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Infinity-enhancing of Leibniz algebras

Sylvain Lavau∗1 and Jakob Palmkvist†2,3

1IMJ-PRG, Université Paris Diderot, Paris, France.2School of Science and Technology, Örebro University, Örebro, Sweden.

3Mathematical Sciences, Chalmers University of Technology and University of Gothenburg,Göteborg, Sweden.

August 17, 2020

Abstract

We establish a correspondence between infinity-enhanced Leibniz algebras, recently in-troduced in order to encode tensor hierarchies [1], and differential graded Lie algebras, whichhave been already used in this context. We explain how any Leibniz algebra gives rise toa differential graded Lie algebra with a corresponding infinity-enhanced Leibniz algebra.Moreover, by a theorem of Getzler, this differential graded Lie algebra canonically inducesan L∞-algebra structure on the suspension of the underlying chain complex. We explicitlygive the brackets to all orders and show that they agree with the partial results obtainedfrom the infinity-enhanced Leibniz algebras in [1].

Contents

1 Introduction 1

2 Embedding tensors and Leibniz algebras 32.1 From differential graded Lie algebras to Leibniz algebras . . . . . . . . . . . . 42.2 Embedding tensors and Lie-Leibniz triples . . . . . . . . . . . . . . . . . . . . 52.3 From Leibniz algebras to differential graded Lie algebras . . . . . . . . . . . . 72.4 Construction from a universal Z-graded Lie superalgebra . . . . . . . . . . . . 8

3 Infinity-enhanced Leibniz algebras 9

4 The L∞-algebra induced by the dgLa 134.1 L∞-algebras and Getzler’s theorem . . . . . . . . . . . . . . . . . . . . . . . . 134.2 Comparing the L∞-algebra structures . . . . . . . . . . . . . . . . . . . . . . 154.3 Example: the (1, 0) superconformal model in six dimensions . . . . . . . . . . 16

1 Introduction

Leibniz algebras (also known as Loday algebras or Leibniz-Loday algebras [2]) have duringthe last years attracted attention for their applications to gauge theories where the gaugevariation δxy of one parameter y with respect to another x is not antisymmetric under theinterchange of x and y (but where the symmetrisation leads to a parameter that acts triviallyon the fields). Such situations occur for example in the embedding tensor formulation of

∗lavau@math.univ-lyon1.fr (corresponding author)†jakob.palmkvist@oru.se

1

gauged supergravity [3–7] as well as in extended geometry [8–32], and give rise to a tensorhierarchy of gauge parameters, potentials and field strengths.

In [1] it was shown that the general gauge theory based on any Leibniz algebra leads toan extension of it to an infinity-enhanced Leibniz algebra, and it was proposed that this is themost general algebraic structure that enables the construction of the full tensor hierarchy. Itis a further development of the notion of an enhanced Leibniz algebra, originally introducedin [33, 34], to encode general gauge-invariant action functionals for coupled 1- and 2-formgauge fields with kinetic terms. In this paper we will show that any Leibniz algebra canbe ‘infinity-enhanced’ in the sense that it canonically gives rise to a differential gradedLie algebra concentrated in non-negative degrees, and that any such differential graded Liealgebra structure in turn implies the axioms of an infinity-enhanced Leibniz algebra. Thissettles the problem of existence of such a structure, and provides an alternative algebraicstructure encoding the tensor hierarchy that seems much simpler.

The idea of encoding the tensor hierarchy by a differential graded Lie algebra is notnew. It was originally proposed in [35, 36], and was based on the relation between tensorhierarchies and certain Borcherds-Kac-Moody superalgebras [29, 37–40]. The constructionof such a structure from a Leibniz algebra V , together with a Lie algebra g that acts onit and an embedding tensor Θ : V → g satisfying certain compatibility conditions wasgiven in [41]1. In the present paper, we will show that for any Leibniz algebra V there isa canonical choice of Lie algebra g, namely gl(V ), and an embedding tensor Θ : V → g

satisfying these compatibility conditions. With this choice, the construction in [41] leadsto the extension of any Leibniz algebra to a differential graded Lie algebra concentrated innon-negative degrees, without explicit reference to a Lie algebra or an embedding tensor asinitial data. As we will show, the axioms defining an infinity-enhanced Leibniz algebra followfrom this differential graded Lie algebra structure. Conversely, we will also show that anyinfinity-enhanced Leibniz algebra gives rise to a differential graded Lie algebra concentratedin non-negative degrees by the canonical choice of Lie algebra g and embedding tensorΘ : V → g. This correspondence encourages us to think that the most natural and usefulalgebraic structure encoding the tensor hierarchy is the differential graded Lie algebra definedby the Leibniz algebra V .

Due to the fact that the skew-symmetric part of the Leibniz product is not a Lie bracket(it does not satisfy the Jacobi identity), the gauge theory based on a Leibniz algebra V doesnot behave as the usual Yang-Mills gauge theory. Rather, covariance of the field strengthsis guaranteed at the cost of adding higher fields, which leads to the famous tensor hierarchy.The algebraic structure that the gauge parameters form turns out to be a (non-negativelygraded) L∞-algebra, rather than a Lie algebra. An L∞-algebra is a generalization of adifferential graded Lie algebra, where the Jacobi identity is weakened to be satisfied onlyup to homotopy. This implies that higher brackets have to be introduced, satisfying ‘higherJacobi identities’. The number of brackets can be infinite if the chain complex hostingthe L∞-algebra structure is not bounded (see [42, 43] for an introduction to this topic,and [28, 29, 32, 44–50] for more recent reviews and applications). Hence, it is tempting toconstruct an L∞-algebra from a Leibniz algebra V . Such a construction was presented in [53],but it does not give back V itself when V is a Lie algebra. In [1], a construction from aninfinity-enhanced Leibniz algebra was given, but not pushed further than to the 4-bracket.In this paper however, using the fact that any Leibniz algebra canonically gives rise to adifferential graded Lie algebra, we provide explicit formulas for the brackets of this L∞-algebra, to all orders. These formulas are obtained using a theorem by Getzler [51], whichis a special case of a more general result obtained by Fiorenza and Manetti [52]. This factis another argument in favor of using differential graded Lie algebras over infinity-enhancedLeibniz algebras to encode tensor hierarchies. The result also has the following consequence:that the skew-symmetric part of the Leibniz product of any Leibniz algebra V can be lifted

1Unfortunately, the differential graded Lie algebra in [41] was also called ‘tensor hierarchy algebra’, but thereis an important difference: the tensor hierarchy algebra in [35] is a priori not a differential graded Lie algebra,but a Z-graded Lie superalgebra with a subspace at degree −1 accommodating all possible embedding tensorssatisfying the representation constraint. Restricting it to a one-dimensional subspace spanned by one particularembedding tensor leads to the differential graded Lie algebra called ‘tensor hierarchy algebra’ in [41].

2

to an L∞-algebra structure in a canonical way, such that if the Leibniz product is fullyskew-symmetric (i.e., if V is a Lie algebra), then this L∞-extension is V itself.

The content of this paper can be summarized in the following diagram:

differential gradedLie algebra

Leibniz algebra L∞-algebra

infinity-enhancedLeibniz algebra

Section 3 of [1]

Section 2 (and [41])

Section 3

Section 6 of [1]

Section 4 (and [51])

Dashed lines symbolize the results obtained in [1], which were only partial: first, it was notshown that any Leibniz algebra induces an infinity-enhanced Leibniz algebra of order higherthan 2 [34]; second, the associated L∞-algebra was only given up to the 4-brackets. Onthe other hand, the solid lines symbolize complete and canonical derivations that we willdescribe2. First, in Section 2, we will explain the relation between Leibniz algebras and dif-ferential graded Lie algebras, and show that any Leibniz algebra V canonically gives rise to adifferential graded Lie algebra, which, in turn, induces a Leibniz product on V that coincideswith the original one. Second, in Section 3 we explain in detail the correspondence betweendifferential graded Lie algebras and infinity-enhanced Leibniz algebras. It is functorial, butit does not define an equivalence of categories. Finally, in Section 4, we provide Getzler’stheorem and the explicit formulas for the brackets of the L∞-algebra that was investigatedin [1]. We end Section 4 with an application to the tensor hierarchy appearing in the (1, 0)superconformal model in six dimensions [54].

To conclude, given that there is a correspondence between differential graded Lie algebrasconcentrated in non-negative degrees and infinity-enhanced Leibniz algebras, and althoughthe latter notion arises more directly in the approach of [1], we see several advantages torather use the former to mathematically encode tensor hierarchies:

• the algebraic structure of differential graded Lie algebras is much simpler;

• any Leibniz algebra induces such a structure, in a canonical way;

• such a structure induces an L∞-algebra, in a canonical and explicit way.

Moreover, the differential graded Lie algebra can in many interesting cases be seen as comingfrom a tensor hierarchy algebra [35] (see footnote 1) which represents an intriguing class ofnon-contragredient Lie superalgebras [55]. Beyond the differential graded Lie algebra struc-ture, the tensor hierarchy algebras seem to possess crucial information about supergravityand extended geometry [36, 56–58].

