Graded Lie superalgebras from embedding tensors

TL;DR

This paper explores the relationships between different constructions of $ Z$-graded Lie superalgebras, highlighting the role of embedding tensors and Leibniz algebra structures in their formation.

Contribution

It introduces a unified perspective on graded Lie superalgebras using embedding tensors, connecting algebraic structures through quadratic constraints.

Findings

Establishes connections between various graded Lie superalgebra constructions.

Shows how embedding tensors induce Leibniz algebra structures.

Provides a framework linking Lie superalgebras, modules, and quadratic constraints.

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.