The geometric classification of nilpotent $\mathfrak{CD}$-algebras

TL;DR

This paper provides a detailed geometric classification of complex 4-dimensional nilpotent -algebras, revealing the structure of their algebraic variety and showing the absence of rigid algebras.

Contribution

It offers the first geometric classification of these algebras, identifying their irreducible components and describing the algebraic variety structure.

Findings

The geometric variety has dimension 18.

It decomposes into 2 irreducible components.

There are no rigid 4-dimensional nilpotent -algebras.

Abstract

We give a geometric classification of complex $4$-dimensional nilpotent $\mathfrak{CD}$-algebras. The corresponding geometric variety has dimension 18 and decomposes into $2$ irreducible components determined by the Zariski closures of a two-parameter family of algebras and a four-parameter family of algebras (see Theorem 2). In particular, there are no rigid $4$-dimensional complex nilpotent $\mathfrak{CD}$-algebras.