2 Embedding tensors and Leibniz algebras

A Leibniz algebra is a vector space V equipped with a bilinear operation ◦ satisfying thederivation property, or Leibniz identity [2]:

x ◦ (y ◦ z) = (x ◦ y) ◦ z + y ◦ (x ◦ z) (2.1)

2The solid lines would correspond to functors in the language of category theory. Then, given the diagram,there would exist a canonical functor from the category of Leibniz algebras to the category of L∞-algebras (thatrestricts to the identity functor on the full subcategory of Lie algebras). However we do not address this questionexplicitly in this paper, because that would significantly increase its length and obscure our original motivation.See also [53] for another point of view on this question.

3

for all x, y, z ∈ V . We can split the product ◦ of a Leibniz algebra V into its symmetric part{. , .} and its skew-symmetric part [ . , . ]:

x ◦ y = {x, y} + [x, y] (2.2)

where{x, y} =

12

(x ◦ y + y ◦ x

)and [x, y] =

12

(x ◦ y − y ◦ x

)(2.3)

for any x, y ∈ V . As a consequence of (2.1), the Leibniz product is a derivation of bothbrackets.

The subspace U ⊆ V generated by the set of elements of the form {x, x} contains allsymmetric elements of the form {x, y}, since they can always be written as a sum of squares.Using (2.1), one can check that U is an ideal of V with respect to the Leibniz product, i.e.,V ◦ U ⊆ U . We call this subspace the ideal of squares of V . By (2.1), the left action of U onV is trivial:

U ◦ V = 0. (2.4)

We say that this ideal is central, in the sense that it is included in the center Z of V (withrespect to the Leibniz product),

Z ={

x ∈ V∣∣ x ◦ y = 0 for all y ∈ V

}. (2.5)

An important remark here is that even if the bracket [ . , . ] is skew-symmetric, it does notsatisfy the Jacobi identity since, using (2.1), we have

[x, [y, z]

]+

[y, [z, x]

]+

[z, [x, y]

]= −

13

({x, [y, z]

}+

{y, [z, x]

}+

{z, [x, y]

}), (2.6)

which is not necessarily zero for all x, y, z ∈ V . Hence the skew-symmetric bracket [ . , . ] isnot a Lie bracket, but since the left hand side of (2.6) (the Jacobiator) takes values in theideal of squares U , its action on V is trivial.

2.1 From differential graded Lie algebras to Leibniz algebras

An important class of examples of Leibniz algebras come from differential graded Lie algebras,or dgLa for short. A differential graded Lie algebra

T = · · · ⊕ T−2 ⊕ T−1 ⊕ T0 ⊕ T1 ⊕ T2 ⊕ · · · (2.7)

is in fact a Z-graded Lie superalgebra (and not a Lie algebra, as the term ‘differential gradedLie algebra’ unfortunately suggests), where the Z-grading is consistent with the Z2-grading.This means that it is equipped with a bilinear bracket which is graded skew-symmetric andsatisfying the graded Jacobi identity:

Ja, bK = −(−1)ℓ(a)ℓ(b)Jb, aK, (2.8)

Ja, Jb, cKK = JJa, bK, cK + (−1)ℓ(a)ℓ(b)Jb, Ja, cKK, (2.9)

where ℓ(a) denotes the Z-degree of a homogeneous element a ∈ Tℓ(a). As a differential gradedLie algebra, T is in addition equipped with a differential

∂ =(∂i : Ti+1 → Ti

)i∈Z, (2.10)

which is an odd derivation of T that squares to zero. The odd derivation property meansthat ∂ acts by the Leibniz rule

∂(Ja, bK

)= J∂(a), bK + (−1)ℓ(a)Ja, ∂(b)K. (2.11)

One can then define the following degree −1 derived bracket on T1 [59]:

x ◦ y ≡ J∂(x), yK (2.12)

4

for any x, y ∈ T1. The Jacobi identity (2.9) and the Leibniz rule (2.11) together ensure thatthis product is a derivation of itself, i.e., that it is a Leibniz product on T1.

In the category of graded vector spaces, the grading of T can be shifted by −1 by usingthe suspension operator s. It is defined as follows:3

(sT )i ≡ Ti+1 (2.13)

In the present paper, since using too much mathematical notations could be obfuscating, wewill not write s(a) for the suspension of an element a ∈ T , but we will stick to the notation a.However, to avoid any confusion between the grading on T and the grading on sT , we chooseto denote by ℓ(a) the degree of a as seen as an element of T , and |a| the degree of a as seenas an element of sT . Hence:

|a| = ℓ(a) − 1 (2.14)

From now on, we assume (if nothing else explicitly stated) that the differential gradedLie algebras are concentrated in non-negative degrees, i.e., T = T0 ⊕ T1 ⊕ T2 ⊕ · · · . We alsodefine T to be the subspace of T that consists of strictly positively graded elements, i.e.,

T = T1 ⊕ T2 ⊕ · · · (2.15)

We call dgLa>0 the category of non-negatively graded differential graded Lie algebras, withtheir associated morphisms. Then, we may identify the differential ∂ with the adjoint actionof an element, which we denote by Θ, in an additional one-dimensional subspace T−1 thatwe may add to T by a direct sum, such that

JΘ, ΘK = 0. (2.16)

This quadratic constraint on Θ ensures that the operator JΘ, −K : T• → T•−1 squares tozero. The derivation property (2.11) translates to a Jacobi identity that Θ has to satisfy:

JΘ, Ja, bKK = JJΘ, aK, bK + (−1)ℓ(a)Ja, JΘ, bKK (2.17)

for any a, b ∈ T . Hence the differential graded Lie algebra structure on T0 ⊕ T1 ⊕ · · · canbe interpreted as a Z-graded Lie superalgebra structure on T−1 ⊕ T0 ⊕ T1 ⊕ · · · . With thisidentification, (2.12) implies

JJΘ, xK, yK = x ◦ y. (2.18)

The reverse construction, i.e., starting from a Leibniz algebra and building a differentialgraded Lie algebra such that the induced Leibniz product (2.18) coincides with the originalone, has been given in [41]. It relies on the notion of Lie-Leibniz triples and has been inspiredby the gauging procedure in supergravity, in particular by the embedding tensor formalismand the correspondence between embedding tensors and Leibniz algebras [30, 53]. This willbe the topic of the next two subsections.

2.2 Embedding tensors and Lie-Leibniz triples

An example of a Leibniz algebra arises in the embedding tensor approach to gauged super-gravity [3–7]. In the gauging procedure, a subgroup H of the global duality symmetry groupG is promoted to a local symmetry group. Covariance under G can be maintained with thehelp of an embedding tensor, which is a linear map Θ : V → g from a g-module V (usuallyfundamental) to the Lie algebra g of the original global symmetry group G, describing howH is embedded into G. For consistency, this embedding tensor has to satisfy a representationconstraint and a quadratic constraint. One may now define a Leibniz algebra structure on V

by using the action of h = Im(Θ) ⊆ g on V :

x ◦ y ≡ ρΘ(x)(y), (2.19)

3Obviously, the suspension operator has an inverse. It is called the desuspension operator and is denoted s−1.

5

where the representation ρ : g → gl(V ) defines the g-module structure on V . The Leibnizidentity is then a consequence of the quadratic constraint

Θ(ρΘ(x)(y)

)=

[Θ(x), Θ(y)

]g

(2.20)

since

x ◦ (y ◦ z) − y ◦ (x ◦ z) = ρΘ(x)

(ρΘ(y)(z)

)− ρΘ(y)

(ρΘ(x)(z)

)

=[ρΘ(x), ρΘ(y)

]gl(V )

(z)

= ρ[Θ(x),Θ(y)]g(z)

= ρΘ(ρΘ(x)(y))(z)

= ρΘ(x◦y)(z) = (x ◦ y) ◦ z. (2.21)

Conversely, one can show that a Leibniz algebra (V, ◦) canonically defines an embeddingtensor, since any vector space V is a g-module with g ≡ gl(V ) and ρ being the identity map.We then let Θ be the map sending any element x ∈ V to its associated left-multiplicationoperator xL:

Θ : V −−−−−−→ gl(V ) (2.22)

x 7−−−−−−→ xL : y 7→ x ◦ y.

The image of the map Θ is a subspace h of gl(V ) generated by all endomorphisms of the typexL for x ∈ V . It turns out to be stable under the Lie bracket of endomorphisms [ . , . ]gl(V ),since the Leibniz identity (2.1) implies that [xL, yL]gl(V ) = (x ◦ y)L. This equality can berewritten as the condition that Θ : V → gl(V ) is a homomorphism of Leibniz algebras:

Θ(x ◦ y) =[Θ(x), Θ(y)

]gl(V )

(2.23)

where one considers(gl(V ), [ . , . ]gl(V )

)as a Leibniz algebra with fully skew-symmetric prod-

uct. One can thus restrict the action of gl(V ) on V to h, turning V into a h-module. TheLeibniz product is then compatible with the embedding tensor in the same sense as in (2.19):

Θ(x)(y) = x ◦ y. (2.24)

Inserting (2.24) into (2.23) implies that Θ satisfies the quadratic constraint (2.20). Moreover,from (2.23) we see that the kernel of Θ coincides with the center Z of V . This means thatthe gauge algebra h = Im(Θ) is isomorphic to the Lie algebra V

/Z .

We have seen that any embedding tensor defines a Leibniz algebra, and that any Leibnizalgebra defines an embedding tensor in a canonical way. One can generalize the discussionand allow for more general Lie algebras and embedding tensors, that would satisfy the twoconstraints (2.23) and (2.24). These are consistency conditions that encode compatibilitybetween the embedding tensor and the Leibniz algebra structure [41].

