# Isomorphism classes of four dimensional nilpotent associative algebras over a field

## TL;DR

This paper classifies four-dimensional nilpotent associative algebras over various fields by analyzing isomorphism classes and regular subgroups of the affine group, providing explicit representatives for different types of fields.

## Contribution

It offers a comprehensive classification of four-dimensional nilpotent associative algebras over finite, real, and algebraically closed fields, including explicit representatives.

## Key findings

- Explicit classification of isomorphism classes over finite fields
- Explicit classification over the real field R
- Explicit classification over algebraically closed fields

## Abstract

In this paper we classify the isomorphism classes of four dimensional nilpotent associative algebras over a field F, studying regular subgroups of the affine group AGL_4(F). In particular we provide explicit representatives for such classes when F is a finite field, the real field R or an algebraically closed field.