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.