Embedding tensors and tensor hierarchy in super/dg-Lie theory
Embedding tensors and tensor hierarchy in super/dg-Lie theory
We spell out aspects of the formalization of the concept of embedding tensors and their tensor hierarchies in terms of super Lie algebras/dg-Lie algebras.
The super Lie algebra of multi-endomorphisms
The algebra of embedding tensors and their tensor hierarchies turns out to be neatly captured by structure found in or induced from the following super Lie algebra.
The following construction is briefly highlighted in Palmkvist 09, 2.3 Palmkvist 13, 3.1 (reviewed more clearly in Lavau-Palmkvist 19, 2.4) where it is attributed to Kantor 70:
Definition
(super Lie algebra of multi-endomorphisms)
Let be a finite-dimensional vector space over some ground field .
Define a -graded vector space
concentrated in degrees , recursively as follows:
For we set
For , the component space in degree is taken to be the vector space of linear maps from to the component space in degree :
Hence:
(1)
Consider then the direct sum of these component spaces as a super vector space with the even number/odd number-degrees being in super-even/super-odd degree, respectively.
On this super vector space consider a super Lie bracket defined recusively as follows:
For all we set
For and we set
(2)
Finally, for and we set
(3)
It remains to show that:
Proof
We proceed by induction:
By Remark we have that the super Jacobi identity holds for all triples with .
Now assume that the super Jacobi identity has been shown for triples and , for any . The following computation shows that then it holds for :
(Fine, but is this sufficient to induct over the full range of all three degrees?)
Example
For (1) we have that the bracket on in Def. restricts to
(by combining (3) with (2)).
This is the Lie bracket of the general linear Lie algebra , as indicated on the right in (1).
Embedding tensors
Definition
(embedding tensor)
Given
then an embedding tensor is a linear map
such that for all the following condition (“quadratic constraint”) is satisfied:
(4)
where on the left we have the Lie bracket of .
The idea of this definition goes back to Nicolai-Samtleben 00, with many followups in the literature on tensor hierarchies in gauged supergravity. The above mathematical formulation is due to Lavau 17.
Proposition
(embedding tensors are square-0 elements in )
Let be a ground field of characteristic zero.
An element in degree -1 of the super Lie algebra from Def. ,
which by Example is identified with a linear map
from to the general linear Lie algebra on , is square-0 precisely if it is an embedding tensor (Def. ), in that:
Here on the right, denotes the Lie bracket in , while denotes the canonical Lie algebra action of on .
Proof
By unwinding of the definition (2) and (3) and using again Example we compute as follows:
Tensor hierarchies
(embedding tensors induce tensor hierarchies)
In view of the relation between super Lie algebras and dg-Lie algebras (above), Prop. says that every choice of an embedding tensor for a faithful representation on a vector space induces a dg-Lie algebra .
According to Palmkvist 13, 3.1, Lavau-Palmkvist 19, 2.4 this dg-Lie algebra (or some extension of some sub-algebra of it) is the tensor hierarchy associated with the embedding tensor.