Deformation of the Okubic Albert algebra and its relation to the Okubic affine and projective planes

TL;DR

This paper explores a deformation of the Okubic Albert algebra, establishing a correspondence between its rank-1 idempotents and points in the Okubic projective plane, extending classical geometric-algebraic relations.

Contribution

It introduces a deformation of the Okubic Albert algebra linking algebraic elements to geometric points, extending known relations from octonionic to Okubic structures.

Findings

Deformation of Okubic Albert algebra constructed.

Rank-1 idempotents correspond to points in Okubic projective plane.

Extension of classical algebraic-geometry correspondence.

Abstract

We present a deformation of the Okubic Albert algebra introduced by Elduque whose rank-1 idempotent elements are in biunivocal correspondence with points of the Okubic projective plane, thus extending to Okubic algebras the correspondence found by Jordan, Von Neumann, Wigner between the octonionic projective plane and the rank-3 Exceptional Jordan Algebra.