#liegroup

I've been getting tons of mail from people in response to the announcement of the mapping of the E8 Lie group, asking what a Lie group is, what E8 is, and why the mapping of E8 is such a big deal?

Let me start by saying that this is way outside of my area of expertise. So I fully expect that I'll manage to screw something up as I try to figure it out and explain it - so do follow the comments, where I'm sure people who know this better than I do will correct whatever errors I make.

What's a Lie group?

Let's start with the easy part. What's a Lie group? Informally, it's a group whose objects form a manifold, and whose group operation is a continuous function. We can break that down a bit, to make it a little bit clearer.

A group is a set of objects/values with a single binary operator that has a certain set of basic properties: associativity, existence of inverse, existence of identity. It's one of the simplest constructions of abstract algebra. What's really fascinating about it is that that simple construction - the set plus one operation will a simple set of properties - defines the entire concept of symmetry.

Groups don't normally require any structure on their members beyond what's required to make the group operator work properly. You can define a group whose values are a set of points, a set of numbers, a set of coins - very nearly anything you want.

But there are certain structured sets of values that we care about, which you can use as the objects for a group. One of those is a topological space.

A topological space is just a collection of objects which have a kind of nearness/adjacency relationship between the objects in the collection. So a group on a topological space is interesting, because what it does is define symmetry on a set of values that preserves the nearness/adjacency relationships of the objects in the space.

Even more interesting, we can define a particular kind of topological space: a manifold, which is a sort of "smooth" topological space: a manifold is a topological space where the structure of the nearness/adjacency relations makes every small finite region of the space appear to be Euclidean.

So a Lie group is a group whose objects form a manifold, and whose group operations preserve the manifold structure of the nearness/adjacency relations.

Moving on - what's E8?

Many lie groups are based on topological spaces that whose values are representable as some collection of matrices or groups. E8 is one of those - it's a group based on something called a root system. The root system for E8 consists of a set of 8-dimensional vectors, which fall into two families. One family consists of all 8-dimensional vectors, with 2 unit-length elements, and 6 0-length elements; things like (1, 1, 0, 0, 0, 0, 0, 0), (1, 0, 0, 1, 0, 0, 0, 0), (0, -1, 0, 0, 0, 0, 0, 1), etc. The other family consists of all of the 8-dimensional vectors whose elements are all either +1/2 or -1/2, where the sum of all of the elements are even. So (1/2, 1/2, 1/2, 1/2, -1/2, -1/2, -1/2, -1/2) is a member of the root system, since the sum of those elements is 0; (1/2, 1/2, -1/2, 1/2, 1/2, -1/2, 1/2, -1/2) is not an element of the root, system, since it's sum is 1. The beautiful image over to the right is the image of the root system of E8.

The E8 Lie group is based on that root system - it's a massive structure with one complex dimension (complex as in complex numbers - it's value in each dimension is a complex number) for each of the members of the root system. So its a manifold with 248 complex dimensions, or 496 real dimensions.

There are two reasons that having mapped E8 is so important. The practical one is that E8 has major applications: mathematical analysis of the most recent versions of string theory and supergravity theories all keep revealing structure based on E8. E8 seems to be part of the structure of our universe.

The other reason is just that the complete mapping of E8 is the largest mathematical structure ever mapped out in full detail by human beings. It takes 60 gigabytes to store the map of E8. If you were to write it out on paper in 6-point print (that's really small print), you'd need a piece of paper bigger than the island of Manhattan. This thing is huge.