Graded Lie superalgebras from embedding tensors
Sylvain Lavau, Jakob Palmkvist
TL;DR
The paper addresses the problem of relating two graded Lie superalgebra constructions arising from embedding tensors: Kantor’s universal graded Lie superalgebra and tensor hierarchy (or tensor hierarchy) algebras built from an embedding tensor. It shows that, under the assumptions that $\mathfrak{g}$ is simple, $V$ is faithful, and $\Theta\neq0$, the tensor hierarchy algebra $P(V[-1],T_{-1})$ is isomorphic to the canonical Lie superalgebra $L(\mathfrak{g},V,\Theta)$ after quotienting degree $3$ and higher, thereby unifying the two approaches. The authors develop a detailed account of Kantor’s prolongation, the reduced prolongation, and the role of embedding tensors in generating a differential graded Lie algebra via the inner differential $d_\Theta=[\Theta,\cdot]$, placing the construction in a broader DG-Lie algebra context. They further relate these structures to Lie–Leibniz triples, showing how a canonically associated Lie superalgebra arises and how the quadratic constraint on $\Theta$ induces a Leibniz structure on $V$, with potential implications for higher-algebraic formulations and the Coquecigrue problem.
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.