The structure of Leibniz superalgebras admitting a multiplicative basis

In the literature, most of the descriptions of different classes of Leibniz superalgebras (L=L0¯⊕L1¯,[⋅,⋅]) have been made by giving the multiplication table on the elements of a graded basis B={vk}k∈K of L, in such a way that for any i,j∈K we have [vi,vj]=λi,j[vj,vi]∈Fvk for some k∈K, where F denotes the base field and λi,j∈F. In order to give a unifying viewpoint of all these classes of algebras we introduce the category of Leibniz superalgebras admitting a multiplicative basis and study its structure. We show that if a Leibniz superalgebra L=L0¯⊕L1¯ admits a multiplicative basis then it is the direct sum L=⨁αIα with any Iα=Iα,0¯⊕Iα,1¯ being a well described ideal of L admitting a multiplicative basis inherited from B. Also the B-simplicity of L is characterized in terms of J-connections.