On naturally graded Lie and Leibniz superalgebras

Abstract: In general, the study of gradations has always represented a cornerstone in algebra theory. In particular, \textit{naturally graded} seems to be the first and the most relevant gradation when it comes to nilpotent algebras, a large class of solvable ones. In fact, many families of relevant solvable algebras have been obtained by extensions of naturally graded nilpotent algebras, i.e. solvable algebras with a well-structured nilradical. Thus, the aim of this work is introducing the concept of naturally graded for superalgebra structures such as Lie and (non-Lie) Leibniz. After having defined naturally graded Lie and Leibniz superalgebras, we characterize natural gradations on a very important class of each of them, that is, those with maximal super-nilindex.