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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 Dynkin diagram of E_8 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). 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) -
\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}
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