# E8 (mathematics)

In mathematics, E8 is the name of a Lie group and also its Lie algebra [itex]\mathfrak{e}_8[itex]. 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

Missing image
Dynkin_diagram_E8.png
Dynkin diagram of E_8

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

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

as well as

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

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

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

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 [itex]\begin{pmatrix}8\\2\end{pmatrix}[itex] permutations of

[itex](\pm 1,\pm 1,0,0,0,0,0,0)[itex]

and all of the following vectors

[itex](\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})[itex]

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

There are 240 roots in all.

(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

[itex]

\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}[itex]

 E6 | E7 | E8 | F4 | G2

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