E8 (mathematics)

In mathematics, E₈ is the name of a Lie group and also its Lie algebra . 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 group as the automorphism group of the Lie algebra. This algebra has a 120-dimensional subalgebra generated by as well as 128 new generators that transform as a Weyl-Majorana spinor of . These statements determine the commutators

as well as

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

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 permutations of
:

and all of the following vectors

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

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