Definition. A Lie-Leibniz triple is a triple (g, V, Θ) where:

1. g is a Lie algebra,

2. V is a g-module equipped with a Leibniz algebra structure ◦, and

3. Θ : V → g is a linear map called the embedding tensor, that satisfies two compatibilityconditions. The first one is the linear constraint:

x ◦ y = ρΘ(x)(y) (2.25)

where ρ : g → gl(V ) denotes the action of g on V . The second one is called thequadratic constraint:

Θ(x ◦ y) =[Θ(x), Θ(y)

]g

(2.26)

where [ . , . ]g is the Lie bracket on g.

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The two conditions that Θ has to satisfy guarantee the compatibility between the Leibnizalgebra, g-module structure on V and the Lie bracket of g. In particular, the quadraticconstraint (2.26) says that Θ has to be a homomorphism of Leibniz algebras, consideringthe Lie algebra g as a Leibniz algebra whose product is fully skew-symmetric. Given thesedata, we deduce that h ≡ Im(Θ) is a Lie subalgebra of g. We call it the gauge algebra of theLie-Leibniz triple (g, V, Θ). Moreover, the linear constraint (2.25) implies that the kernel ofΘ is contained in the center,

Ker(Θ) ⊆ Z. (2.27)

We have equality when the representation of h on V is faithful (as in the case of the embeddingtensor canonically defined by any Leibniz algebra as in (2.22)). On the other hand, thequadratic constraint (2.26) implies that the ideal of squares U is contained in the kernel,

U ⊆ Ker(Θ), (2.28)

but a priori we do not have equality here either.

2.3 From Leibniz algebras to differential graded Lie algebras

In [41] it was shown that a Lie-Leibniz triple (g, V, Θ) gives rises to a dgLa

T = T0 ⊕ T1 ⊕ T2 ⊕ · · · , (2.29)

which satisfies several properties that makes it unique (up to equivalence). Among otherproperties [41], the differential graded Lie algebra

(T, J . , . K, ∂

)induced by the Lie-Leibniz

triple (g, V, Θ) is such that

1. the subalgebra T0 is equal to h as a Lie algebra,

2. the space T1 is equal to V ,

3. for any i > 1, the space Ti is a g-module with a representation ρ given by

Jg, aK ≡ ρg(a), (2.30)

for any g ∈ h = T0 and a ∈ Ti,

4. the differential satisfies

∂(x) ≡ Θ(x) (2.31)

for any x ∈ T1.

The combination of (2.30) and (2.31), together with the fact that the embedding tensor Θsatisfies (2.25), imply the following identity:

J∂(x), yK = JΘ(x), yK = ρΘ(x)(y) = x ◦ y (2.32)

for any x, y ∈ T1 = V . Hence, the Leibniz product defined by the dgLa structure as in(2.18) coincides with the original one, and this is a particular feature of this differentialgraded Lie algebra. Since any Leibniz algebra V canonically induces a Lie-Leibniz triple(gl(V ), V, Θ : x 7→ xL), we deduce that any Leibniz algebra gives rise to such a dgLa.

Schematically, the construction in [41] of the dgLa T associated to the Lie-Leibniz triple(g, V, Θ) goes as follows: one sets X0 = V , then one chooses X1 by noticing that the sym-metric bracket { . , . } : S2(V ) → V has a kernel, which is a h-module (where h = Im(Θ)),but not necessarily a g-module. Let K ⊆ Ker

({ . , . }

)be the biggest g-module contained in

Ker({ . , . }

). Then one sets X1 = S2(V )

/K and X1 naturally inherits the quotient g-module

structure. Usually, S2(V ) is completely reducible with respect to the action of g so thatthe quotient is a direct sum of irreducible representations. In that case, X1 is the smallestg-submodule of S2(V ) through which the symmetric bracket { . , . } : S2(V ) → V factors:

7

X1

S2(V ) V

P1

{. , .}

Here P1 is the projection on X1, and the vertical arrow is uniquely defined by requiringthat the diagram commutes, given the projection P1. Elements of X1 are considered tohave degree +1. In gauged supergravity, X1 is the space in which 2-form potentials takevalues and is determined by the representation constraint. More generally, in the hierarchy,Xp is the g-module in which (p + 1)-form potentials take values. We extend P1 to a mapP1 : S•(V ) → S•−2(V ) ⊗ X1 and we set X2 to be the cokernel of this map when appliedto S3(V ):

X2

V ⊗ X1 X1

S3(V ) S2(V ) V

P1

P2

P1

{. , .}

Here P2 is the quotient map onto X2. We extend it to P2 : S(V ⊕X1) → S(V ⊕X1)⊗X2

so that we can define X3 as the cokernel of P1 + P2 : S2(V ) ⊗ X1 → V ⊗ X2 ⊕ X1 ∨ X1.We repeat the same construction at each iteration, and we obtain a tower of graded spacesX = X0 ⊕ X1 ⊕ X2 ⊕ · · · , together with their respective projections: Pi : S2(X) → Xi.The detailed construction of the hierarchy can be found in [41]. Notice that if V is a Liealgebra, then the kernel of the symmetric bracket coincides with S2(V ) and then X1 is thezero vector space. This implies in turn that all other spaces Xp are zero.

The differential graded Lie algebra T is obtained as follows: one first shifts the degreeof each space by +1, i.e., T = s−1X . In particular we have T1 = V and Ti = Xi−1 forany i > 1. Then, we add h = Im(Θ) at degree 0, i.e., T0 = h, with its associated Liebracket. After some technical considerations, the projection maps Pi induce a graded Liebracket J . , . K on T = T0 ⊕ T1 ⊕ T2 ⊕ · · · . One can show that the vertical arrows in theabove diagram canonically define a differential ∂ on T such that the triple

(T, J . , . K, ∂

)is a

differential graded Lie algebra [41]. Notice that the presence of T0 = h is crucial so that theLeibniz rule (2.11) is satisfied. If V is a Lie algebra, then T = T0 ⊕ T1 and there is no spaceof degree higher than or equal to 2.

2.4 Construction from a universal Z-graded Lie superalgebra

Another way of constructing a dgLa associated to a Leibniz algebra V (corresponding tothe canonical Leibniz-Lie triple) is to first consider the universal Z-graded Lie superalgebraU(V ) associated to V , where V is considered as a Z2-graded vector space with a trivialeven part [35,60,61]. This means that we set U1 = V and define vector spaces U−i for i > 0recursively so that U−i = Hom(V, U−i+1), consisting of all linear maps V → U−i+1. Then the

8

direct sum⊕

i>0 U−i is a consistently Z-graded Lie superalgebra with the Lie superbracketdefined recursively by

Ja, bK(x) = Ja, b(x)K − (−1)ℓ(a)ℓ(b)Jb, a(x)K, (2.33)

where ℓ(a) is the Z-degree of a homogeneous element a ∈ Uℓ(a), and Ja, xK should be readas a(x) if x ∈ U1 = V and ℓ(a) 6 0. In particular, this means that the subalgebra U0 isgl(V ). The Jacobi identity can then be shown to hold by induction. We can extend thisLie superalgebra to positive degrees as well, by letting

⊕i>1 Ui be the free Lie superalgebra

generated by the odd vector space U1 = V . The Lie superbracket of an element at a positivedegree with an element at a non-positive degree can be defined by the Jacobi identity fromthe relations

Ja, xK = −Jx, aK = a(x) (2.34)

for x ∈ U1 = V and ℓ(a) 6 0. In this way we obtain a consistently Z-graded Lie superalgebra

U(V ) =⊕

i>0

U−i ⊕⊕

i>1

Ui =⊕

i∈Z

Ui (2.35)

for any vector space V . When V happens to be an algebra with a product x ◦ y, there is adistinguished element Θ in U−1 defined by JJΘ, xK, yK = x ◦ y for any x, y ∈ U1 = V , and thecondition that V be a Leibniz algebra is then equivalent to the condition JΘ, ΘK = 0.

Consider now the subalgebra R =⊕

i∈ZRi of U(V ) generated by V = U1 and Θ ∈ U−1.

At degree 0 in R, we have all linear maps of the form JΘ, xK = xL for x ∈ V , acting ony ∈ V by xL(y) = x ◦ y. The bracket of any such element xL with Θ does not give any newelements at degree −1 since

JΘ, xLK = JΘ, JΘ, xKK =12JJΘ, ΘK, xK = 0. (2.36)

Thus R−1 is one-dimensional, spanned by Θ, and R−i for i > 2 are trivial since JΘ, ΘK = 0.Thus the subalgebra R of U(V ) generated by V = U1 and Θ ∈ U−1 is a Lie superalgebraconcentrated in degrees > −1 with a one-dimensional subspace U−1. As we have seen inSection 2.1, it can then be identified with a dgLa concentrated in non-negative degrees. ThedgLa constructed from the canonical Lie-Leibniz triple in Section 2.3 may then be obtainedas a quotient. We leave the study of the precise relation for future work.

