Integral Orders for the Okubo Algebra and Idempotent Geometries of the E8 Lattice
Abstract
We study the Coxeter-Dickson E8 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 E8 and the vertices of the Gosset polytope 421. 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 E8 geometry unseparated; the $57$ para-octonionic idempotents reproduce the E7 contact decomposition 1+56+126+56+1; the $12$ Okubo idempotents split into four mutually orthogonal oriented A2 triangles and hence determine canonically an A24 subsystem. Choosing one triangle as the external A2 then yields the A2+E6 Magic Star, while the remaining nine idempotents determine the trinification subsystem A23⊂ E6. This provides an arithmetic interpretation of the E8⊃ E7 and E8⊃ E6+A2 decompositions directly from integral idempotents.
Related
- Measures of noncompactness in Lipschitz Banach algebra and application to quadratic fractional integral equation
- A note on the ideal lattice of a weighted Leavitt path algebra
- Gribov Copies and Gauge Invariance: A Henstock–Kurzweil Continuum Integral on the SU(3) Lattice
- The variety generated by all additively idempotent semirings of order four
- Idempotent radicals and factorization systems
BibTeX
@misc{corradetti2026integral,
title = {{Integral Orders for the Okubo Algebra and Idempotent Geometries of the $E_8$ Lattice}},
author = {Daniele Corradetti and Alessio Marrani},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.27762},
eprint = {2608.27762},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.27762},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Integral Orders for the Okubo Algebra and Idempotent Geometries of the E_8 Lattice
AU - Daniele Corradetti
AU - Alessio Marrani
PY - 2026
DA - 2026/08/27
PB - arXiv
DO - 10.48550/arxiv.2608.27762
UR - https://arxiv.org/abs/2608.27762
AB - 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⊂ E_6. This provides an arithmetic interpretation of the E_8⊃ E_7 and E_8⊃ E_6+A_2 decompositions directly from integral idempotents.
ER -