Three generations of colored fermions with S_3 family symmetry from Cayley–Dickson sedenions

Abstract

An algebraic representation of three generations of fermions with SU(3)_C color symmetry based on the Cayley–Dickson algebra of sedenions {mathbb {S}} is constructed. Recent constructions based on division algebras convincingly describe a single generation of leptons and quarks with Standard Model gauge symmetries. Nonetheless, an algebraic origin for the existence of exactly three generations has proven difficult to substantiate. We motivate mathbb {S} as a natural algebraic candidate to describe three generations with SU(3)_C gauge symmetry. We initially represent one generation of leptons and quarks in terms of two minimal left ideals of mathbb {C}ell (6), generated from a subset of all left actions of the complex sedenions on themselves. Subsequently we employ the S_3 automorphism of order three, which is an automorphism of mathbb {S} but not of mathbb {O}, to generate two additional generations. Given the relative obscurity of sedenions, efforts have been made to present the material in a self-contained manner.