The algebraic and geometric classification of nilpotent right alternative algebras

TL;DR

This paper classifies all 4-dimensional complex nilpotent right alternative algebras, revealing only 9 non-isomorphic types and analyzing their geometric structure with 13-dimensional variety.

Contribution

It provides the first complete algebraic and geometric classification of 4-dimensional nilpotent right alternative algebras, identifying rigid algebras and their geometric relations.

Findings

Only 9 non-isomorphic nilpotent right alternative algebras exist in 4 dimensions.

The geometric variety has dimension 13 and is characterized by rigid algebras.

The classification includes a one-parametric family of algebras.

Abstract

We present algebraic and geometric classifications of the $4$-dimensional complex nilpotent right alternative algebras. Specifically, we find that, up to isomorphism, there are only $9$ non-isomorphic nontrivial nilpotent right alternative algebras. The corresponding geometric variety has dimension $13$ and it is determined by the Zariski closure of $4$ rigid algebras and one one-parametric family of algebras.