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

TL;DR

This paper explores the structure of the octonionic root system of E_7, demonstrating its automorphism group as the Chevalley group G_2(2), and discusses implications for M-theory compactification.

Contribution

It establishes the automorphism group of the octonionic E_7 root system as G_2(2) and identifies its maximal subgroups related to other Lie algebra root systems.

Findings

Automorphism group of E_7 root system is G_2(2) of order 12096.

Identifies maximal subgroups related to E_6xU(1), SU(2)xSO(12), and SU(8).

Potential applications in M-theory compactification.

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.