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In gauged supergravity the gauging of a subgroup of a global U-duality symmetry group (via a choice of embedding tensor) in general requires the existence of a hierarchy of differential p-form-fields to ensure the gauge covariance of the resulting field strengths. This is known as the tensor hierarchy (de Wit-Samtleben 05, deWit-Samtleben 08)


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:


(super Lie algebra of multi-endomorphisms)

Let VV be a finite-dimensional vector space over some ground field kk.

Define a \mathbb{Z}-graded vector space

MultEnd(V)Vect k , MultEnd(V) \;\in \; Vect_k^{\mathbb{Z}} \,,

concentrated in degrees 1\leq 1, recursively as follows:

For n=1n =1 we set

MultEnd(V) 1V. MultEnd(V)_{1} \;\coloneqq\; V \,.

For n0n \leq 0 \in \mathbb{Z}, the component space in degree n1n-1 is taken to be the vector space of linear maps from VV to the component space in degree nn:

MultEnd(V) n1Hom k(V,MultEnd(V) n). MultEnd(V)_{n-1} \;\coloneqq\; Hom_k( V, MultEnd(V)_n ) \,.


(1)MultEnd(V) 1 =V MultEnd(V) 0 =Hom k(V,V)=𝔤𝔩(V) MultEnd(V) 1 =Hom k(V,Hom k(V,V))Hom k(VV,V) MultEnd(V) 2 =Hom k(V,Hom k(V,Hom k(V,V)))Hom k(VVV,V) \begin{aligned} MultEnd(V)_1 & = V \\ MultEnd(V)_0 & = Hom_k(V,V) = \mathfrak{gl}(V) \\ MultEnd(V)_{-1} & = Hom_k(V, Hom_k(V,V)) \simeq Hom_k(V \otimes V, V) \\ MultEnd(V)_{-2} & = Hom_k(V, Hom_k(V, Hom_k(V,V))) \simeq Hom_k(V \otimes V \otimes V, V) \\ \vdots \end{aligned}

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 v 1,v 2MultEnd(V) 1=Vv_1, v_2 \in MultEnd(V)_1 = V we set

[v 1,v 2]=0. [v_1, v_2] \;=\; 0 \,.

For fMultEnd(V) n0f \in MultEnd(V)_{n \leq 0} and vMultEnd(V) 1=Vv \in MultEnd(V)_1 = V we set

(2)[f,v]f(v) [f, v] \;\coloneqq\; f(v)

Finally, for fMultEnd(V) deg(f)0f\in MultEnd(V)_{ deg(f) \leq 0 } and gMultEnd(V) deg(g)0g\in MultEnd(V)_{deg(g) \leq 0} we set

(3)[f,g] :v[f,g(v)](1) deg(f)deg(g)[g,f(v)] \begin{aligned} [f, g] & \colon\; v \;\mapsto\; [f, g(v)] - (-1)^{ deg(f) deg(g) } [ g, f(v) ] \\ \end{aligned}

By (2) the definition (3) is equivalent to

[[f,g],v]=[f,[g,v]](1) deg(f)deg(g)[g,[f,v]] [ [f,g],v ] \;=\; [f, [g,v] ] - (-1)^{ deg(f) deg(g) } [ g, [f,v] ]

Hence (3) is already implied by (2) if the bracket is to satisfy the super Jacobi identity

It remains to show that:


Def. indeed gives a super Lie algebra in that the bracket (3) satisfies the super Jacobi identity.


We proceed by induction:

By Remark we have that the super Jacobi identity holds for all triples f 1,f 2,f 3MultEnd(V)f_1, f_2, f_3 \in MultEnd(V) with deg(f 3)0deg(f_3) \geq 0.

Now assume that the super Jacobi identity has been shown for triples (f 1,f 2,f 3(v))(f_1, f_2, f_3(v)) and (f 1,f 3,f 2(v))(f_1, f_3, f_2(v)), for any vVv \in V. The following computation shows that then it holds for (f 1,f 2,f 3)(f_1, f_2, f_3):

