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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeOct 20th 2009

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeJan 11th 2017

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.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeApr 29th 2019

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeDec 8th 2019

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeMay 25th 2020

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

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeMay 25th 2020
• (edited May 25th 2020)

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.

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeDec 11th 2020
• (edited Dec 11th 2020)