User:Fropuff/Draft 12
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[edit] E8
[edit] Overview
The e8 Lie algebra[1] is a complex Lie algebra of dimension 248 and rank 8. It is a simple Lie algebra meaning that it has no non-trivial ideals.
It is said that E8 has rank (the maximum number of mutually commutative degrees of freedom) 8, and dimension (as a manifold) 248. This means that a maximal torus of the compact Lie group E8 has dimension 8. The vectors of the root system are in eight dimensions, and are specified later in this article. The Weyl group of E8, which acts as a symmetry group of the maximal torus by means of the conjugation operation from the whole group, is of order 696729600.
The compact Lie group E8 is simply connected and its center is the trivial subgroup. Its outer automorphism group is the trivial group, meaning that all its automorphisms are inner automorphisms. Its fundamental representation is the 248-dimensional adjoint representation (in other words conjugation acting on tangent vectors at the identity element).
[edit] Real forms
As well as the complex Lie group E8, of complex dimension 248 or real dimension 496, there are three 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.
| Real form | Maximal compact subalgebra |
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[edit] Geometry
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.[2]
[edit] Constructions
One can construct the (compact form of the) E8 group as the automorphism group of the corresponding e8 Lie algebra. This algebra has a 120-dimensional subalgebra so(16) generated by Jij as well as 128 new generators Qa that transform as a Weyl-Majorana spinor of spin(16). These statements determine the commutators
![[J_{ij},J_{k\ell}]=\delta_{jk}J_{i\ell}-\delta_{j\ell}J_{ik}-\delta_{ik}J_{j\ell}+\delta_{i\ell}J_{jk}](../../../math/1/2/2/1228e886cdc52aebf0972174498179f7.png)
as well as
![[J_{ij},Q_a] = \frac 14 (\gamma_i\gamma_j-\gamma_j\gamma_i)_{ab} Q_b,](../../../math/8/7/1/8715b3afb5587c891c9a869679c42978.png)
while the remaining commutator (not anticommutator!) is defined as
![[Q_a,Q_b]=\gamma^{[i}_{ac}\gamma^{j]}_{cb} J_{ij}.](../../../math/5/c/7/5c7d805ebe3f0a51722c0054393cea37.png)
It is then possible to check that the Jacobi identity is satisfied.
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