The algebraic and geometric classification of nilpotent anticommutative algebras

TL;DR

This paper classifies all 6-dimensional complex nilpotent anticommutative algebras algebraically and geometrically, revealing 14 families and 130 classes, with the variety being irreducible and lacking rigid algebras.

Contribution

It provides the first comprehensive algebraic and geometric classification of 6-dimensional nilpotent anticommutative algebras, identifying all isomorphism classes and their geometric structure.

Findings

14 one-parameter families of algebras

130 isomorphism classes identified

The geometric variety is irreducible and has no rigid algebras

Abstract

We give algebraic and geometric classifications of $6$-dimensional complex nilpotent anticommutative algebras. Specifically, we find that, up to isomorphism, there are $14$ one-parameter families of $6$-dimensional nilpotent anticommutative algebras, complemented by $130$ additional isomorphism classes. The corresponding geometric variety is irreducible and determined by the Zariski closure of a one-parameter family of algebras. In particular, there are no rigid $6$-dimensional complex nilpotent anticommutative algebras.