The algebraic and geometric classification of Zinbiel algebras

TL;DR

This paper provides a comprehensive algebraic and geometric classification of complex 5-dimensional Zinbiel algebras, detailing the structure of their variety, irreducible components, and rigid algebras.

Contribution

It offers the first complete classification of 5-dimensional Zinbiel algebras, including the dimension, irreducible components, and rigid algebras of their variety.

Findings

The variety of complex 5-dimensional Zinbiel algebras has dimension 24.

It is defined by 16 irreducible components.

There are 11 rigid algebras.

Abstract

This paper is devoted to the complete algebraic and geometric classification of complex $5$-dimensional Zinbiel algebras. In particular, we proved that the variety of complex $5$-dimensional Zinbiel algebras has dimension $24$, it is defined by $16$ irreducible components and it has $11$ rigid algebras.