On the structure of graded $3-$Leibniz algebras

Abstract

We study the structure of a $3-$Leibniz algebra $T$ graded by an arbitrary abelian group $G,$ which is considered of arbitrary dimension and over an arbitrary base field $\bbbf.$ We show that $T$ is of the form $T=\uu\oplus\sum_jI_j,$ with $\uu$ a linear subspace of $T_1,$ the homogeneous component associated to the unit element $1$ in $G,$ and any $I_j$ a well described graded ideal of $T,$ satisfying $$[I_j,T,I_k]=[I_j,I_k,T]=[T,I_j,I_k]=0,$$ if $j\ne k.$ In the case of $T$ being of maximal length, we characterize the gr-simplicity of the algebra in terms of connections in the support of the grading.

arXiv admin note: text overlap with arXiv:1603.08426 by other authors