Three fermion generations with two unbroken gauge symmetries from the complex sedenions

Abstract

Abstract We show that three generations of leptons and quarks with unbroken Standard Model gauge symmetry $$SU(3)_c\times U(1)_{em}$$ SU(3)c×U(1)em can be described using the algebra of complexified sedenions $${\mathbb {C}}\otimes {\mathbb {S}}$$ C⊗S . A primitive idempotent is constructed by selecting a special direction, and the action of this projector on the basis of $${\mathbb {C}}\otimes {\mathbb {S}}$$ C⊗S can be used to uniquely split the algebra into three complex octonion subalgebras $${\mathbb {C}}\otimes {\mathbb {O}}$$ C⊗O . These subalgebras all share a common quaternionic subalgebra. The left adjoint actions of the 8 $${\mathbb {C}}$$ C -dimensional $${\mathbb {C}}\otimes {\mathbb {O}}$$ C⊗O subalgebras on themselves generates three copies of the Clifford algebra $${\mathbb {C}}\ell (6)$$ Cℓ(6) . It was previously shown that the minimal left ideals of $${\mathbb {C}}\ell (6)$$ Cℓ(6) describe a single generation of fermions with unbroken $$SU(3)_c\times U(1)_{em}$$ SU(3)c×U(1)em gauge symmetry. Extending this construction from $${\mathbb {C}}\otimes {\mathbb {O}}$$ C⊗O to $${\mathbb {C}}\otimes {\mathbb {S}}$$ C⊗S naturally leads to a description of exactly three generations.