The tensor hierarchy algebra introduced in [35] is defined in a similar way, from theuniversal Z-graded Lie superalgebra associated to a g-module V , but in that construction therelevant subalgebra is not generated by V = U1 and a single element Θ ∈ U−1, but by V = U1

and a whole subspace of U−1, which is the g-module containing all allowed embedding tensorsaccording to the representation constraint. The tensor hierarchy algebra is then obtained byfactoring out the maximal ideal of this subalgebra contained in the subspaces at degree 2and higher. Choosing a particular embedding tensor Θ ∈ U−1 amounts to defining a Leibnizalgebra structure on V and restricting the tensor hierarchy algebra to a dgLa. This dgLathen coincides with the one constructed from the Lie-Leibniz triple (g, V, Θ) in Section 2.3up to possible differential ideals in the latter contained in the subspaces at degree strictlyhigher than 2.

3 Infinity-enhanced Leibniz algebras

In [33,34], the concept of an enhanced Leibniz algebra was introduced as a first step towardsa mathematical formalization of higher gauge theories. It did not allow for gauge fields ofform degree higher than 2 though, so that the notion of an infinity-enhanced Leibniz algebrawas eventually proposed in [1] as the most general structure encoding the tensor hierarchy.

Definition. An infinity-enhanced Leibniz algebra is defined as an N-graded vector space

X = X0 ⊕ X1 ⊕ X2 ⊕ · · · = X0 ⊕ X (3.1)

9

together with a Leibniz product ◦ : X0 ⊗ X0 → X0,

x ◦ (y ◦ z) = (x ◦ y) ◦ z + y ◦ (x ◦ z), (3.2)

a degree +1 graded symmetric product • : Xi ⊗ Xj → Xi+j+1 (i, j > 0),

a • b = (−1)|a||b|b • a (3.3)

(where |a| denotes the Z-degree of a homogeneous element a ∈ X|a|) and a linear mapD =

(Di : Xi+1 → Xi

)i>0 satisfying in addition the following axioms:

1. Du ◦ x = 0 (u ∈ X1, x ∈ X0),

2. D(x • y) = x ◦ y + y ◦ x (x, y ∈ X0),

3. D(x • (y • z)

)= (x ◦ y) • z + (x ◦ z) • y − (y ◦ z + z ◦ y) • x (x, y, z ∈ X0),

4. D(x[1•(x2]•u)

)= 2 x[2•D

(x1]•u

)+x[2•

(x1]•D(u)

)+[x1, x2]•u (x1, x2 ∈ X0, u ∈ X),

5. D(u • v) + (Du) • v + (−1)|u|u • D(v) = 0 (u, v ∈ X),

6. (−1)|a|a • (b • c) + (a • b) • c + (−1)|b||c|(a • c) • b = 0 (a, b, c ∈ X),

7. D(D(a)) = 0 (a ∈ X).

where in the latter condition, D is assumed to vanish on X0.A morphism of infinity-enhanced Leibniz algebras is a degree preserving linear map that

is compatible in the usual way with the products and the differentials. More precisely, fortwo infinity-enhanced Leibniz algebra (X, ◦,D, •) and (Y, ◦′,D′, •′), a morphism f : X → Y

satisfies the following three conditions (a, b ∈ X):

f(a ◦ b) = f(a) ◦′ f(b), f(a • b) = f(a) •′ f(b), f(D(a)

)= D′

(f(a)

). (3.4)

Let ∞−enLeib be the category of infinity-enhanced Leibniz algebras, with the above de-fined morphisms.

An important operator introduced in [1] is the generalized Lie derivative of an elementx of X0. It acts on the entirety of the chain complex X = X0 ⊕ X and is defined by thefollowing two equations:

Lx(y) ≡ x ◦ y (y ∈ X0), (3.5)

Lx(u) ≡ x • D(u) + D(x • u) (u ∈ X). (3.6)

It defines an action of X0 = V on each subspace Xi, and it turns out that the above axiomsimply that any previously defined operators ◦,D, • are covariant under the action of thisgeneralized Lie derivative. That is to say,

Lx(y ◦ z) = Lx(y) ◦ z + y ◦ Lx(z) (y, z ∈ X0), (3.7)

Lx

(D(u)

)= D

(Lx(u)

)(u ∈ X), (3.8)

Lx(a • b) = Lx(a) • b + a • Lx(b) (a, b ∈ X). (3.9)

Moreover, this generalized Lie derivative automatically satisfies a closure condition:[Lx, Ly

]= L[x,y] (3.10)

where, on the right hand side, [x, y] is the skew-symmetric part of the Leibniz product x ◦ y

(and the left hand side is just a commutator of linear maps).This section is devoted to showing that any dgLa T = T0 ⊕ T1 ⊕ T2 ⊕ · · · = T0 ⊕ T canon-

ically induces an infinity-enhanced Leibniz algebra structure on X = s T and, conversely,that any infinity-enhanced Leibniz algebra X = X0 ⊕ X1 ⊕ X2 ⊕ · · · canonically gives rise toa dgLa structure on T = T0 ⊕ s−1X , where T0 is the image of the embedding tensor definedin (2.22).

First, let us show that the axioms above follow from any dgLa T = T0 ⊕ T1 ⊕ T2 ⊕ · · ·(extended to a graded Lie algebra T−1 ⊕ T0 ⊕ T1 ⊕ · · · with a one-dimensional subspace T−1,

10

identifying the differential with a basis element Θ of T−1 squaring to zero, see (2.16)). Tothis end, for every i > 0, we set Xi = Ti+1, as well as the following operations:

x ◦ y ≡ JJΘ, xK, yK (x, y ∈ T1 = s−1X0), (3.11)

a • b ≡ (−1)ℓ(a)Ja, bK (a, b ∈ T = s−1X), (3.12)

D ≡ −JΘ, − K (3.13)

where elements on the left hand side are considered as elements of X , whereas elements onthe right hand side are considered as elements of T . It then follows that (3.3) is satisfied:

a • b = (−1)ℓ(a)Ja, bK

= −(−1)ℓ(b)ℓ(a)+ℓ(a)Jb, aK

= (−1)(ℓ(a)−1)(ℓ(b)−1)+ℓ(b)Jb, aK

= (−1)|a||b|b • a (3.14)

since |a| = ℓ(a) − 1 (and likewise for b). The identity (3.2) follows from the Leibniz rule(2.11) and the Jacobi identity (2.17), and it is easy to see that the other basic conditions on◦, • and D are satisfied. Let us now go through the additional six axioms.

Axiom 1: By (3.11) and (3.13) we have

Du ◦ x = JJΘ,DuK, xK = −JJΘ, JΘ, uKK, xK = −12JJJΘ, ΘK, uK, xK, (3.15)

which vanishes by (2.16).Axiom 2: Using (3.12) and (3.13), and recalling that ℓ(x) = 1, the left hand side of

Axiom 2 is JΘ, Jx, yKK. By the Jacobi identity (2.17) on T , we have

JΘ, Jx, yKK = JJΘ, xK, yK − Jx, JΘ, yKK = JJΘ, xK, yK + JJΘ, yK, xK. (3.16)

Using (3.11), this is equal to x ◦ y + y ◦ x, which is precisely the right hand side of Axiom 2.Axiom 3: Using (3.12) and (3.13), the left hand side of Axiom 3 is −JΘ, Jx, Jy, zKKK.

Then the Jacobi identity (2.17) implies

−JΘ, Jx, Jy, zKKK = −JJΘ, xK, Jy, zKK + Jx, JJΘ, yK, zKK − Jx, Jy, JΘ, zKKK

= −JJJΘ, xK, yK, zK − Jy, JJΘ, xK, zKK

+ JJJΘ, yK, zK, xK + JJJΘ, zK, yK, xK

= −Jx ◦ y, zK − Jy, x ◦ zK + Jy ◦ z, xK + Jz ◦ y, xK, (3.17)

where we used (3.11) between the second line and the last one. By using (3.12), the righthand side of (3.17) can be written as (x ◦ y) • z + (x ◦ z) • y − (y ◦ z + z ◦ y) • x, which is theright hand side of Axiom 3.

Axiom 4: Using (3.12) and (3.13), the left hand side of Axiom 4 is −JΘ, Jx[1, Jx2], uKKK.Then (2.17) implies

−JΘ, Jx[1, Jx2], uKKK = −JJΘ, x[1K, Jx2], uKK + Jx[1, JΘ, Jx2], uKKK

= −JJJΘ, x[1K, x2]K, uK − Jx[2, JJΘ, x1]K, uKK − Jx[2, JΘ, Jx1], uKKK

= −J[x1, x2], uK − 2Jx[2, JΘ, Jx1], uKKK − Jx[2, Jx1], JΘ, uKKK, (3.18)

which, by (3.12) and (3.13), gives back [x1, x2] • u + 2x[2 • D(x1] • u

)+ x[2 •

(x1] • D(u)

),

that is, the right hand side of Axiom 4.Axiom 5: Recalling that |u| = ℓ(u) − 1, the left hand side is equal to (−1)ℓ(u)+1 times

JΘ, Ju, vKK − JJΘ, uK, vK − (−1)ℓ(u)Ju, JΘ, vKK, (3.19)

which is zero according to (2.17).Axiom 6: Since ℓ(a • b) = ℓ(a) + ℓ(b) − 2, the left hand side is equal to (−1)ℓ(b)+1 times

Ja, Jb, cKK − JJa, bK, cK + (−1)ℓ(b)ℓ(c)JJa, cK, bK, (3.20)

11

which is zero according to the Jacobi identity (2.9).Axiom 7: The nilpotency of D follows from the nilpotency of Θ (by the Jacobi identity),

D2 = 12JJΘ, ΘK, −K = 0, hence turning (X,D) into a chain complex.

