Division Algebras, Triality, and Exceptional Magic

Abstract

We describe in explicit detail the rich relationship between division algebras, split composition algebras, triality, Clifford algebras, spinors, triality Lie algebras, generalized reflections, exceptional magic square Lie algebras, triality eigenspaces, 3-symmetric spaces, Vinberg theta algebras, and Exceptional Unification in particle physics.

arXiv comment50 pages, 9 figures. Published in Advances in Applied Clifford Algebras. Originally submitted to arXiv on April 24, 2025

References in corpus (13)

• Octonions, E6, and Particle Physics
• Exceptional quantum geometry and particle physics II
• Three fermion generations with two unbroken gauge symmetries from the complex sedenions
• The standard model, the Pati-Salam model, and "Jordan geometry"
• Division algebraic symmetry breaking
• Left-Right symmetric fermions and sterile neutrinos from complex split biquaternions and bioctonions
• Octions: An $E_8$ description of the Standard Model
• Beyond the standard model with six-dimensional spinors
• Spin(11,3), particles and octonions
• An octonionic construction of $E_8$ and the Lie algebra magic square
• Dixon-Rosenfeld Lines and the Standard Model
• The GraviGUT Algebra Is not a Subalgebra of $E_8$, but $E_8$ Does Contain an Extended GraviGUT Algebra
• Three Generations and a Trio of Trialities