Integral elements of Okubo algebra and the E8-lattice
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915998556225536 |
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| author | Corradetti, Daniele |
| author_facet | Corradetti, Daniele |
| contents | In this work we study the interplay between the Coxeter-Dickson $E_{8}$-order, the para-octonions, and the real Okubo algebra. We prove that the Coxeter-Dickson order remains closed for the para-octonionic product, so that one recovers a genuine $\mathbb{Z}$-integral system with underlying lattice $E_{8}$. Intriguingly, the Okubo product behaves in a different and more arithmetic way: it forces $\mathbb{Q}(\sqrt{3})$-coefficients and does not preserve the same $\mathbb{Z}$-order. After a diagonal $2$-adic scaling we obtain a closed $\mathbb{Z}[\sqrt{3}]$-order, whose direct metric shadow is a $2$-primary conductor sublattice of $E_{8}$, not $E_{8}$ itself. The lattice $E_{8}$ is recovered only by $2$-adic saturation, equivalently by gluing, and this recovery is metric-arithmetic rather than multiplicative. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_09333 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Integral elements of Okubo algebra and the E8-lattice Corradetti, Daniele Rings and Algebras In this work we study the interplay between the Coxeter-Dickson $E_{8}$-order, the para-octonions, and the real Okubo algebra. We prove that the Coxeter-Dickson order remains closed for the para-octonionic product, so that one recovers a genuine $\mathbb{Z}$-integral system with underlying lattice $E_{8}$. Intriguingly, the Okubo product behaves in a different and more arithmetic way: it forces $\mathbb{Q}(\sqrt{3})$-coefficients and does not preserve the same $\mathbb{Z}$-order. After a diagonal $2$-adic scaling we obtain a closed $\mathbb{Z}[\sqrt{3}]$-order, whose direct metric shadow is a $2$-primary conductor sublattice of $E_{8}$, not $E_{8}$ itself. The lattice $E_{8}$ is recovered only by $2$-adic saturation, equivalently by gluing, and this recovery is metric-arithmetic rather than multiplicative. |
| title | Integral elements of Okubo algebra and the E8-lattice |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2605.09333 |