## 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 , , , and . There are also five extras that don’t fit into those four families, called , , , , and . 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 .

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 , which has now been solved.

I am not a mathematician but still I try to comprehend. Are there 2 or 3 dimensional diagrams of these simple Lie groups?

Comment by eliza | March 24, 2007 |

Well, sort of. The lower ones down, at least. For example, B1 is the collection of all rotations of three-dimensional space. In general, Bn is made up of rotations in (2n+1)-dimensional space, and the D series gives rotations in even-dimensional spaces. Unfortunately, due to a technicality, rotations in the plane aren’t considered a simple Lie group.

If you’re thinking of a diagram like the one that ran alongside all the news reports, it’s actually

notthe Lie group they’re talking about. It’s a sort of tool used in Cartan’s classification called a “root system”, and the picture is a 2-dimensional rendering of an 8-dimensional (for E8) shape. There are pictures like these for all Lie groups, and John Baez has a bunch of them in his most recent column.Comment by John Armstrong | March 24, 2007 |

Can you please dumb it down further and explain any possible practical application for this? Maybe cite something Star Trek or Star Wars and a maybe a reference to a weapon or some cool space ship? Or even cooler some invading alien force? I understand what your saying but I have no real frame of reference for it because I am not a mathematician but I do understand stuff like “fundamental underlying principle to teleporation” or “fundamental underlying principle to big explosions”. Or even “fundamental underlying principle for making space craft fly”.

I just don’t have a frame of reference for this formula or how it matters.

Comment by Kenny Coffin | March 25, 2007 |

Kenny, that’s a great question and it deserves its own post. I’m going to mull it over and write it up in the next day or so.

Comment by John Armstrong | March 25, 2007 |

Cool! Thanks for the explanation. Seems like the press could certainly have explained that. So can this be applied to computer graphics?

Comment by SomeGuy | March 26, 2007 |

I’m actually not sure what this can directly apply to. I just like it ’cause it’s pretty (in an intellectual sense). I’ve just put up another post linking to other sketches by the people directly involved, and my thoughts on why (in a real-world sense) we care about this.

Comment by John Armstrong | March 26, 2007 |

[...] I gave a quick overview of the idea of a Lie group, a Lie algebra, and a representation; a rough overview for complete neophytes of what, exactly, had been calculated; and an attempt to explain why we [...]

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