Okubo algebras with isotropic norm

TL;DR

This paper characterizes Okubo algebras with isotropic norm through special gradings and describes their automorphism groups, revealing new structural insights into these nonunital composition algebras of dimension 8.

Contribution

It introduces a characterization of Okubo algebras with isotropic norm via special gradings and identifies their automorphism groups in the split case.

Findings

Okubo algebras with isotropic norm are characterized by a special grading.

The automorphism group of the most symmetric grading is PU(3,4).

Structural properties of these algebras are elucidated through grading analysis.

Abstract

Okubo algebras form an important class of nonunital composition algebras of dimension 8. Contrary to what happens for unital composition algebras, they are not determined by their multiplicative norms. Okubo algebras with isotropic norm are characterized here by the existence of a special grading. In the split case, and under some restrictions on the ground field, the automorphism group of the most symmetric of these gradings is the projective unitary group PU(3,4), whose structure is showcased by the grading.