The identities (3.5) and (3.6) characterizing the generalized Lie derivative can also beunified in the dgLa setting. For any x ∈ X0 we set

Lx ≡ JJΘ, xK, − K. (3.21)

Then (3.5) is nothing but (3.11), whereas (3.6) is implied by the Jacobi identity (2.17).Covariance and closure conditions, which are proven from the Axioms in [1], follow herefrom the Jacobi identity on T . This construction is functorial, for every morphism of dgLaφ : T → T ′ canonically induces a morphism of infinity-Leibniz algebras fφ : X → X ′.Let F : dgLa>0 → ∞-enLeib be the functor that associates an infinity-enhanced Leibnizalgebra X to any non-negatively graded dgLa T .

Conversely, given an infinity-enhanced Leibniz algebra X = X0 ⊕ X1 ⊕ · · · = X0 ⊕ X,we will now show that one can canonically define a dgLa structure on some non-negativelygraded vector space T = T0 ⊕T1 ⊕· · · = T0 ⊕T . First, let us shift the degree of all subspacesby +1, i.e., T = s−1X . That is to say, Ti = Xi−1 for every i > 1, and in particularT1 = X0 = V . Let us set

Ja, bK ≡ (−1)|a|+1a • b, (3.22)

∂ ≡ −D, (3.23)

for a, b ∈ X and ∂ acting on s−1X . The nilpotency of the operator D implies that ∂ acts as adifferential on T . As it is,

(T1 ⊕T2 ⊕· · · , J . , .K, ∂

)is not a dgLa, since the Leibniz rule might

not be satisfied on a bracket of two elements in T1, since ∂(T1) = −D(X0) = 0. Therefore,at degree 0 we let T0 be the Lie algebra generated by the generalized Lie derivatives4, wherethe Lie bracket is given by the closure condition (3.10). The bracket between T0 and anyother Ti, for i > 1, is induced by the T0-module structure on Ti, as defined in (3.5) and (3.6):

JLx, aK ≡ Lx(a) (3.24)

for any x ∈ T1 and a ∈ T . We enforce the skew-symmetry by setting Ja, LxK = −Lx(a).Finally, the differential ∂ can be extended to T1 by setting

∂(x) ≡ Lx (3.25)

for any x ∈ T1.Then, the skew-symmetry of the graded bracket J . , . K on T (see (2.8)) is a consequence of

(3.3). Axiom 6 implies that the bracket satisfies the Jacobi identity (2.9) on T . Covarianceand closure conditions imply that the graded bracket J . , . K satisfies the Jacobi identity alsowhen any one or two elements in T0 is involved. The Jacobi identity on T0 is automaticallysatisfied because T0 is a Lie algebra. Proving the Leibniz rule (2.11) is a bit more involved.Axiom 2 is the Leibniz rule for two elements of T1. Axiom 5 is the Leibniz rule for twoelements of T . The Leibniz rule for one element x of T1 and any other element u of T isinduced by (3.6) and the definition of the differential ∂ : T1 → T0, x 7→ Lx. The last Leibnizidentities we need to check are those involving at least one element of T0. The Leibniz rulefor an element of T0 and any element in T is induced by the covariance of D. The Leibnizrule for an element of T0 and any element of T1 is induced by the covariance of ◦.

Hence, we have proven that (T, J . , . K, ∂) is a differential graded Lie algebra that is canon-ically induced from the data defining the infinity-enhanced Leibniz algebra. Also this con-struction is functorial, for any morphism of infinity-enhanced Leibniz algebras f : X → X ′

4By (3.5), we see that the generalized Lie derivatives are endomorphisms of gl(V ), so T0 can be seen as asub-Lie-algebra of gl(V ). Moreover, by (2.22), we have that Lx(y) = x ◦ y = xL(y) for every x, y ∈ V . Hence,Lx and xL define the same endomorphism on V , however it could well happen that for some x ∈ Z, the righthand side of (3.6) is not vanishing, implying that T0 may be bigger than V

/Z . This is consistent with the fact

that the map x 7−→ Lx defines an embedding tensor Θ : V → gl(V ). Then, by (2.27), we obtain again thatT0 = Im(Θ) ≃ V

/Ker(Θ) may have a bigger dimension than V

/Z .

12

canonically induces a morphism of dgLa φf : T → T ′. The correspondence is obvious on T ,and on T0 = {Lx | x ∈ X0}, the obvious definition is φf (Lx) = Lf(x). Then by (3.5)–(3.6)and the fact that f is a morphism in the category of infinity-enhanced Leibniz algebras,φf

(Jx, aK

)= Jφf (x), φf (a)K for every x ∈ T0 and a ∈ T . This proves that φf : T → T ′ is a

well-defined morphism in the category of dgLa. Let G : ∞-enLeib → dgLa>0 be the func-tor that associates a dgLa T to any infinity-enhanced Leibniz algebra X by the constructionabove.

Moreover, notice that according to the discussion in Section 2.1, the dgLa T can then becanonically extended to a dgLa T−1 ⊕T0 ⊕T1 ⊕· · · , where T−1 is a one-dimensional subspacespanned by the embedding tensor Θ. To conclude, since any non-negatively graded dgLaT induces a Leibniz product on T1 (see Section 2.1), and a compatible infinity-enhancedLeibniz algebra (this section), and since any Leibniz algebra V gives rise to a non-negativelygraded dgLa (see Section 2.3) whose induced Leibniz product coincides with the original one,we have an explicit construction of an infinity-enhanced Leibniz algebra starting from onlythe Leibniz algebra V (see the diagram in the introduction).

The question of the functoriality of both constructions is worth some attention. Onenotices that for any infinity-enhanced Leibniz algebra X , we have F (G(X)) = X , but notevery non-negatively graded dgLa satisfies G(F (T )) = T – and in general it is not true. Thisis due to the fact that F forgets the degree 0 part of T . Hence in full generality F and G donot define an equivalence of categories. However, the image of the functor G defines a fullsubcategory of dgLa>0, on which the composition of G and F is the identity, thus definingan equivalence of categories with ∞-enLeib. Then, non-negatively graded dgLa provide anatural way of coding infinity-enhanced Leibniz algebras, without any loss of data5. One ofthe advantages with the dgLa structure is that it gives rise to an L∞-algebra structure in acanonical and explicit way, as we will see in the next section.

4 The L∞-algebra induced by the dgLa

The link between tensor hierarchies and L∞-algebras goes back to [44, 45], where an L3-algebra structure was induced from the field strengths of a superconformal (1, 0) model insix dimensions [54]. We will come back to this example in the end of this section. Our aim isto show how the dgLa structure of the tensor hierarchy gives rise to a canonical L∞-algebrastructure. This result is a consequence of a theorem by Getzler [51], which in turn is a specialcase of a more general theorem by Fiorenza and Manetti [52]. We then show that the firstfew brackets of this L∞-algebra are the ones defined in [1].

This section also implies a more general result for Leibniz algebras: the skew-symmetricpart [ . , . ] of the Leibniz product is in general not a Lie bracket since it does not satisfythe Jacobi identity, see (2.6). The material presented in this section implies however thatthis bracket actually fits precisely in the L∞-algebra structure that is defined by Getzler’stheorem from the dgLa induced by the Leibniz algebra. This means in particular that theskew-symmetric part of the Leibniz product of any Leibniz algebra can be canonically liftedto an L∞-algebra structure (such that if the Leibniz algebra is a Lie algebra, then this lift istrivial in the sense that the L∞-algebra reduces to an Lie 1-algebra which is this Lie algebraitself). This result differs from the one presented in [53], where the L∞-algebra constructedfrom the Leibniz algebra V does not coincide with V itself when V is a Lie algebra.

4.1 L∞

-algebras and Getzler’s theorem

The notion of an L∞-algebra generalizes the notion of a differential graded Lie algebraby weakening the Jacobi identity, and allowing it to be satisfied only up to homotopy [42,43]. The precise definition of L∞-algebras can be found in many papers, see for example[48], and we will only recall the basics here. We emphasize that there are two different

5There are actually many more non-negatively graded dgLa than there are infinity-enhanced Leibniz algebras,because the degree 0 part of such a dgLa T is forgotten by the functor G. This opens the question of the physicalinformation that may be contained in T0, and that cannot be captured by the associated infinity-enhanced Leibnizalgebra G(T ).

13

conventions, with graded symmetric and graded skew-symmetric brackets, respectively. Wewill use the usual skew-symmetric convention here, as it it more adapted to our setting. Thesymmetric convention for L∞-algebras is given in [62], and the correspondence between theskew-symmetric and the symmetric brackets is rigorously defined in Remark 1.1 of [52], see(4.3). A discussion about this correspondence can also be found in [29].

An L∞-algebra is a Z-graded vector space X =⊕

i∈ZXi that is equipped with a family

(lk)k>1 of degree (k − 2) graded skew-symmetric k-multilinear ‘brackets’ that satisfy thehigher Jacobi identities, written symbolically as6

n∑

i+j=n+1

(−1)i(j−1)lj ◦ li = 0. (4.2)

In particular, l1 is a differential on X , and it is a derivation of the 2-bracket l2. We say thatX is a Lie n-algebra if it is non-negatively graded and concentrated in degrees 0, . . . , n − 1:X = X0 ⊕ · · · ⊕ Xn−1. In particular, this implies that a Lie algebra is a Lie 1-algebra.

