On identities of $2$-dimensional algebras

TL;DR

This paper explores polynomial identities in finite-dimensional algebras, focusing on classifying all 2-dimensional algebras based on well-known identities to understand their structural properties.

Contribution

It provides a comprehensive list of polynomial identities and classifies all 2-dimensional algebras according to these identities, offering new insights into their algebraic structure.

Findings

Classification of 2-dimensional algebras based on polynomial identities

List of well-known single polynomial identities for finite-dimensional algebras

Identification of algebraic structures satisfying specific identities

Abstract

In the paper we provide some polynomial identities for finite-dimensional algebras. A list of well known single polynomial identities is exposed and the classification of all $2$-dimensional algebras with respect to these identities is given.