A characterization of endo-commutativity of 3-dimensional curled algebras

TL;DR

This paper characterizes when 3-dimensional curled algebras are endo-commutative by establishing necessary and sufficient conditions based on their linear basis properties.

Contribution

It provides a complete criterion for endo-commutativity in 3D curled algebras, a class of non-associative algebras, over any field.

Findings

Derived necessary and sufficient conditions for endo-commutativity

Expressed conditions in terms of basis properties

Applicable to algebras over arbitrary fields

Abstract

A curled algebra is a non-associative algebra in which $x$ and $x^2$ are linearly dependent for every element $x$. An algebra is called endo-commutative, if the square mapping from the algebra to itself preserves multiplication. In this paper, we provide a necessary and sufficient condition for a 3-dimensional curled algebra over an arbitrary field to be endo-commutative, expressed in terms of the properties of its underlying linear basis.