Toward a Three Generation Model of Standard Model Fermions Based on Cayley–Dickson Sedenions

Abstract: We describe an algebraic model, currently being developed, based on the Cayley–Dickson algebra of sedenions, S, with the goal of describing three generations of Standard Model fermions. Although recent algebraic constructions based on the four (normed) division algebras convincingly describe a single generation of leptons and quarks, they offer little insight on how to generalize the results to three generations. We motivate S as a natural algebraic candidate to describe three generations, and argue that Cayley–Dickson algebras beyond the sedenions are unlikely to offer additional physical insights. In our approach, the three generations correspond to three intersecting octonion subalgebras of S. Depending on which maximal compact subgroup of G ₂ = Aut(O) is chosen, the generations are differentiated by either their SU(3)_C color symmetry or their chiral SU(2)_L weak gauge symmetry.