The geometric classification of nilpotent algebras

TL;DR

This paper provides a comprehensive geometric classification of various types of nilpotent algebras, establishing their irreducibility, dimensions, and explicit generic families, with applications to anticommutative algebra length.

Contribution

It introduces a novel geometric framework for classifying nilpotent algebras, detailing their varieties and explicit generic families, advancing understanding of their structure.

Findings

Proved irreducibility of algebraic varieties of nilpotent algebras

Determined dimensions of these varieties

Described explicit generic algebra families

Abstract

We give a geometric classification of $n$-dimensional nilpotent, commutative nilpotent and anticommutative nilpotent algebras. We prove that the corresponding geometric varieties are irreducible, find their dimensions and describe explicit generic families of algebras which define each of these varieties. We show some applications of these results in the study of the length of anticommutative algebras.