A sedenion algebraic representation of three colored fermion generations

Abstract

Three generations of fermions with SU(3) C symmetry are represented algebraically in terms of the algebra of sedenions, 𝕊, generated from the octonions, 𝕆, via the Cayley-Dickson process. Despite significant recent progress in generating the Standard Model gauge groups and particle multiplets from the four normed division algebras, an algebraic motivation for the existence of exactly three generations has been difficult to substantiate. In the sedenion model, one generation of leptons and quarks with SU(3) C symmetry is represented in terms of two minimal left ideals of ℂℓ(6), generated from a subset of all left actions of the complex sedenions on themselves. Subsequently, the finite group S 3, which are automorphisms of 𝕊 but not of 𝕆, is used to generate two additional generations. The present paper highlight the key aspects and ideas underlying this construction.