A theorem by Getzler [51] shows that a differential graded Lie algebra structure onT = T0 ⊕ T1 ⊕ T2 ⊕ · · · = T0 ⊕ T canonically induces a L∞-algebra structure on X = s T .In [51] formulas for the brackets of all orders are given, but there appears to be an error inthe formulas that can be corrected by reversing the sign of the higher odd brackets. It is alsomentioned in [51] that the theorem is a special case of a more general result already foundby Fiorenza and Manetti [52]7. To pass from the symmetric convention for L∞-algebrasused in [51] to the usual skew-symmetric convention, one uses the rigorous formula given inRemark 1.1 of [52]:

ln(a1, . . . , an) = (−1)n(n−2)+∑

n−1

i=1(n−i)|ai|{

a1, . . . , an

}n

(4.3)

where {. . .}n is the n-bracket in the symmetric convention, and where as usual |a| is thedegree of a when seen as an element of X . In particular, the differential and the oddbrackets inherit an additional minus sign when performing this transformation.

Let us now turn to the result of Getzler, translated to the convention of skew-symmetricbrackets and with reversed signs of the odd brackets of order 3 and higher, correcting thesign error in the original statement of the theorem.

Theorem (Getzler [51]). A differential graded Lie algebra (T, J . , . K, ∂) gives rise to thefollowing L∞-algebra structure on X = s T :

1. the 1-bracket is l1 ≡ −∂ on X8 and 0 on X0, where X = X0 ⊕ X;

2. the 2-bracket is defined by

l2(a, b) ≡(−1)|a|

2

(JD(a), bK + (−1)ℓ(a)ℓ(b)JD(b), aK

), (4.4)

where D : T → T is the operator that is equal to ∂ on T1, and 0 in any other degree;

3. the k-bracket for k > 3 is given by

lk(a1, . . . , ak) ≡ βk

∑

σ∈Sk

χσa1,...,ak

JJ. . . JJD(aσ(1)), aσ(2)K, aσ(3)K . . .Kaσ(k)K, (4.5)

6The rigorous formula is

∑

i+j=n+1

(−1)i(j−1)∑

σ∈Un(i,n−i)

ǫσx1,...,xn

lj

(li(xσ(1), . . . , xσ(i)), xσ(i+1), . . . , xσ(n)

)= 0 (4.1)

where Un(i, n − i) is the set of (i, n − i)-unshuffles and where ǫσx1,...,xn

is the sign induced by the permutation ofthe elements x1, . . . , xn in the exterior algebra of X, i.e., x1 ∧ . . . xn = ǫσ

x1,...,xnxσ(1) ∧ . . . ∧ xσ(n).

7See for example Remark 5.4 in [52] and there replace the differential ∂ with the map D defined in [51].Unfortunately, they also made a mistake in this application: for n > 2 the right hand side of the formulacomputing the n + 1 bracket should inherit a minus sign.

8Since Getzler’s 1-bracket is {a} = ∂(a), we pick up a minus sign from (4.3).

14

where βk = (−1)∑

k

i=1(k−i)|ai| Bk−1

(k−1)! , with Bk−1 being the (k − 1)-th Bernoulli number9

and the sign χσa1,...,ak

is the Koszul sign of the permutation σ with respect to the degree in

T , that is: a∨b = (−1)ℓ(a)ℓ(b)b∨a so that χ(1 2)a,b = (−1)ℓ(a)ℓ(b). Since B3 = B5 = · · · = 0

there is no k-bracket for k even and greater than 3.

The occurrence of Bernoulli numbers in this context was first noted by Bering [63], andthey also show up in a similar way in the L∞ algebra encoding the gauge structure ofgeneralised diffeomorphisms in extended geometry [29].

4.2 Comparing the L∞

-algebra structures

Using Getzler’s theorem, let us compute the L∞-algebra (X, lk) that is associated to thedifferential graded Lie algebra (T, J . , . K, JΘ, −K) induced by a Leibniz algebra V (see Section2.3). First, notice that the underlying graded vector space defined in Getzler’s theorem is thesame as the one used in [1], that is: X = s T . We will now show that the L∞-algebra definedon X by using Getzler’s theorem gives back all brackets defined from the infinity-enhancedLeibniz algebra structure on X in [1].

First, the 1-bracket on X is l1 = −JΘ, −K = D, which is the same differential as in [1].Following Getzler, we define the operator D : T → T as D = JΘ, −K on T1 = V , and 0elsewhere. Then, the 2-bracket of two elements x, y of X0 = V , which are considered ashaving degree 0 in X so that |x| = |y| = 0, is given by (4.4):

l2(x, y) =12

(JJΘ, xK, yK − JJΘ, yK, xK

)=

12

(x ◦ y − y ◦ x

)= [x, y], (4.6)

where we have used (2.18). Hence, the 2-bracket of two elements of X0 is precisely theskew-symmetric part of the Leibniz product. Now, let us compute the bracket of an elementx ∈ X0 = V and another element u ∈ X . Since |u| > 0, then D(u) = 0 so that only the firstterm of (4.4) appears:

l2(x, u) =12JJΘ, xK, uK =

12

Lx(u), (4.7)

where we have used (3.21). The bracket with two elements in X vanishes because the operatorD vanishes on s−1X. Thus, we see that the 2-bracket defined by Getzler’s theorem from thedgLa induced by a Leibniz algebra coincides with the one of [1].

For the 3-bracket, the formula (4.5) gives:

l3(a, b, c) =(−1)2|a|+|b|

12

(JJD(a), bK, cK + (−1)ℓ(b)ℓ(a)JJD(b), aK, cK (4.8)

+ (−1)ℓ(a)(ℓ(b)+ℓ(c))JJD(b), cK, aK + (−1)ℓ(a)(ℓ(b)+ℓ(c))+ℓ(b)ℓ(c)JJD(c), bK, aK

+ (−1)(ℓ(a)+ℓ(b))ℓ(c)JJD(c), aK, bK + (−1)ℓ(b)ℓ(c)JJD(a), cK, bK)

.

For three elements x1, x2, x3 of X0 = V , we have |xi| = 0 = ℓ(xi) − 1, so that we obtain

l3(x1, x2, x3) =12Jx[1 ◦ x2, x3]K = −

12

(x[1 ◦ x2) • x3]. (4.9)

Including one element un ∈ Xn for some n > 1, so that |un| = n = ℓ(n) − 1, we get

l3(x1, x2, un) =16

(J[x1, x2], unK + (−1)nJLx[2

(un), x1]K)

(4.10)

= −16

[x1, x2] • un −16

Lx[2(un) • x1]. (4.11)

9Getzler’s definition of higher brackets does not satisfy the definition of a L∞-algebra, but it does if we reversethe sign of the odd brackets of order 3 and higher (even brackets of order 4 and higher are vanishing anyway sinceB3 = B5 = · · · = 0). Hence, when passing to the skew-symmetric convention, the odd brackets inherit a plus sign,because the sign brought by the translation from the symmetric convention to the skew-symmetric convention(see (4.3)) cancels the sign that we added to correct Getzler’s formula for odd brackets of order 3 and higher.

15

Finally, for the 3-bracket of one element x ∈ X0 = V and two other elements un ∈ Xn andum ∈ Xm (m, n > 1), we have

l3(x, un, um) =(−1)n

12

(JLx(un), umK + (−1)ℓ(n)ℓ(m)JLx(um), unK

)(4.12)

= −(−1)ℓ(n)

12

(JLx(un), umK − Jun, Lx(um)K

)(4.13)

= −112

(Lx(un) • um − un • Lx(um)

)(4.14)

=112

(un • Lx(um) − (−1)|n||m|um • Lx(un)

), (4.15)

and with only elements in X, the 3-bracket vanishes in the same way as the 2-bracket.We observe that the 3-brackets are exactly the ones defined in [1]. Given that Bernoulli

numbers Bk−1 vanish for k even and greater than 3, all even brackets vanish, as was foundfor the 4-bracket in [1]. The L∞-algebra structure on X obtained from Getzler’s theorem,and the one defined on X [1] thus coincide up to that order. However, Getzler’s theoremprovides us with the precise formulas for higher brackets, that were not given in [1].

This result is mathematically very deep because it says that given any Leibniz algebraV , one can always find a (non-negatively graded) L∞-algebra that ‘lifts’ the skew-symmetricpart of the Leibniz product in a non-trivial way (recall that this bracket does not satisfythe Jacobi identity, see (2.6)). More precisely, the vector space at degree 0 is V and the2-bracket of the L∞-algebra at degree 0 is [ . , . ]. This L∞-algebra has the particularity thatif V is a Lie algebra, i.e., if the symmetric part of the Leibniz product is zero, then X = V .This can be explained by the construction of the dgLa induced by the Leibniz algebra V

in Section 2.3 and with more details in [41]: if V is a Lie algebra, the dgLa does not haveany space of degree higher than 1. Hence, that is why the L∞-algebra X defined from aLeibniz algebra V by applying Getzler’s theorem to the dgLa induced by V can be called theL∞-extension of the Leibniz algebra V . We postpone to further research the study of suchalgebraic structures.

