Graded Lie superalgebras from embedding tensors

Abstract

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.

21 pages