Subalgebras, idempotents, ideals and quasi-units of two-dimensional algebras

TL;DR

This paper classifies all subalgebras, ideals, and quasi-units of two-dimensional algebras, providing explicit polynomial conditions and listing isomorphism classes, including simple algebras.

Contribution

It offers a complete classification of subalgebras, ideals, and quasi-units in two-dimensional algebras, with explicit polynomial criteria and enumeration of isomorphism classes.

Findings

Explicit forms of polynomials determining subalgebras and ideals

Classification of all simple two-dimensional algebras

Enumeration of algebras based on subalgebra and ideal counts

Abstract

All subalgebras, idempotents, left(right) ideals and left quasi-units of two-dimensional algebras are described. Classification of algebras with given number of subalgebras, left(right) ideals are provided. In particular, a list of isomorphism classes of all simple two-dimensional algebras is given. In the study of ideals and subalgebras the number of them depend on roots of certain system of polynomials at structure constants of the algebra. We also give explicit forms of the polynomials.