The algebraic and geometric classification of derived Jordan and bicommutative algebras

TL;DR

This paper introduces a new method for classifying low-dimensional derived Jordan and bicommutative algebras, providing algebraic and geometric classifications for 3-dimensional cases.

Contribution

It develops a novel classification method for n-dimensional derived Jordan and bicommutative algebras, applied specifically to 3-dimensional cases.

Findings

Classified 3-dimensional derived Jordan algebras.

Classified 3-dimensional metabelian commutative algebras.

Classified 3-dimensional bicommutative algebras.

Abstract

We developed a new proper method for classifying $n$-dimensional derived Jordan algebras, and apply it to the classification of $3$-dimensional derived Jordan algebras. As a byproduct, we have the algebraic classification of $3$-dimensional metabelian commutative algebras and $3$-dimensional derived commutative associative algebras. After that, we introduced a method of classifying $n$-dimensional bicommutative algebras, based on the classification of $n$-dimensional derived commutative associative algebras, and applied it to the classification of $3$-dimensional bicommutative algebras. The second part of the paper is dedicated to the geometric classification of $3$-dimensional metabelian commutative, derived commutative associative, derived Jordan and bicommutative algebras.