The algebraic and geometric classification of nilpotent left-symmetric algebras

TL;DR

This paper provides a comprehensive algebraic and geometric classification of complex 4-dimensional nilpotent left-symmetric algebras, revealing their structure, irreducible components, and the absence of rigid algebras.

Contribution

It offers the first complete classification of 4-dimensional nilpotent left-symmetric algebras, detailing their geometric variety and decomposition.

Findings

The geometric variety has dimension 15.

Decomposition into 3 irreducible components.

No rigid 4-dimensional nilpotent left-symmetric algebras.

Abstract

This paper is devoted to the complete algebraic and geometric classification of complex $4$-dimensional nilpotent left-symmetric algebras. The corresponding geometric variety has dimension $15$ and decomposes into $3$ irreducible components determined by the Zariski closures of two one-parameter families of algebras and a two-parameter family of algebras (see Theorem B). In particular, there are no rigid $4$-dimensional complex nilpotent left symmetric algebras.