Abelian groups gradings on null-filiform and one-parametric filiform Leibniz algebras

TL;DR

This paper classifies all abelian group gradings on specific Leibniz algebras, revealing that null-filiform cases are toral while some one-parametric filiform cases are non-toral, advancing understanding of algebra gradings.

Contribution

It provides a complete classification of abelian group gradings on null-filiform and one-parametric filiform Leibniz algebras, highlighting differences in grading types.

Findings

All gradings on null-filiform Leibniz algebras are toral.

Existence of non-toral gradings on one-parametric filiform Leibniz algebras.

Classification up to equivalence of all such gradings.

Abstract

We classify, up to equivalences, all abelian groups gradings on null-filiform and one-parametric filiform Leibniz algebras. Any grading on a null-filiform Leibniz algebra is toral but there are non-toral gradings on one-parametric filiform Leibniz algebras.