Three Generations and a Trio of Trialities

Abstract

We identify the Standard Model's $\mathfrak{su}(3)\oplus \mathfrak{su}(2)\oplus \mathfrak{u}(1)$ internal symmetries within the triality symmetries $\mathfrak{tri}(\mathbb{C})\oplus \mathfrak{tri}(\mathbb{H})\oplus \mathfrak{tri}(\mathbb{O})$. From here, the corresponding Standard Model group action is applied to the triality triple $\left({\Psi }_{+},{\Psi }_{-},V\right)$ for ${\Psi }_{+},{\Psi }_{-},V\in \mathbb{C}\otimes \mathbb{H}\otimes \mathbb{O}$. Together, ${\Psi }_{+}$ and ${\Psi }_{-}$ provide the correct irreducible representations for two generations. Owing to a certain Cartan Factorization, which we define, $V$ provides the irreducible representations for a third generation. Said more explicitly in another way, division algebraic multiplication merges a third generation of spinor representations into a set of scalar bosons. This set of scalar bosons includes the familiar Standard Model Higgs representation.

Comments: 11 pages, 6 figures