E8 (mathematics) - Definition and Overview

In mathematics, E₈ is the name of a Lie group and also its Lie algebra <math>\mathfrak{e}_8<math>. It is the largest of the five exceptional simple Lie groups. It is also one of the simply laced groups. E₈ has rank 8 and dimension 248. Its center is the trivial subgroup. Its outer automorphism group is the trivial group. Its fundamental representation is the 248-dimensional adjoint.

The Dynkin diagram of the E₈ algebra is

One can construct the <math>E_8<math> group as the automorphism group of the <math>E_8<math> Lie algebra. This algebra has a 120-dimensional subalgebra <math>so(16)<math> generated by <math>J_{ij}<math> as well as 128 new generators <math>Q_a<math> that transform as a Weyl-Majorana spinor of <math>spin(16)<math>. These statements determine the commutators

<math>[J_{ij},J_{kl}]=\delta_{jk}J_{il}-\delta_{jl}J_{ik}-\delta_{ik}J_{jl}+\delta_{il}J_{jk}<math>

as well as

<math>[J_{ij},Q_a] = \frac 14 (\gamma_i\gamma_j-\gamma_j\gamma_i)_{ab} Q_b<math>,

while the remaining commutator (not anticommutator!) is defined as

<math>[Q_a,Q_b]=\gamma^{[i}_{ac}\gamma^{j]}_{cb} J_{ij}.<math>

It is then possible to check that the Jacobi identity is satisfied.

This group frequently appears in string theory and supergravity, for example as the U-duality group of supergravity on an eight-torus (a noncompact version), or as a part of the gauge group of the heterotic string (the compact version).

Root system

All <math>\begin{pmatrix}8\\2\end{pmatrix}<math> permutations of

<math>(\pm 1,\pm 1,0,0,0,0,0,0)<math>

and all of the following vectors

<math>(\pm{1\over 2},\pm{1\over 2},\pm{1\over 2},\pm{1\over 2},\pm{1\over 2},\pm{1\over 2},\pm{1\over 2},\pm{1\over 2})<math>

for which the sum of all the eight coordinates is even.

There are 240 roots in all.

Simple roots:

(0,0,0,0,0,0,1,-1)

(0,0,0,0,0,0,1,1)

(0,0,0,0,0,1,-1,0)

(0,0,0,0,1,-1,0,0)

(0,0,0,1,-1,0,0,0)

(0,0,1,-1,0,0,0,0)

(0,1,-1,0,0,0,0,0)

(1/2,-1/2,-1/2,-1/2,-1/2,-1/2,-1/2,1/2)

Cartan matrix

<math>

\begin{pmatrix}

2 & -1 &  0 &  0 &  0 &  0 &  0 & 0 \\

-1 & 2 & -1 & 0 & 0 & 0 & 0 & 0 \\

0 & -1 &  2 & -1 &  0 &  0 &  0 & -1 \\
0 &  0 & -1 &  2 & -1 &  0 &  0 & 0 \\
0 &  0 &  0 & -1 &  2 & -1 &  0 & 0 \\
0 &  0 &  0 &  0 & -1 &  2 & -1 & 0 \\
0 &  0 &  0 &  0 &  0 & -1 &  2 & 0 \\
0 &  0 & -1 &  0 &  0 &  0 &  0 & 2

\end{pmatrix}<math>

Example Usage of (mathematics)