Two-dimensional left (right) unital algebras over algebraically closed fields and $\mathbb{R}$

TL;DR

This paper classifies all two-dimensional unital, left unital, and right unital algebra structures over algebraically closed fields and the real numbers, providing a comprehensive tabulation of these algebras.

Contribution

It offers a complete classification of two-dimensional unital algebra structures over algebraically closed fields and the reals, including explicit tabulations.

Findings

Classified all two-dimensional unital algebras over algebraically closed fields.

Provided explicit descriptions and isomorphism classes.

Tabulated all algebras with units.

Abstract

In this paper we describe all, up to isomorphism, left unital, right unital and unital algebra structures on two-dimensional vector space over any algebraically closed field and $\mathbb{R}$. We tabulate the algebras with the units.