I have added some discussion (here) of the super Lie algebra of multilinear maps $V^{\otimes_n} \to V$ for any finite-dimensional vector space $V$, following Palmkvist 13, 3.1, Lavau-Palmkvist 19, 2.4 (my proof of the super-Jacobi identity remains incomplete, despite the lengthy computation).

The discussion culminates in the observation that “embedding tensors” are equivalently elements of square=0 and degree = -1 in this algebra, and hence are equivalently the datum to turn it into a dg-Lie algebra – supposedly (up to some restrictions and extensions) the “tensor hierarchy” induced by the embedding tensor. Which is a neat observation.

This super Lie algebra ought to have some standard name? Palmkvist attributes it to

- Isaiah L. Kantor,
*Graded Lie algebras*, Trudy Sem. Vektor. Tenzor. Anal 15 (1970): 227-266.

but I haven’t found an actual copy of this article yet.

]]>added the statement (here) that super Lie algebras equipped with a lift to $\mathbb{Z}$-grading and with a choice of square-0 element in degree -1 are equivalently dg-Lie algebras

]]>added doi for:

- Victor Kac,
*Lie superalgebras*, Advances in Math. 26 (1977), no. 1, 8–96 (doi:10.1016/0001-8708(77)90017-2)

added some minimum cross-link with *Whitehead product*

I have expanded the Definition-section at *super Lie algebra*, stating more variants of equivalent definitions. This is material taken from the more comprehensive lecture notes at *geometry of physics – superalgebra*.

added a few references and links to super Lie algebra

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