A Geometrical Interpretation of Okubo Spin Group

TL;DR

This paper introduces a novel geometric interpretation of the Spin group associated with the real Okubo algebra by constructing affine and projective planes over it, revealing new connections between algebra and geometry.

Contribution

It is the first to define affine and projective geometries over the Okubo algebra and interpret its Spin group as a group of collineations preserving geometric structures.

Findings

Established affine and projective planes over Okubo algebra

Demonstrated the Spin(O) group as collineations preserving the plane's axis

Provided a bijection between two geometric constructions

Abstract

In this work we define, for the first time, the affine and projective plane over the real Okubo algebra, showing a concrete geometrical interpretation of its Spin group. Okubo algebra is a flexible, composition algebra which is also a not unital division algebra. Even though Okubo algebra has been known for more than 40 years, we believe that this is the first time the algebra was used for affine and projective geometry. After showing that all axioms of affine geometry are verified, we define a projective plane over Okubo algebra as completion of the affine plane and directly through the use of Veronese coordinates. We then present a bijection between the two constructions. Finally we show a geometric interpretation of Spin(O) as the group of collineations that preserve the axis of the plane.