. We call this a “matrix representation” of the group. Again, some representations can be broken into simpler pieces, and we’re concerned with the simple ones that can’t be broken down anymore.

What the Atlas project is trying to do is build up a classification of all the simple representations of all the simple Lie groups, and the hardest chunk is $E_8$, which has now been solved.

March 22, 2007 Posted by | Atlas of Lie Groups | 7 Comments