E8 (mathematics)

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In mathematics, E₈ is the name of a root system and of several associated Lie groups and also their Lie algebras [\mathfrak_8]. These are the largest of the exceptional simple Lie groups. It is also one of the simply laced groups. E₈ has rank 8 and dimension 248. It is simply connected and its center is the trivial subgroup. Its outer automorphism group is the trivial group. Its fundamental representation is the 248-dimensional adjoint representation.

As well as the complex Lie group E₈, of complex dimension 248 or real dimension 496, there are 3 real forms of the group, all of real dimension 248. There is one compact one (which is usually the one meant if no other information is given), one split one, and a third one.

One can construct the (compact form of the) [E_8] group as the automorphism group of the corresponding[E_8] Lie algebra. This algebra has a 120-dimensional subalgebra [\operatorname(16)] generated by [J_] as well as 128 new generators [Q_a] that transform as a Weyl-Majorana spinor of [\operatorname(16)]. These statements determine the commutators

[[J_,J_]=\delta_J_-\delta_J_-\delta_J_+\delta_J_]

as well as

[[J_,Q_a] = \frac 14 (\gamma_i\gamma_j-\gamma_j\gamma_i)_ Q_b,]

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

[[Q_a,Q_b]=\gamma^_gamma^_ J_.]

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

The compact real form of E₈ is the isometry group of a 128-dimensional Riemannian manifold known informally as the 'octooctonionic projective plane' because it can be built using an algebra that is the tensor product of the octonions with themselves. This can be seen systematically using a construction known as the 'magic square', due to Hans Freudenthal and Jacques Tits.

The group E₈ 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).

Contents

1 Algebra

2 References
3 See also

Algebra

All [\begin8\\2\end] permutations of
[(\pm 1,\pm 1,0,0,0,0,0,0).]

and all of the following vectors

[\left(\pm,\pm,\pm,\pm,\pm,\pm,\pm,\pm\right)]

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 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]

References

John Baez, The Octonions, Section 4.6: E₈, [Bull. Amer. Math. Soc. 39 (2002), 145-205]. Online HTML version at

http://math.ucr.edu/home/baez/octonions/node19.html.

E8 (mathematics)

Algebra

[\begin 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]

References

See also