Classification of three-dimensional Jordan superalgebras

TL;DR

This paper classifies all three-dimensional Jordan superalgebras over an algebraically closed field of characteristic not two, aiming to identify minimal dimensions of exceptional cases and distinguish between special and exceptional superalgebras.

Contribution

It provides the first complete classification of small-dimensional Jordan superalgebras, aiding in understanding their structure and the minimal dimension of exceptional superalgebras.

Findings

Complete list of 3D Jordan superalgebras provided

Identified which superalgebras are special or exceptional

Set bounds on minimal dimension of exceptional Jordan superalgebras

Abstract

In this paper, we classify Jordan superalgebras of dimension up to three over an algebraically closed field of characteristic different of two. Our main motivation to obtain such classification comes out from the intention to give an answer to the problem of determining the minimal dimension of exceptional Jordan superalgebras. Our strategy to provide a lower bounded for this dimension is to determine the complete list of Jordan superalgebras of small dimensions and verify which ones are special or exceptional.