Abelian subalgebras and ideals of maximal dimension in Zinbiel algebras

TL;DR

This paper investigates the structure of abelian subalgebras and ideals of maximal dimension in finite-dimensional Zinbiel algebras, focusing on their codimensions and specific subclasses like supersolvable and filiform Zinbiel algebras.

Contribution

It provides a detailed comparison of maximal abelian subalgebras and ideals in Zinbiel algebras, including classifications and examples for low-dimensional cases.

Findings

Maximal abelian subalgebras can have codimension 1 or 2 in certain Zinbiel algebras.

Classification of low-dimensional Zinbiel algebras over the complex field.

Examples illustrating the values of parameters for classified Zinbiel algebras.

Abstract

In this paper, we compare the abelian subalgebras and ideals of maximal dimension for finite-dimensional Zinbiel algebras. We study Zinbiel algebras containing maximal abelian subalgebras of codimension $1$ and supersolvable Zinbiel algebras in which such subalgebras have codimension $2$, and we also analyze the case of filiform Zinbiel algebras. We give examples to clarify some results, including listing the values for $\alpha$ and $\beta$ for the low dimensional Zinbiel algebras over the complex field that have been classified.