Three fermion generations with two unbroken gauge symmetries from the complex sedenions
Adam Gillard (Department of Mathematical Sciences, Xi’an Jiaotong-Liverpool University, 111 Ren’ai Road, Suzhou HET, Jiangsu, 215123, China); Niels Gresnigt (Department of Mathematical Sciences, Xi’an Jiaotong-Liverpool University, 111 Ren’ai Road, Suzhou HET, Jiangsu, 215123, China)
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 $${\mathbb {C}}\ell (6)$$ . It was previously shown that the minimal left ideals of $${\mathbb {C}}\ell (6)$$ 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.
