I’ve been getting tons of mail from people in response to the announcement of the mapping of

the E_{8} Lie group, asking what a Lie group is, what E_{8} is, and why the mapping of E_{8} 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.

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 E_{8}?

Many lie groups are based on topological spaces that whose values are representable as some collection of matrices or groups. E_{8} is one of those – it’s a group based on something called a *root system*. The root system for E_{8} 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 E_{8}.

The E_{8} 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 E_{8} is so important. The practical one is that E_{8} has major applications: mathematical analysis of the most recent versions of string theory and supergravity theories all keep revealing structure based on E_{8}. E_{8} seems to be part of the structure of our universe.

The other reason is just that the complete mapping of E_{8} is the largest mathematical structure ever mapped out in full detail by human beings. It takes *60 gigabytes* to store the map of E_{8}. 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*.

**Update: ** For those who claim that mathematicians have no sense of himor, I heard via Gooseana that the title of the formal presentation where they’ll be talking about the E_{8} map is: “The Character Table for E8, or How We Wrote Down a 453,060 x 453,060 Matrix and Found Happiness”.