4.3 Example: the (1, 0) superconformal model in six dimensions

The first levels of the tensor hierarchy appearing in the (1, 0) superconformal model in sixdimensions have been given in [54]. Its mathematical aspects were investigated in [44, 45].The algebra of global symmetries of this model is g ≡ e5(5) = so(5, 5) [7]. The model involvesa set of p-forms (for p = 1, 2, 3, . . . , 6) taking values, respectively, in the following g-modules:X0 = 16, X1 = 10, X2 = 16, X3 = 45, X4 = 144 and X5 = 10 ⊕ 126 ⊕ 320 [7]. Theembedding tensor Θ : X0 → g is considered as a spurionic object, i.e., that is only fixed atthe end of the computations, thus immediately fixing any tensors appearing in the hierarchy.In the (1,0) superconformal model, the rank of Θ is constant, so that we can formally seth ≡ Im(Θ) ⊂ g, without further addressing the content of this Lie subalgebra.

The beginning of the hierarchy is governed by a set of constants haI , gIt, f c

ab = −f cba,

dIab = dI

ba, bIta, subject to the following relations:

2(dJ

c(adIb)s − dI

csdJab

)hs

J = 2fc(asdI

b)s − bJscdJabg

Is, (4.16)(dJ

rsbIut + dJrtbIsu + 2dK

rubKstδJI

)hu

J = frsubIut + frt

ubIsu + gJubIurbJst, (4.17)

f[abrfc]r

s −13

hsIdI

r[afbc]r = 0, (4.18)

haI gIt = 0, (4.19)

frbahr

I − dJrbh

aJhr

I = 0, (4.20)

gJshrIbKsr − 2hs

KhrIdJ

rs = 0, (4.21)

−frtsgIt + dJ

rthsJgIt − gItgJsbJtr = 0, (4.22)

bJt(adJbc) = 0. (4.23)

16

Following (2.19), the g-module X0 = 16 can be equipped with a Leibniz algebra structure◦ that is defined between any two basis elements ea, eb ∈ X0 by

ea ◦ eb ≡ −Xabc ec. (4.24)

We call Xabc = −fab

c +dIabh

cI the structure constants of the Leibniz algebra. The symmetric

and skew-symmetric brackets are then defined by

[ea, eb] = fabc ec and {ea, eb} = −dI

abhcI ec. (4.25)

Then, the skew-symmetric part of the Leibniz product does not satisfy the Jacobi identity,but rather (2.6), which is precisely (4.18).

The second g-module X1 = 10 is a sub-representation of S2(X0) that is fixed by therepresentation constraint, and that turns out to be the smallest g-submodule of S2(X0)through which the symmetric bracket factors (see Section 2.3). We set {eI} to be a basisof X1, and following the notations in [54], we label these elements by capital letters of themiddle of the alphabet, not to confuse them with the generators of X0. The action of h onX1 is defined by

ρΘ(ea)(eI) ≡ −XaIJ eJ , (4.26)

where XaIJ = 2hc

IdJac −gJsbIsa, and where ea is a basis element of X0. Going further up, we

reach the g-module X2 = 16 in which 3-form fields take values. In the (1, 0) superconformalmodel, the 3-forms are dual to the 1-forms, that is why we use latin letters of the end of thealphabet as labels for basis elements {et} of X2. By duality, the action of h on a generatoret is defined by

ρΘ(ea)(et) ≡ Xas

t es, (4.27)

where Xast = −fas

t + dIasht

I , and where ea is a basis element of X0.As explained in [54], the hierarchy of p-forms (for p = 1, 2, 3) can be extended one step

further by adding a set of 4-forms that take values in the g-module X3 = 45. Three newtensors kα

t , cαIJ = −cαJI and ctαa have to be introduced so that this extension is consistent.

They obey a set of additional conditions:

gKtkαt = 0, (4.28)

4dJabcαIJ − bItact

αb − bItbctαa = 0, (4.29)

kαt cαIJ − ha

[IbJ]ta = 0, (4.30)

kαt cs

αa − ftas + bJtagJs − dJ

tahsJ = 0. (4.31)

The tensor hierarchy is not investigated higher than this level, so we will only consider thestructures that are induced by (4.16)–(4.23) and (4.28)–(4.31), keeping in mind that thehierarchy formally goes higher up.

The construction of the tensor hierarchy presented in Section 2.3 and with more details in[41] gives the following brackets on the tensor hierarchy T = h⊕s−1X0⊕s−1X1⊕s−1X2⊕· · · :

JXa, XbK = −2 dIab XI , (4.32)

JXa, XIK = −bIta Xt, (4.33)

JXI , XJK = −2 cαIJ Xα, (4.34)

JXa, XtK = ctαa Xα (4.35)

where Xa = s−1ea, XI = s−1eI and Xt = s−1et (and similarly for Xb and XJ). Notice thatelements of T1 = s−1X0 have degree +1, elements of T2 = s−1X1 have degree +2, elementsof T3 = s−1X2 have degree +3, etc. so that the brackets are indeed consistent with thesymmetries of the tensors aforementioned. The bracket of h and any other element if thehierarchy is given by the action of h:

Jg, aK = ρg(a) (4.36)

17

for any g ∈ h and a ∈ T . Skew-symmetry is enforced by setting Ja, gK = −ρg(a). To completethe differential graded Lie algebra structure on T , the differential ∂ is given by:

∂(Xa) = Θ(Xa), (4.37)

∂(XI) = haI Xa, (4.38)

∂(Xt) = −gIt XI , (4.39)

∂(Xα) = kαt Xt. (4.40)

Notice that it is the opposite sign convention that has been chosen in Section 4.3 in [41].One can check that the homological property ∂2 = 0, the Jacobi identity (2.9) and the

Leibniz rule (2.11) are equivalent to (4.16)–(4.23) and (4.28)–(4.31). For example, the Jacobiidentity between elements Xa, Xb ∈ T1 and XI ∈ T2 is:

JXa, JXb, XIKK − JJXa, XbK, XIK + JXb, JXa, XiKK =

= 2JX(a|, −bIt|b)XtK + 2dJ

abJXJ , XIK

=(

− 2ctα(a|bIt|b) − 4dJ

abcαJI

)Xα, (4.41)

which vanishes by skew-symmetry of the I − J indices of the tensor cαJI , and by (4.29).Another example of Jacobi identity is the one between an element Θ(Xa) of T0 = h = Im(Θ)and two elements Xb, Xc of T1:

JΘ(Xa), JXb, XcKK − JJΘ(Xa), XbK, XcK − JJXb, Θ(Xa), XcKK =

= −2dIbcJΘ(Xa), XIK + Xab

rJXr, XcK + XacrJXb, XrK

= 2(

dIbcXaI

J − 2Xa(brdJ

c)r

)XJ , (4.42)

which vanishes by using the definition of Xabr and XaI

J , and by (4.16).Now let us turn to some Leibniz identities. We have

∂(JXI , XJK

)− J∂(XI), XJK − JXI , ∂(XJ)K = −2 cαIJkα

t Xt − 2ha[IJXa, XJ]K

= −2(

cαIJ kαt − ha

[IbJ]ta

)Xt, (4.43)

which vanishes by (4.30). Another example using Θ is

∂(JXa, XtK

)− J∂(Xa), XtK + JXa, ∂(Xt)K = ct

αakαs Xs − JΘ(Xa), XtK − gItJXa, XIK

=(

ctαakα

s − Xast + gItbIsa

)Xs, (4.44)

which vanishes by the definition of Xast = −fas

t + dKasht

K and (4.31).Let us now give the L∞-algebra that is induced from this dgLa, following Getzler’s

theorem presented in Section 4.1. First, the L∞-algebra structure is defined on the gradedvector space X = s T = X0 ⊕ X1 ⊕ · · · . We will use the same basis elements for X as above,that is: ea, eb, ec, ed, ee are basis elements of X0, eI , eJ , eK are basis elements of X1, es, et

are basis elements of X2, etc. Then, Item 1. of Getzler’s theorem states that l1 ≡ −∂, sothat we have

l1(eI) = −haI ea , l1(et) = gIt eI and l1(eα) = −kα

t et. (4.45)

The 2-bracket is defined as in (4.4):

l2(ea, eb) = fabcec, (4.46)

l2(ea, eI) = −12

XaIJeJ , (4.47)

l2(ea, et) =12

Xastes. (4.48)

18

If the 2-bracket does not involve at least one element from X0 then it is vanishing. Following(4.5), the 3-bracket satisfies at lower degrees:

l3(ea, eb, ec) = −f[abrdI

c]r eI , (4.49)

l3(ea, eb, eI) = −16

(fab

cbItc + X[a|IKbKt|b]

)et, (4.50)

l3(ea, eb, et) = −16

(− fab

cctαc + X[a|s

tcsα|b]

)eα, (4.51)

l3(ea, eI , eJ) = −13

Xa(I|KcαK|J) eα. (4.52)

Obviously there are 2-brackets and 3-brackets involving elements of the spaces X3, X4 andX5, but we only focus here on the beginning of the hierarchy, using only the tensors thatwere given in [54]. Thus, the only 5-bracket that is computable given the available tensorsis the one involving five elements of X0:

l5(ea, eb, ec, ed, ee) =13

fabrdI

rcbItdctαe eα, (4.53)

where underlined indices are fully-antisymmetric. There is only one 7-bracket l7 : ∧7X0 →X5, but since not all tensors were given in [54] to define the whole hierarchy, we cannotwrite it down. The 4- and 6-brackets are vanishing because the Bernoulli numbers B3 andB5 vanish. As a final remark, since the hierarchy in the (1, 0) superconformal model stopsat degree 5, we will actually obtain a Lie 6-algebra.

