The geometric classification of $2$-step nilpotent algebras and applications

TL;DR

This paper provides a geometric classification of complex 2-step nilpotent algebras, identifying irreducible components and rigid algebras, and applies these results to classify 5-dimensional nilpotent associative algebras.

Contribution

It introduces a geometric classification framework for complex 2-step nilpotent algebras and determines the structure of their irreducible components and rigid algebras.

Findings

Identified the number of irreducible components for the classification.

Determined the dimensions of these components.

Classified 5-dimensional nilpotent associative algebras with 14 components and 9 rigid algebras.

Abstract

We give a geometric classification of complex $n$-dimensional $2$-step nilpotent (all, commutative and anticommutative) algebras. Namely, has been found the number of irreducible components and their dimensions. As a corollary, we have a geometric classification of complex $5$-dimensional nilpotent associative algebras. In particular, it has been proven that this variety has $14$ irreducible components and $9$ rigid algebras.