On the structure of graded $3-$Leibniz algebras

TL;DR

This paper investigates the structure of graded 3-Leibniz algebras over arbitrary abelian groups, providing a decomposition into a linear subspace and graded ideals, and characterizing simplicity in maximal length cases.

Contribution

It introduces a structural decomposition of graded 3-Leibniz algebras and characterizes their simplicity based on support connections in the grading.

Findings

Decomposition of T into a linear subspace and graded ideals.

Conditions for ideals to be orthogonal in the algebra.

Characterization of gr-simplicity via support connections.

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 $j\neq k.$ 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.