Hom–Leibniz algebras admitting a multiplicative basis

A lot of the study of diverse classes of Hom–Leibniz algebras of arbitrary dimension [Formula: see text] in the literature are given by the multiplication table on the elements of a basis [Formula: see text] of [Formula: see text] thus, for any [Formula: see text] we obtain [Formula: see text] and [Formula: see text] for certain [Formula: see text] where [Formula: see text] denotes the arbitrary base field and [Formula: see text]. So as to give a unified view of all these classes of Hom-algebras, we introduce the more general category of Hom–Leibniz algebras admitting a multiplicative basis and discuss its structure. We prove that if a Hom–Leibniz algebra [Formula: see text] admits a multiplicative basis, then it can be decomposed into the direct sum [Formula: see text], where [Formula: see text] is a well-described ideal of [Formula: see text] admitting a multiplicative basis inherited from [Formula: see text]. In addition, the [Formula: see text]-simplicity of [Formula: see text] is characterized in terms of the multiplicative basis.