Standard Model Symmetries and the Nested Embeddings of $\mathbb{R}\subset \mathbb{C}\subset \mathbb{H}\subset \mathbb{O}$

Abstract

Where does the Standard Model's internal structure come from? Treating $\mathbb{O}\oplus \mathbb{H}\oplus \mathbb{C}\oplus \mathbb{R}$ as a module for its own multiplication algebra enables a particular origin story for the Standard Model's pre-Higgs, ${\mathfrak{g}}_{SM}:=\mathfrak{su}(3{)}_C\oplus \mathfrak{su}(2{)}_L\oplus \mathfrak{u}(1{)}_Y,$ and post-Higgs, ${\mathfrak{g}}_{LE}:=\mathfrak{su}(3{)}_C\oplus \mathfrak{u}(1{)}_Q,$ symmetries. We recognize the endomorphisms and their modules alike as ${\mathbb{Z}}_2^n$-graded algebras. Then, annihilating certain highest grade (volume) elements, and enacting an equal-trace condition on anti-hermitian operators leads precisely to ${\mathfrak{g}}_{SM}$ and ${\mathfrak{g}}_{LE}$. Weak hypercharge and electric charge operators, $Y$ and $Q,$ take on a remarkably simple form: $\sum \frac{1}{n}{\mathbb{I}}_{n\times n}$. Upon the introduction of Cayley-Dickson imaginary units, this 15 $\mathbb{R}$ dimensional $\mathbb{O}\oplus \mathbb{H}\oplus \mathbb{C}\oplus \mathbb{R}$ embeds naturally as a vector space into several well-studied 16 $\mathbb{R}$ dimensional algebras, which we generically refer to as $\mathbb{V}.$ With this embedding, the Standard Model's internal symmetries may then be seen to arise in part from the sequence of nested inclusions: $\mathbb{R}\subset \mathbb{C}\subset \mathbb{H}\subset \mathbb{O}\subset \mathbb{V}.$ We define the notion of endomorphic models of particle physics, and connect $En{d}_{\mathbb{R}}(\mathbb{V})\simeq Cl(0,8)$ to the earlier ideas of Bott Periodic Particle Physics. We comment on a possible connection between the existence of multiple complex structures and the baryon asymmetry problem. In closing, we identify an appearance in this model of the fully connected tree of division algebraic Hopf fibrations that starts at $S^{15},$ and simultaneously involves the four parallelizable spheres $S^7,S^3,S^1,S^0.$

Comments: 13 pages, 8 figures, presented at the Exceptional Structures and Standard Model Workshop, University of Edinburgh, 2026.07.13. Includes a new section on nested division algebraic Hopf fibrations