Classification of nilpotent associative algebras of small dimension

TL;DR

This paper classifies all nilpotent associative algebras of dimension up to 4 over any field by constructing them as central extensions of smaller algebras, extending methods from nilpotent Lie algebra classification.

Contribution

It provides a complete classification of small-dimensional nilpotent associative algebras using a novel extension approach.

Findings

Complete list of nilpotent associative algebras up to dimension 4

Methodology based on central extensions of lower-dimensional algebras

Applicable over any field

Abstract

We classify nilpotent associative algebras of dimensions up to 4 over any field. This is done by constructing the nilpotent associative algebras as central extensions of algebras of smaller dimension, analogous to methods known for nilpotent Lie algebras.