The algebraic and geometric classification of nilpotent terminal algebras

TL;DR

This paper classifies 4-dimensional complex nilpotent terminal algebras algebraically and geometrically, revealing the structure, parameter families, and the absence of rigid algebras within this class.

Contribution

It provides the first comprehensive algebraic and geometric classification of 4-dimensional nilpotent terminal algebras, including parameter families and irreducible components.

Findings

41 one-parameter families of algebras

18 two-parameter families of algebras

No rigid algebras found

Abstract

We give algebraic and geometric classifications of $4$-dimensional complex nilpotent terminal algebras. Specifically, we find that, up to isomorphism, there are $41$ one-parameter families of $4$-dimensional nilpotent terminal (non-Leibniz) algebras, $18$ two-parameter families of $4$-dimensional nilpotent terminal (non-Leibniz) algebras, $2$ three-parameter families of $4$-dimensional nilpotent terminal (non-Leibniz) algebras, complemented by $21$ additional isomorphism classes. The corresponding geometric variety has dimension 17 and decomposes into 3 irreducible components determined by the Zariski closures of a one-parameter family of algebras, a two-parameter family of algebras and a three-parameter family of algebras. In particular, there are no rigid $4$-dimensional complex nilpotent terminal algebras.