The classification of $2$-dimensional rigid algebras

TL;DR

This paper classifies all 2-dimensional rigid, conservative, and terminal algebras over an algebraically closed field of characteristic 0, providing both algebraic and geometric classifications and proposing open problems.

Contribution

It offers the first comprehensive algebraic and geometric classification of 2-dimensional rigid, conservative, and terminal algebras, expanding understanding of their structure.

Findings

Complete algebraic classification of 2D rigid algebras

Geometric classification of the variety of 2D terminal algebras

Formulation of open problems based on geometric insights

Abstract

Using the algebraic classification of all $2$-dimensional algebras, we give the algebraic classification of all $2$-dimensional rigid, conservative and terminal algebras over an algebraically closed field of characteristic 0. We have the geometric classification of the variety of $2$-dimensional terminal algebras, and based on the geometric classification of these algebras we formulate some open problems.