Chemistry - E8 (mathematics)

In mathematics, E₈ is the name of a Lie group and also its Lie algebra $\mathfrak{e}_8$ . 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 $E 8$ group as the automorphism group of the $E 8$ Lie algebra. This algebra has a 120-dimensional subalgebra $s o (16)$ generated by $J i j$ as well as 128 new generators $Q a$ that transform as a Weyl-Majorana spinor of $s p i n (16)$ . These statements determine the commutators

[J i j, J k l] = δ j k J i l - δ j l J i k - δ i k J j l + δ i l J j k

as well as

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

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

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

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 $\begin{pmatrix}8\\2\end{pmatrix}$ permutations of

$(\pm 1,\pm 1,0,0,0,0,0,0)$

and all of the following vectors

$(\pm{1\over 2},\pm{1\over 2},\pm{1\over 2},\pm{1\over 2},\pm{1\over 2},\pm{1\over 2},\pm{1\over 2},\pm{1\over 2})$

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

$\begin{pmatrix} 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\ -1 & 2 & -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & -1 & 2 & -1 & 0 & 0 & 0 & -1 \\ 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 & 2 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 & 0 & 2 \end{pmatrix}$

Categories: Lie groups