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

Abstract

We classify, up to equivalences, all abelian groups gradings on null-filiform and oneparametric 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.

Non-associative algebraic structures: classification and structure

“…The present part is based on the papers written together with Antonio Jesús Calderón, Artem Lopatin, Bakhrom Omirov, Gulkhayo Solijanova, Luisa Camacho, Pasha Zusmanovich and Yury Popov [44,59,159,180].…”

“…In our work [44] we begin the study of gradings on Leibniz algebras by classifying, up to equivalence, all abelian groups gradings of null-filiform and one-parametric filiform Leibniz algebras. Let us define all the considered algebras.…”

“…We described all abelian gradings of null-filiform Leibniz algebras and one-parametric filiform Leibniz algebras in [44,Theorems 13,17 and 19].…”

Varieties of Null-Filiform Leibniz Algebras Under the Action of Hopf Algebras

“…The support of a G-grading is the set Supp(G) = {g ∈ G | A g = 0}. It has been proved in [12] that any G-grading on a null-filiform Leibniz algebra, where G is an abelian group, is cyclic. In particular, the next result holds.…”