[f 1,[f 2,f 3]](v) =[f 1,[f 2,f 3](v)](1) deg(f 1)(deg(f 2)+deg(f 3))[[f 2,f 3],f 1(v)] =[f 1,[f 2,f 3(v)]] =(1) deg(f 2)deg(f 3)[f 1,[f 3,f 2(v)]] =(1) deg(f 1)(deg(f 2)+deg(f 3))[f 2,[f 3,f 1(v)]] =+(1) deg(f 1)(deg(f 2)+deg(f 3))+deg(f 2)deg(f 3)[f 3,[f 2,f 1(v)]] =[f 1,[f 2,f 3(v)]](1) deg(f 1)deg(f 2)[f 2,[f 1,f 3(v)]] =(1) deg(f 2)deg(f 3)([f 1,[f 3,f 2(v)]](1) deg(f 1)deg(f 3)[f 3,[f 1,f 2(v)]]) =(1) deg(f 1)(deg(f 2)+deg(f 3))([f 2,[f 3,f 1(v)]](1) deg(f 1)deg(f 3)[f 2,[f 1,f 3(v)]]) =+(1) deg(f 1)deg(f 2)+deg(f 1)deg(f 3)+deg(f 2)deg(f 3)([f 3,[f 2,f 1(v)]](1) deg(f 1)deg(f 2)[f 3,[f 1,f 2(c)]]) =+(1) deg(f 1)deg(f 2)(+[f 2,[f 1,f 3(v)]][f 2,[f 1,f 3(v)]]=0) =+(1) deg(f 1)deg(f 3)+deg(f 2)deg(f 3)([f 3,[f 1,f 2(v)]][f 3,[f 1,f 2(v)]]=0) =[[f 1,f 2],f 3(v)] =(1) deg(f 2)deg(f 3)[[f 1,f 3],f 2(c)] =+(1) deg(f 1)deg(f 2)[f 2,[f 1,f 3](v)] =(1) deg(f 3)(deg(f 1)+deg(f 2))[f 3,[f 1,f 2](v)] =[[f 1,f 2],f 3](v)+(1) deg(f 1)deg(f 2))[f 2,[f 1,f 3]](v) \begin{aligned} [f_1, [f_2, f_3] ] (v) & = [ f_1, [f_2, f_3](v) ] - (-1)^{deg(f_1)(deg(f_2) + deg(f_3))} [ [ f_2, f_3 ], f_1(v) ] \\ & = [ f_1, [ f_2, f_3(v) ] ] \\ & \phantom{=} - (-1)^{deg(f_2)deg(f_3)} [ f_1, [ f_3, f_2(v) ] ] \\ & \phantom{=} - (-1)^{deg(f_1)(deg(f_2) + deg(f_3))} [ f_2, [ f_3, f_1(v) ] ] \\ & \phantom{=} + (-1)^{deg(f_1)(deg(f_2) + deg(f_3)) + deg(f_2)deg(f_3)} [ f_3, [ f_2, f_1(v) ] ] \\ & = [ f_1, [ f_2, f_3(v) ] ] - (-1)^{deg(f_1) deg(f_2)} { \color{green} [ f_2, [ f_1, f_3(v) ] ] } \\ & \phantom{=} - (-1)^{deg(f_2) deg(f_3)} \big( [ f_1, [ f_3, f_2(v) ] ] - (-1)^{deg(f_1) deg(f_3)} { \color{orange} [ f_3, [ f_1, f_2(v) ] ] } \big) \\ & \phantom{=} - (-1)^{deg(f_1)(deg(f_2) + deg(f_3))} \big( [ f_2, [ f_3, f_1(v) ] ] - (-1)^{deg(f_1)deg(f_3)} { \color{blue} [ f_2, [ f_1, f_3(v) ] ] } \big) \\ & \phantom{=} + (-1)^{deg(f_1) deg(f_2 ) + deg(f_1) deg(f_3) + deg(f_2) deg(f_3)} \big( [ f_3, [ f_2, f_1(v) ] ] - (-1)^{deg(f_1) deg(f_2)} { \color{cyan} [ f_3, [ f_1, f_2(c) ] ] } \big) \\ & \phantom{=} + (-1)^{deg(f_1) deg(f_2)} \big( \underset{ = 0 }{ \underbrace{ + { \color{green} [ f_2, [ f_1, f_3(v) ] ] } - { \color{blue} [ f_2, [ f_1, f_3(v) ] ] } } } \big) \\ & \phantom{=} + (-1)^{deg(f_1) deg(f_3) + deg(f_2) deg(f_3)} \big( \underset{ = 0 }{ \underbrace{ { \color{orange} [ f_3, [ f_1, f_2(v) ] ] } - { \color{cyan} [ f_3, [ f_1, f_2(v) ] ] } } } \big) \\ & = \big[ [f_1, f_2], f_3(v) \big] \\ & \phantom{=} - (-1)^{ deg(f_2) deg(f_3) } \big[ [f_1, f_3], f_2(c) \big] \\ & \phantom{=} + (-1)^{ deg(f_1) deg(f_2) } \big[ f_2, [f_1, f_3](v) \big] \\ & \phantom{=} - (-1)^{ deg(f_3)( deg(f_1) + deg(f_2) ) } \big[ f_3, [f_1, f_2](v) \big] \\ & = \big[ [f_1, f_2], f_3 \big](v) + (-1)^{deg(f_1)deg(f_2))} \big[ f_2, [f_1, f_3] \big](v) \end{aligned}

(Fine, but is this sufficient to induct over the full range of all three degrees?)


For f,gMultEnd(V) 0=Hom k(V,V)f,g \in MultEnd(V)_0 = Hom_k(V,V) (1) we have that the bracket on MultEnd(V)MultEnd(V) in Def. restricts to

[f,g](v)=[f,g(v)][g,f(v)]=f(g(v))g(f(v)) [f,g](v) \;=\; [f,g(v)] - [g,f(v)] \;=\; f(g(v)) - g(f(v))

(by combining (3) with (2)).

