We show how various constructions of $\mathbb{ Z}$-graded Lie superalgebras are related to each other. These Lie superalgebras have a Lie algebra $\mathfrak{ g}$ as the subalgebra at degree 0, an odd $\mathfrak{ g}$-module V as the subspace at degree 1, and an embedding tensor as an element at degree -1. This is a linear map from V to $\mathfrak{ g}$ satisfying a quadratic constraint, which equips V with the structure of a Leibniz algebra.
Categories: math.RT hep-th math-ph math.MP math.RA