Sedenions, Clifford Algebras, and Three Fermion Generations: A Focused Review

ABSTRACT

The existence of three fermion generations remains one of the unexplained structural features of the Standard Model. This article provides a focused review of an algebraic framework in which an intrinsic family symmetry relates three gauge‐equivalent fermion sectors. The framework is motivated by division algebra and Clifford algebra constructions in which the complex octonions and organise the colour and electromagnetic quantum numbers of one generation. Continuing the Cayley–Dickson sequence, the sedenions provide an intrinsic automorphism structure, while their complexified left‐multiplication operators generate an associative endomorphism algebra isomorphic to . In the formulation, the order‐three family action produces three linearly independent fermion sectors while leaving a single gauge algebra. Extension to incorporates a single gauge algebra, and each generation includes a right‐handed neutrino sterile under the Standard Model gauge interactions. We compare this construction with representative algebraic and family‐symmetry approaches to three generations, including triality‐based proposals, and explain why the family symmetry used here is distinct from standard triality despite the common appearance of and . The framework remains algebraic and representation theoretic rather than a complete dynamical theory; family‐symmetry breaking, realistic fermion masses and mixing, and a dynamical account of the gauge and matter sectors remain open problems.