Leibniz superalgebras with a set grading

Abstract

Consider a Leibniz superalgebra $\mathfrak{L}$ additionally graded by an arbitrary set $I$ (set grading). We show that $\mathfrak{L}$ decomposes as the sum of well-described graded ideals plus (maybe) a suitable linear subspace. In the case of $\mathfrak{L}$ being of maximal length, the simplicity of $\mathfrak{L}$ is also characterized in terms of connections.