E8 (mathematics)

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

Related Topics:
Mathematics - Lie group - Lie algebra - Simple Lie group - Simply laced group - Center - Trivial subgroup - Outer automorphism group - Trivial group - Fundamental representation - Adjoint

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The Dynkin diagram of the E8 algebra is

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

Related Topics:
Automorphism group - Weyl-Majorana spinor

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:=delta_{jk}J_{il}-delta_{jl}J_{ik}-delta_{ik}J_{jl}+delta_{il}J_{jk}

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as well as

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: = rac 14 (gamma_igamma_j-gamma_jgamma_i)_{ab} Q_b,

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

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:=gamma^{}_{cb} J_{ij}.

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

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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).

Related Topics:
String theory - Supergravity - U-duality - Heterotic string

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