Okubo algebras and Exceptional Periodicity: tetracode gradings, root-lattice obstructions and Clifford cones
Abstract
The Okubo algebra brings together ternary gradings, integral root frames and rank-three geometry. We examine how these structures extend to the complex Exceptional Periodicity algebras. With N=4(n+1), a tetracode character gives eight nonzero homogeneous components of dimension tn=8n+3+24n. On a tip, a compatible grading has zero component of dimension 8n−5 and eight nonzero components of dimension 1+24n−3. The arithmetic behaves differently: for n≥2, the lattice DN+ has no full-rank orthogonal frame of unscaled A2 lattices. We exhibit failures of the Jacobi and Jordan identities and determine the derivations and full automorphism group of the ambient algebra above level one. We also give a Clifford-module proof, with the normalization fixed by root products, of the published realization of EP tips as special Vinberg H-algebras. This identifies a positive real form and yields explicit formulas for its cone, cubic, Cholesky coordinates and rational dual invariant. The cone is symmetric exactly at n=1. Its determinant-one hypersurface belongs to the known $L(8n,1,0)$ series, whose r- and c-maps give homogeneous scalar manifolds for supergravity theories with eight real supercharges.
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BibTeX
@misc{marrani2026okubo,
title = {{Okubo algebras and Exceptional Periodicity: tetracode gradings, root-lattice obstructions and Clifford cones}},
author = {Alessio Marrani and Daniele Corradetti},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18643},
eprint = {2610.18643},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18643},
}
RIS
TY - UNPB
TI - Okubo algebras and Exceptional Periodicity: tetracode gradings, root-lattice obstructions and Clifford cones
AU - Alessio Marrani
AU - Daniele Corradetti
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.18643
UR - https://arxiv.org/abs/2610.18643
AB - The Okubo algebra brings together ternary gradings, integral root frames and rank-three geometry. We examine how these structures extend to the complex Exceptional Periodicity algebras. With N=4(n+1), a tetracode character gives eight nonzero homogeneous components of dimension t_n=8n+3+2^4n. On a tip, a compatible grading has zero component of dimension 8n−5 and eight nonzero components of dimension 1+2^4n−3. The arithmetic behaves differently: for n≥2, the lattice D_N^+ has no full-rank orthogonal frame of unscaled A_2 lattices. We exhibit failures of the Jacobi and Jordan identities and determine the derivations and full automorphism group of the ambient algebra above level one. We also give a Clifford-module proof, with the normalization fixed by root products, of the published realization of EP tips as special Vinberg H-algebras. This identifies a positive real form and yields explicit formulas for its cone, cubic, Cholesky coordinates and rational dual invariant. The cone is symmetric exactly at n=1. Its determinant-one hypersurface belongs to the known $L(8n,1,0)$ series, whose r- and c-maps give homogeneous scalar manifolds for supergravity theories with eight real supercharges.
ER -