Construction of exceptional Lie algebra G2 and non-associative algebras using Clifford algebra

TL;DR

This paper presents a simplified Clifford algebra-based method to construct the exceptional Lie algebra G2 and related non-associative algebras, revealing new algebraic structures and symmetries without relying on traditional Lie brackets.

Contribution

It introduces a novel Clifford algebra approach to derive G2 and related algebras, simplifying previous exterior algebra methods and uncovering new symmetry relations.

Findings

Derived G2 from Clifford algebra without Lie brackets

Constructed 100 new algebras in 15 dimensions including sedenions

Linked Spin(7) calibrations to algebraic ideals and symmetries

Abstract

This article uses Clifford algebra of definite signature to derive octonions and the Lie exceptional algebra G2 from calibrations using Pin(7). This is simpler than the usual exterior algebra derivation and uncovers a subalgebra of Spin(7) that enables G2 and an invertible element used to classify six other algebras which are found to be related to the symmetries of G2 in a way that breaks the symmetry of octonions. The 4-form calibration terms of Spin(7) are related to an ideal with three idempotents and provides a direct construction of G2 for each of the 480 representations of the octonions. Clifford algebra thus provides a new construction of G2 without using the Lie bracket. This result is extended to 15 dimensions generating another 100 algebras as well as the sedenions.