Automorphism groups of discrete octonions and possible applications in physics

Abstract

Octonions constituting the root system of E₈ form a closed algebra where the root system of E₇ can be represented by pure imaginary octonions. It has been shown that the automorphism group of the octonionic root system of E₇ is the adjoint Chevalley group G₂(2) of order 12096. The maximal subgroups of G2(2) preserve the octonionic root systems of the maximal Lie algebras of E₇ with one exception.

It has been shown that the automorphism group of the octonionic set ±ei (i=1,...,7) (called Moufang loop) which represents the roots of SU(2)⁷ is the non-split extension of the elementary abelian group 2³ of order 8 by the Klein's group L₂(7) of order 168. Matrix generators of the 7-dimensional irreducible representations of the split and non-split extensions of the groups of order 1344 have been constructed. The Weyl group of D₄ turns out to be maximal in the split 1344 and the Demazure-Tits' subgroups of A₂, G₂ and B₂ are naturally embedded in the groups of concern.

There are indications that the parity, P, CP, and CPT violations can be explained in a group theoretical way as C, P, T generate an elementary abelian group of order 8 while acting on the bilinear Dirac fields.

Construction of the E8 root system with the icosians is also briefly mentioned.