The Chevalley group G_{2}(2) of order 12096 and the octonionic root system of E_{7}

Mehmet Koca; Ramazan Koc; Nazife O. Koca

doi:10.1016/j.laa.2006.12.011

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arXiv · hep-th/0509189

The Chevalley group G_{2}(2) of order 12096 and the octonionic root system of E_{7}

Mehmet Koca·Ramazan Koc·Nazife O. Koca

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Abstract

The octonionic root system of the exceptional Lie algebra E_8 has been constructed from the quaternionic roots of F_4 using the Cayley-Dickson doubling procedure where the roots of E_7 correspond to the imaginary octonions. It is proven that the automorphism group of the octonionic root system of E_7 is the adjoint Chevalley group G_2(2) of order 12096. One of the four maximal subgroups of G_2(2) of order 192 preserves the quaternion subalgebra of the E_7 root system. The other three maximal subgroups of orders 432,192 and 336 are the automorphism groups of the root systems of the maximal Lie algebras E_6xU(1), SU(2)xSO(12), and SU(8) respectively. The 7-dimensional manifolds built with the use of these discrete groups could be of potential interest for the compactification of the M-theory in 11-dimension.

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Mehmet Koca, Ramazan Koc, Nazife O. Koca. 2005-10-20. The Chevalley group G_{2}(2) of order 12096 and the octonionic root system of E_{7}. https://doi.org/10.1016/j.laa.2006.12.011

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