Revisiting the universal linear algebraic model for the characteristic two case

TL;DR

This paper extends a universal linear algebraic model for homogeneous conformal geometries to the characteristic 2 case, using virtual quadratic spaces, and compares it with non-characteristic 2 cases.

Contribution

It provides a detailed study of the characteristic 2 case within the universal linear algebraic model using virtual quadratic spaces, highlighting differences from other fields.

Findings

Achieved similar results for finite fields of characteristic 2 as for other fields.

Identified specific differences in the characteristic 2 case.

Extended the formalism to better understand geometries in characteristic 2.

Abstract

In a previous article, a universal linear algebraic model was proposed for describing homogeneous conformal geometries, such as the spherical, Euclidean, hyperbolic, Minkowski, anti-de Sitter and Galilei planes. This formalism was independent from the underlying field, providing an extension and general approach to other fields, such as finite fields. Some steps were taken even for the characteristic $2$ case. In this article, we undertake the study of the characteristic $2$ case in more detail. In particular, the concept of virtual quadratic spaces is used, defined in a previous article by the author, and a similar result is achieved for finite fields of characteristic $2$ as for other fields. Some differences from the non-characteristic $2$ case are also pointed out.