Central extensions of filiform Zinbiel algebras

TL;DR

This paper classifies central extensions of complex null-filiform and filiform Zinbiel algebras, showing that non-split extensions produce higher-dimensional null-filiform algebras and identifying all non-isomorphic quasi-filiform Zinbiel algebras.

Contribution

It provides a complete description of central extensions of these algebras and classifies all quasi-filiform Zinbiel algebras up to isomorphism.

Findings

Non-split central extensions produce higher-dimensional null-filiform Zinbiel algebras.

All pairwise non-isomorphic quasi-filiform Zinbiel algebras are obtained.

Classification results are up to isomorphism.

Abstract

In this paper we describe central extensions (up to isomorphism) of all complex null-filiform and filiform Zinbiel algebras. It is proven that every non-split central extension of an $n$-dimensional null-filiform Zinbiel algebra is isomorphic to an $(n+1)$-dimensional null-filiform Zinbiel algebra. Moreover, we obtain all pairwise non isomorphic quasi-filiform Zinbiel algebras.