Classifications of two-dimensional Jordan algebras over the algebraically closed fields and $\mathbb{R}$

TL;DR

This paper classifies all two-dimensional Jordan algebras over algebraically closed fields and the real numbers, providing a comprehensive understanding of their structure through matrices of structure constants.

Contribution

It offers a complete classification of two-dimensional Jordan algebras over algebraically closed fields and , using matrix representations of their structure constants, which was not previously established.

Findings

Complete classification of 2D Jordan algebras over algebraically closed fields.

Complete classification of 2D Jordan algebras over .

Representation of algebras via matrices of structure constants.

Abstract

The classification, up to isomorphism, of two-dimensional (not necessarily commutative) Jordan algebras over algebraically closed fields and $\mathbb{R}$ is presented in terms of their matrices of structure constants.