Construction of Nilpotent Jordan Algebras Over any Arbitrary Fields

TL;DR

This paper introduces a computational method based on second cohomology to classify nilpotent Jordan algebras over arbitrary fields, extending classification results up to dimension four and analyzing their associative properties.

Contribution

It develops an analogue of the Skjelbred-Sund method for Jordan algebras and applies it to classify nilpotent Jordan algebras of low dimensions over various fields.

Findings

Classified nilpotent Jordan algebras up to dimension three over any field.

Classified nilpotent Jordan algebras of dimension four over algebraically closed fields and the real field.

Identified 13 nilpotent Jordan algebras of dimension 4 over algebraically closed fields, with 4 non-associative; 17 over the reals, with 5 non-associative.

Abstract

We give a computional method to construct and classify nilpotent Jordan algebras over any arbitrary fields by the second cohomolgy of nilpotent Jordan algebras of low dimension "analogue of Skjelbred-Sund method", we see that every nilpotent Jordan algebras can be constructed by the second cohomolgy of nilpotent Jordan algebras of low dimension. We use this method to classify nilpotent Jordan algebras up to dimension three over any field and nilpotent Jordan algebras of dimension four over an algebraic closed field of characteristic not 2 and over the real field R. Also commutative nilpotent associative algebras are classified, we show that there are up to isomorphism 13 nilpotent Jordan algebras of dimension 4 over an algebraic closed field of characteristic not 2, four of those are not associative, yielding 9 commutative nilpotent associative algebras. Also up to isomorphism there are 17 nilpotent Jordan algebras of dimension 4 over the real field R, five of those are not associative, yielding 12 commutative nilpotent associative algebras.