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 finite group $S_3$, which are automorphisms 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.

arXiv comment18 pages, 1 figure

References in corpus (24)

• The Flavor Symmetry
• Lepton masses, mixings and FCNC in a minimal S_3-invariant extension of the Standard Model
• Three generations, two unbroken gauge symmetries, and one eight-dimensional algebra
• Higgs Potential in Minimal S_3 Invariant Extension of the Standard Model
• Exceptional quantum geometry and particle physics
• Octonions, E6, and Particle Physics
• Generations: Three Prints, in Colour
• A geometric basis for the standard-model gauge group
• Deducing the symmetry of the standard model from the automorphism and structure groups of the exceptional Jordan algebra
• $SU(3{)}_C\times SU(2{)}_L\times U(1{)}_Y\left(\times U(1{)}_X\right)$ as a symmetry of division algebraic ladder operators
• Exceptional quantum geometry and particle physics II
• Dimensional Reduction
• Octonions, exceptional Jordan algebra and the role of the group F_4 in particle physics
• Three fermion generations with two unbroken gauge symmetries from the complex sedenions
• Braids, normed division algebras, and Standard Model symmetries
• The standard model, the Pati-Salam model, and "Jordan geometry"
• Octions: An $E_8$ description of the Standard Model
• The Standard Model particle content with complete gauge symmetries from the minimal ideals of two Clifford algebras
• An Exceptional G(2) Extension of the Standard Model from the Correspondence with Cayley-Dickson Algebras Automorphism Groups
• Octonion Internal Space Algebra for the Standard Model
• A topological model of composite preons from the minimal ideals of two Clifford algebras
• An octonionic construction of $E_8$ and the Lie algebra magic square
• Braided fermions from Hurwitz algebras
• Braided matter interactions in quantum gravity via 1-handle attachment

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