E8 is a name used to refer to several of the most beautiful structures in mathematics.
One of the simplest of these is the E8 lattice. The E8 lattice can be defined as all points in 8-dimensional space that are:
- Either all integers or all integers plus one-half--i.e.(1,1,0,0,0,0,0,0) or (½,½,½,½,½,½,½,½)
- Add up to an even number--i.e. (½,½,½,½,½,½,½,½) is part of E8 because when you add up all 8 coordinates you get 4, but (½,½,½,½,½,½,½,-½) is not because they add up to 3.
This lattice has the unique property of being the only lattice in 8-dimensional space that is both even(taking the square of each point in this lattice and adding them all up will always give an even number) and unimodular(the 8 basis elements of the lattice--you can multiply them by integers to get every member of the lattice--form a matrix with determinant 1).
So-called even unimodular lattices only occur in dimensions that are multiples of 8. In 16 dimensions there are two lattices, one by taking two copies of E8 and one by constructing a D16+ lattice in 16 dimensions similar to the E8 one. In 24 dimensions there are 24 lattices(mostly a coincidence, but the number 24 pops up in a lot of weird places) called Niemeier lattices. In 32 dimensions there are at least millions.
But anyways, back to E8. Remember the definition of an even lattice in the third paragraph? That number you get by adding all the squares is called the norm. There is a very easy formula for calculating the number of points on the E8 lattice with norm 2n.
- Find every divisor of the number n.
- Add the cube of every divisor together.
- Multiply the sum by 240.
This formula is related to modular forms and can be derived for other lattices as well--480 times the sum of seventh powers of divisors works for both 16-dimensional even unimodular lattices!
The most important points in the E8 lattice, however, are the roots of E8, which have norm 2. Since 1 is the only number that divides 2/2=1, there are only 240*1^3=240 roots of E8. There are two types of E8 roots:
- Roots of the type (±1,±1,0,0,0,0,0,0), where the ±1s can be in any two coordinates. There are 8*7*4/2=112 roots of this type.
- Roots of the type (±½,±½,±½,±½,±½,±½,±½,±½) where the number of - signs is even. There are 2^8/2=128 roots of this type.
I'll leave it to you to add those up and show there are 240 roots.
The 240 roots of E8 make up the E8 polytope. It's the biggest semiregular polytope. E8 is the biggest of a lot of things.
The 240 roots of E8 also make up one of the most beautiful E8 structures, the e8 Lie algebra and E8 Lie group. E8 is an exceptional simple Lie group--in fact, it's the biggest exceptional simple Lie group. The five exceptional simple Lie groups are the following:
- the 14-dimensional G2
- the 52-dimensional F4
- the 78-dimensional E6
- the 133-dimensional E7
- the 248-dimensional E8
Now, you may be wondering why it's a 248-dimensional group when there are 240 roots in the E8 lattice. That's because you have a dimension for every root in the lattice, then a dimension for every dimension in the lattice!
Unfortunately, most methods of deriving a Lie group from a root system start with its corresponding Lie algebra. Usually, e8(the Lie algebra is usually written in lowercase, and the group in uppercase) is actually defined in terms of smaller Lie groups, such as this:
e8 = so(16) + S16 +
where so refers to the special orthogonal group and S+ refers to the right-handed spinor. Note that because of the definition of a simple Lie group, this cannot be seen as a direct or even semidirect product but rather both parts of the group act on each other.
Another formulation of e8, which connects it to the triality of Spin(8)(spin groups share a Lie algebra with the corresponding special orthogonal group), is the following:
e8 = so(8) + so(8) + V8×V8 + S+8×S+8 + S-8×S-8
where the first so(8) acts on the first term in the tensor products and the second so(8) on the second term. V represents the vector representation, which in 8 dimensions is isomorphic to either of the two spinors(this is triality)
An interesting feature of e8 is that the smallest interesting representation of the E8 Lie group is that on its own Lie algebra--the adjoint representation. This is a feature unique to E8.
E8 has already found applications in physics, from the use of two copies of E8 in heterotic string theory to Antony Garrett Lisi's use of E8 as a symmetry group of the elementary particles.
In March 2007