Integral Orders for the Okubo Algebra and Idempotent Geometries of the $E_8$ Lattice

[2608.27762] Integral Orders for the Okubo Algebra and Idempotent Geometries of the $E_8$ Lattice Skip to main content Search Submit Donate Log in Mathematics > Rings and Algebras arXiv:2608.27762 (math) [Submitted on 27 Aug 2026] Title: Integral Orders for the Okubo Algebra and Idempotent Geometries of the $E_8$ Lattice Authors: Daniele Corradetti , Alessio Marrani View PDF HTML (experimental) Abstract: We study the Coxeter-Dickson $E_8$ order as a common integral support for the octonionic, para-octonionic and compact real Okubo products. The three algebra structures have the same additive lattice, the same positive composition norm and hence the same $240$ norm-one vectors, namely the roots of $E_8$ and the vertices of the Gosset polytope $4_{21}$. Their multiplicative structures are nevertheless different and this difference is already visible in the idempotents: among the same $240$ roots one finds respectively $1$, $57$ and $12$ nonzero integral idempotents. We show that these three counts organize the common root system in three increasingly polarized ways. The octonionic product leaves the full $E_8$ geometry unseparated; the $57$ para-octonionic idempotents reproduce the $E_7$ contact decomposition $1+56+126+56+1$; the $12$ Okubo idempotents split into four mutually orthogonal oriented $A_2$ triangles and hence determine canonically an $A_2^4$ subsystem. Choosing one triangle as the external $A_2$ then yields the $A_2+E_6$ Magic Star, while the remaining nine idempotents determine the trinification subsystem $A_2^3\subset E_6$. This provides an arithmetic interpretation of the $E_8\supset E_7$ and $E_8\supset E_6+A_2$ decompositions directly from integral idempotents. Comments: Article for the 10th Conference AGACSE 2026 August 24-28, 2026 Beijing, China Subjects: Rings and Algebras (math.RA) Cite as: arXiv:2608.27762 [math.RA] (or arXiv:2608.27762v1 [math.RA] for this version) https://doi.org/10.48550/arXiv.2608.27762 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Daniele Corradetti [ view email ] [v1] Thu, 27 Aug 2026 22:49:35 UTC (13 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.RA < prev | next > new | recent | 2026-08 Change to browse by: math References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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