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 F. We show that T is of the form T = U ⊕ j I j , with U 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, satisfyingIn 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.