On three-dimensional anti-commutative algebras

TL;DR

This paper classifies three-dimensional anti-commutative zero-potent algebras over fields with characteristic not 2, where every element has a square root, advancing understanding of their structure.

Contribution

It provides a complete classification of 3D anti-commutative zero-potent algebras under specified field conditions, a previously unresolved problem.

Findings

Complete classification of 3D anti-commutative zero-potent algebras.

Identification of algebraic structures satisfying the square root condition.

Results applicable over any field with characteristic not 2.

Abstract

This paper is devoted to the classification problem of tree-dimensional anti-commutative(zero-potent) algebras over any base field $\mathbb{F}$ such that $Char(\mathbb{F})\neq 2$ and every element admits a square root.