Standard model symmetries and the nested embeddings of R⊂C⊂H⊂O
Year: 2026
Type: article
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, 𝔤 S M : = 𝔰 𝔲 ( 3 ) 𝐶 ⊕ 𝔰 𝔲 ( 2 ) 𝐿 ⊕ 𝔲 ( 1 ) 𝑌 , and low energy post-Higgs, 𝔤 L E : = 𝔰 𝔲 ( 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 𝑛 𝕀 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 E n d ℝ ( 𝕍 ) ≃ 𝐶 𝑙 ( 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 .
Source: Physics Letters B
Author: N. Furey
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
Funder: Humboldt-Universität zu Berlin
OpenAlex ID: W7170092325