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