Central extensions of 3-dimensional Zinbiel algebras

TL;DR

This paper classifies all central extensions of 3-dimensional complex Zinbiel algebras, leading to a comprehensive classification of certain 4- and 5-dimensional Zinbiel algebras with specific properties.

Contribution

It provides a complete description of central extensions of 3D Zinbiel algebras and classifies 4D and 5D Zinbiel algebras with a 2-dimensional annihilator.

Findings

All central extensions of 3D Zinbiel algebras are described.

Full classification of 4D non-trivial complex Zinbiel algebras.

Classification of 5D Zinbiel algebras with 2-dimensional annihilator.

Abstract

We describe all central extensions of all $3$-dimensional non-zero complex Zinbiel algebras. As a corollary, we have a full classification of $4$-dimensional non-trivial complex Zinbiel algebras and a full classification of $5$-dimensional non-trivial complex Zinbiel algebras with $2$-dimensional annihilator, which gives the principal step in the algebraic classification of $5$-dimensional Zinbiel algebras.