On the structure of graded 3-Leibniz algebras

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 circle plus sigma(j) I-j, with U a linear subspace of T-1, the homogeneous component associated to the unit element 1 in G, and every I-j is 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 not equal k. In the case of T being of maximal length, we characterize the grsimplicity of the algebra in terms of connections in the support of the grading.

Automated summary

We investigate the structure of a T-graded 3-Leibniz algebra T over an arbitrary base field F, with an arbitrary abelian group G as the grading. We prove that T can be expressed as T = U direct sum sigma(j) I-j, where U is a linear subspace of T-1 associated with the unit element 1 in G, and each I-j is a well-defined graded ideal of T satisfying [I-j, T, I-k] = [I-j, I-k, T] = [T, I-j, I-k] = 0, if j not equal k. In the case where T is of maximal length, we characterize the grsimplicity of the algebra in terms of connections in the support of the grading.