Standard Model Symmetries and the Nested Embeddings of ℝ⊂ℂ⊂ℍ⊂𝕆
Abstract
Where does the Standard Model's internal structure come from? Treating 𝕆⊕ℍ⊕ℂ⊕ℝ as a module for its own multiplication algebra enables a particular origin story for the Standard Model's pre-Higgs, gSM:=su(3)C ⊕ su(2)L ⊕ u(1)Y, and post-Higgs, gLE:=su(3)C ⊕ u(1)Q, symmetries. In this model, weak hypercharge and electric charge operators, Y and Q, take on a remarkably simple form: ∑ 1/n𝕀n× n. Upon the introduction of Cayley-Dickson imaginary units, this 15 ℝ dimensional 𝕆⊕ℍ⊕ℂ⊕ℝ embeds naturally as a vector space into several well-studied 16 ℝ dimensional algebras, which we generically refer to as 𝕍. With this embedding, the Standard Model's internal symmetries may then be seen to arise in part from the sequence of nested inclusions: ℝ⊂ℂ⊂ℍ⊂ 𝕆⊂𝕍. In the sedenionic case of 𝕍= 𝕊, the full sequence becomes a Cayley-Dickson tower. We define the notion of endomorphic models of particle physics, and connect EndR(𝕍)≃ 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 S15, and simultaneously involves the four parallelizable spheres S7, S3, S1, S0.
Versions
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BibTeX
@misc{furey2026standard,
title = {{Standard Model Symmetries and the Nested Embeddings of $\mathbbR\subset\mathbbC\subset\mathbbH\subset\mathbbO$}},
author = {N. Furey},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.18450},
eprint = {2607.18450},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.18450},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Standard Model Symmetries and the Nested Embeddings of ℝ⊂ℂ⊂ℍ⊂𝕆
AU - N. Furey
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2607.18450
UR - https://arxiv.org/abs/2607.18450
AB - Where does the Standard Model's internal structure come from? Treating 𝕆⊕ℍ⊕ℂ⊕ℝ as a module for its own multiplication algebra enables a particular origin story for the Standard Model's pre-Higgs, g_SM:=su(3)_C ⊕ su(2)_L ⊕ u(1)_Y, and post-Higgs, g_LE:=su(3)_C ⊕ u(1)_Q, symmetries. In this model, weak hypercharge and electric charge operators, Y and Q, take on a remarkably simple form: ∑ 1/n𝕀_n× n. Upon the introduction of Cayley-Dickson imaginary units, this 15 ℝ dimensional 𝕆⊕ℍ⊕ℂ⊕ℝ embeds naturally as a vector space into several well-studied 16 ℝ dimensional algebras, which we generically refer to as 𝕍. With this embedding, the Standard Model's internal symmetries may then be seen to arise in part from the sequence of nested inclusions: ℝ⊂ℂ⊂ℍ⊂ 𝕆⊂𝕍. In the sedenionic case of 𝕍= 𝕊, the full sequence becomes a Cayley-Dickson tower. We define the notion of endomorphic models of particle physics, and connect End_R(𝕍)≃ 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.
ER -