This is the Lie bracket of the general linear Lie algebra 𝔤𝔩(V)\mathfrak{gl}(V), as indicated on the right in (1).

Embedding tensors


(embedding tensor)


then an embedding tensor is a linear map

Θ:V𝔤 \Theta \;\colon\; V \longrightarrow \mathfrak{g}

such that for all v iVv_i \in V the following condition (“quadratic constraint”) is satisfied:

(4)[Θ(v 1),Θ(v 2)]=Θ(ρ Θ(v 1)(v 2)), [\Theta(v_1), \Theta(v_2)] \;=\; \Theta \big( \rho_{\Theta(v_1)}(v_2) \big) \,,

where on the left we have the Lie bracket of 𝔤\mathfrak{g}.

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.


(Leibniz algebra-structure)

The “quadratic constraint” (4) implies (see this Prop.) that the product

(5)VV V (v 1,v 2) v 1v 2ρ Θ(v 1)(v 2) \array{ V \otimes V &\overset{ }{\longrightarrow}& V \\ (v_1, v_2) &\mapsto& v_1 \cdot v_2 \mathrlap{ \;\coloneqq\; \rho_{\Theta(v_1)}(v_2) } }

makes (the underlying vector space of) VV a Leibniz algebra. Conversely, if a Leibniz algebra structure “\cdot” on VV is already given, we may ask that it coincides with this one induced from the embedding tensor, a condition then called the linear constraint:

(6)v 1v 2=ρ Θ(v 1)(v 2). v_1 \cdot v_2 \;=\; \rho_{\Theta(v_1)}(v_2) \,.

With respect to this induced Leibniz algebra structure, hence equivalently with the “linear constraint” (6) understood, the “quadratic constraint” (4) equivalently says that the embedding tensor is a homomorphism of Leibniz algebras (using that Lie algebras are special cases of a Leibniz algebras):

[Θ(v 1),Θ(v 2)]=Θ(v 1v 2). [\Theta(v_1), \Theta(v_2)] \;=\; \Theta(v_1 \cdot v_2) \,.

(embedding tensors are square-0 elements in MultEnd(V)MultEnd(V))

Let kk be a ground field of characteristic zero.

An element in degree -1 of the super Lie algebra MultEnd(V)MultEnd(V) from Def. ,

ΘMultEnd(V) 1Hom k(V,𝔤𝔩(V)), \Theta \in MultEnd(V)_{-1} \simeq Hom_{k}(V, \mathfrak{gl}(V)) \,,

which by Example is identified with a linear map

Θ:V𝔤𝔤𝔩(V) \Theta \;\colon\; V \longrightarrow \mathfrak{g} \coloneqq \mathfrak{gl}(V)

from VV to the general linear Lie algebra on VV, is square-0 precisely if it is an embedding tensor (Def. ), in that:

[Θ,Θ]=0AAAAAA[Θ(v 1),Θ(v 2)]=Θ(ρ Θ(v 1)(v 2)). [\Theta, \Theta] \;=\; 0 \phantom{AAA} \Leftrightarrow \phantom{AAA} [\Theta(v_1), \Theta(v_2) ] \;=\; \Theta( \rho_{\Theta(v_1)}(v_2) ) \,.

Here on the right, [,][-,-] denotes the Lie bracket in 𝔤𝔩(V)\mathfrak{gl}(V), while ρ\rho denotes the canonical Lie algebra action of 𝔤𝔩(V)\mathfrak{gl}(V) on VV.


By unwinding of the definition (2) and (3) and using again Example we compute as follows:

(12[Θ,Θ](v 1))(v 2) =[Θ,Θ(v 1)](v 2) =[Θ,(Θ(v 1))(v 2)=ρ Θ(v 1)(v 2)][Θ(v 1),Θ(v 2)] =Θ(ρ Θ(v 1)(v 2))[Θ(v 1),Θ(v 2)] \begin{aligned} \big( \tfrac{1}{2} [\Theta,\Theta](v_1) \big)(v_2) & = [\Theta, \Theta(v_1)](v_2) \\ & = [\Theta, \underset{ \mathclap{ = \rho_{\Theta(v_1)}(v_2) } } { \underbrace{ (\Theta(v_1))(v_2) } } ] - [\Theta(v_1), \Theta(v_2)] \\ & = \Theta( \rho_{\Theta(v_1)}(v_2) ) - [ \Theta(v_1), \Theta(v_2) ] \end{aligned}

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 VV induces a dg-Lie algebra (MultEnd(V),[,],[Θ,])(MultEnd(V), [-,-], \partial \coloneqq [\Theta, -]).

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.


The concept originates with

See also

Possible relation to U-duality and M-theory:

Relation to Borcherds algebras:

Relation to Leibniz algebras:

A proposal for understanding tensor hierarchies as dg-Lie algebra/L-infinity algebra-refinenents of Leibniz algebras:

Last revised on May 25, 2020 at 06:02:23. See the history of this page for a list of all contributions to it.