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

G. P. Wilmot

TL;DR

This work presents a Clifford-geometric-algebra framework for deriving octonions and the exceptional Lie algebra ${\rm G2}$ from calibrations in ${\rm Spin}(7)$, avoiding the Lie bracket. By formulating 7D calibrations via primary 3-forms $\Phi_i$ and their duals $\Phi_i^*$ and exploiting $90^{\circ}$-rotations, it constructs a rich enabling algebra that yields automorphisms of the octonions and enumerates all $480$ octonion representations across ${\mathcal G}_7$. A key outcome is the Classification Theorem, which identifies six pseudo-octonion algebras ${\mathbb P}_k$ with distinct non-associative-triple counts, plus zero divisors and connections to sedenions via higher-dimensional generalizations. The ${\rm G2}$ algebra arises as the automorphism group preserving calibration forms, with explicit mappings to Bryant/Cartan representations and a clear relation to the ${\rm Spin}(7)$-calibration structure. The approach yields a unified, geometrically transparent construction of ${\rm G2}$ and related non-associative algebras, and extends to higher dimensions with a wealth of pseudo-sedenion structures, aided by computational tools.

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.

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

TL;DR

This work presents a Clifford-geometric-algebra framework for deriving octonions and the exceptional Lie algebra from calibrations in , avoiding the Lie bracket. By formulating 7D calibrations via primary 3-forms and their duals and exploiting -rotations, it constructs a rich enabling algebra that yields automorphisms of the octonions and enumerates all octonion representations across . A key outcome is the Classification Theorem, which identifies six pseudo-octonion algebras with distinct non-associative-triple counts, plus zero divisors and connections to sedenions via higher-dimensional generalizations. The algebra arises as the automorphism group preserving calibration forms, with explicit mappings to Bryant/Cartan representations and a clear relation to the -calibration structure. The approach yields a unified, geometrically transparent construction of and related non-associative algebras, and extends to higher dimensions with a wealth of pseudo-sedenion structures, aided by computational tools.

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.
Paper Structure (6 sections, 3 theorems, 26 equations, 3 figures, 10 tables)

This paper contains 6 sections, 3 theorems, 26 equations, 3 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Calibrations in GA(7)
  3. Octonion Representations
  4. Non-associative algebras and Zero Divisors
  5. Construction of G2
  6. Summary

Key Result

Lemma 1

The terms of $\{1 + \Phi_i^*\}$ and $\{1 + \Phi_i + \Phi_i^*+ {\rm e}_{1234567}\}$, $i\in{\mathbb N}_1^{30}$, form commuting subalgebras of ${\rm Spin(}7)$ and ${\rm Pin(}7)$, respectively.

Figures (3)

  • Figure 1: The 2-simplex correspondences
  • Figure 2: Primary cross product Fano plane
  • Figure 3: Cartan Root Diagram for ${\rm G2}$

Theorems & Definitions (11)

  • Definition
  • Definition
  • Definition
  • Definition
  • Lemma 1
  • proof
  • Definition
  • Theorem 1
  • proof
  • Theorem 2
  • ...and 1 more