We propose now to give explicit computations of the higher Jacobi identities (4.2) to provethat the above brackets indeed form an L∞-algebra. The first (higher) Jacobi identities arethose involving l1 and l2: (l1)2 = 0 and l1 ◦ l2 − l2 ◦ l1 = 0. They can be checked rather easilyso we turn directly to the first Jacobi identity involving three elements, that is:

3 l2(l2(e[a, eb), ec]

)+ l1

(l3(ea, eb, ec)

)=

(3f[ab|

rfr|c]d + hd

If[abrdI

c]r

)ed, (4.54)

which vanishes by the skew-symmetry of the lower indices of frcd and by (4.18). The second

(higher) Jacobiator that we can compute involves ea, eb and eI , and is defined by

J(ea, eb, eI) ≡ l2(l2(ea, eb), eI

)− 2 l2

(l2(e[a|, eI), e|b]

)

+ l1(l3(ea, eb, eI)

)+ l3

(l1(eI), ea, eb

). (4.55)

We can split it into two parts,

l2(l2(ea, eb), eI

)− 2 l2

(l2(e[a|, eI), e|b]

)= fab

rl2(er, eI) + X[a|IK l2(eK , e|b])

= −12

(fab

rXJrI − X[a|I

KX|b]KJ)

eJ (4.56)

and

l1(l3(ea, eb, eI)

)+ l3

(l1(eI), ea, eb

)= −

16

(fab

rbItr + X[a|IKbKt|b]

)l1(et) − hs

I l3(es, ea, eb)

=[−

gJt

6

(fab

rbItr + X[a|IKbKt|b]

)+ hs

If[sardJ

b]r

]eJ . (4.57)

One can rewrite the last term on the right hand side as

hsIf[sa

rdJb]r =

23

hsIfs[a

rdb]r +13

hsIfab

rdJrs

=23

hsIdK

s[a|hrKdJ

|b]r +16

fabr(2hs

IdJr s)

=13

X[a|IKhr

KdJ|b]r +

16

fabr(XrI

J + gJtbItr), (4.58)

19

where we passed from the first line to the second line by using (4.20), and from the secondline from the third line by using the definition of XaI

K and by (4.19). Then the term in thebrackets on the right hand side of (4.57) becomes

−gJt

6

(fab

rbItr + X[a|IKbKt|b]

)+

13

X[a|IKhr

KdJ|b]r +

16

fabr(XrI

J + gJtbItr)

=16

fabrXrI

J +16

X[a|IKX|b]K

J . (4.59)

Then, adding (4.59) to (4.56), one finds

J(ea, eb, eI) = −13

(fab

rXrIJ − 2X[a|I

KX|b]KJ)

eJ . (4.60)

But the right hand side is nothing but one third times(

ρ[Θ(ea),Θ(eb)] −[ρΘ(ea), ρΘ(eb)

])(eI), (4.61)

which vanishes because X1 is an h-module. One can check as well that the (higher) Jacobiatorof ea, eb and et is one third times

(ρ[Θ(ea),Θ(eb)] −

[ρΘ(ea), ρΘ(eb)

])(et), (4.62)

which vanishes because X2 is an h-module.The next higher Jacobi identity that we can compute is the one involving ea and eI , eJ ,

J(ea, eI , eJ) ≡ 2 l2(l2(ea, e(I), eJ)

)+ l2

(l2(eI , eJ), ea

)

+ l1(l3(ea, eI , eJ)

)− 2 l3

(l1(e(I|), ea, e|J)

). (4.63)

Since the bracket between two elements eK and eL is always zero, the two first terms iden-tically vanish. Now let us show that the two last terms cancel one another,

l1(l3(ea, eI , eJ)

)− 2 l3

(l1(e(I|), ea, e|J)

)

= −13

Xa(I|KcαK|J) l1(eα) + 2hr

(I| l3(er, ea, e|J))

=13

[Xa(I|

KcαK|J)kαt − hr

(I|frasb|J)ts − hr

(I|X[r|J)KbKt|a]

]et. (4.64)

Let us turn our attention to the last two terms:

−hr(I|fra

sb|J)ts − hr(I|X[r|J)

KbKt|a] = −hr(I|d

Krahs

Kb|J)ts +12

hr(I|Xa|J)

KbKtr

= −hs(J|d

Ksahr

Kb|I)tr +12

Xa(JKhr

I)bKtr

= −12

Xa(J|Khr

Kb|I)tr +12

Xa(JKhr

I)bKtr

= Xa(I|K

(−

12

hrKb|J)tr +

12

hr|J)bKtr

). (4.65)

We have passed from the left hand side of the first line to the right hand side by applying(4.20) to the first term, by developing the skew-symmetrization of a and r, and by noticingthat hr

(I|Xr|J)K = 0 by (4.21). The second line is just the first line, rewritten with different

indices. Passing from the second line to the third line uses the definition of Xa(J|K and

(4.19). Passing to the fourth line is done by using the symmetrization between I and J .Then, by inserting the fourth line into (4.64), one obtains

l1(l3(ea, eI , eJ)

)− 2 l3

(l1(e(I|), ea, e|J)

)

=13

Xa(I|K

[kα

t cαK|J) −12

hrKb|J)tr +

12

hr|J)bKtr

]et. (4.66)

20

But the term in the bracket vanishes by (4.30), hence proving the desired higher Jacobiidentity.

One can now compute the higher Jacobi identities mixing the 2- and 3-brackets andstraightforwardly check that l2 ◦ l3 − l3 ◦ l2 identically vanishes, which is consistent withthe fact that the 4-bracket is zero. For example, for four elements ea, eb, ec, ed, we have(underlined indices imply full anti-symmetry on these indices):

6 l3(l2(ea, eb), ec, ed

)−4 l2

(l3(ea, eb, ec), ed

)

= 6fabrl3(er, ec, ed) + 4fab

rdJcrl2(eJ , ed)

=(

− 6fabrf[rc

sdKd]s + 2fab

rdJcrXdJ

K)

eK (4.67)

By expanding f[rcsdK

d]s on the one hand, and by using the identity XdJKdJ

cr − XdcsdK

sr −

XdrsdK

cs = 0 (which is a rewriting of (4.16)) on the other hand, the term in the parenthesiscan be rewritten as:

−4fabrfrc

sdKds −2fab

rfcdsdK

rs + 2fabrdK

srXdcs

︸ ︷︷ ︸= 0

+2fabrdK

csXdrs

= −4fabrfrc

sdKds − 2fab

rfdrsdK

cs + 2fabrhs

JdJrddK

cs

= −4fabrfrc

sdKds + 4fab

rfdrsdK

cs = 0, (4.68)

We passed from the first line to the second line by using the definition of Xdrs, and from the

second line to the last one by using (4.18). The Jacobi identities of the type l3 ◦ l2 − l2 ◦ l3 = 0that involve elements from X1 and X2 can be shown to be satisfied as well.

Now let us show that the Jacobi identity l3 ◦ l3 + l1 ◦ l5 = 0 is satisfied on X0. Forfive elements ea, eb, ec, ed, ee, we have (underlined indices imply full anti-symmetry on theseindices):

10 l3(l3(ea,eb, ec), ed, ee

)+ l1

(l5(ea, eb, ec, ed, ee)

)

= −10f rabd

Icrl3(eI , ed, ee) +

13

fabrdI

crbIsdcsαe l1(eα) (4.69)

=(

53

f rabd

Icr

(fde

sbIts + XdIKbKte

)−

13

fabrdI

crbIsdcsαekα

t

)et (4.70)

By using (4.31) and by noticing that fdes = Xed

s, the term in parenthesis becomes one third

f rabd

Icr

(5Xed

sbIts − 5XeIKbKtd − bIsdXet

s + bIsdgKsbKte

)

= f rabd

Icr

(6Xed

sbIts − 4XeIKbKtd − bIsegKsbKtd

)

= f rabd

Icr

(6Xed

sbIts − 3XeIKbKtd − 2hs

IdKesbKtd

)

= f rab

(6Xed

sdIcrbIts − 3XeI

KdIcrbKtd + 3Xer

sdKcsbKtd

)

= −3Xrab

(2Xed

sdKcrbKts − Xec

sdKrsbKtd

)

= −3(

XrabXed

sdKcrbKts + Xs

abXedrdK

csbKtr + XrabXed

sdKrsbKtc

)

= −9XrabXed

sbKt(sdKcr) = 0 (4.71)

We passed from the first line to second line by using the identity −XeIKbKtd − bIsdXet

s =Xed

sbIts which is a rewriting of (4.17), from the second line to the third by using the definitionof XeI

K , and from the third line to the fourth by noticing that 2f rabd

Icrhs

I = 3f rabXcr

s, whichfollows from (4.18). We passed from the fourth line to the fifth one by using the identity−XeI

KdIcr + Xer

sdKcs = −Xec

sdKrs which is a rewriting of (4.16). From the fifth line to

the sixth line we have just rewritten the first term in the parenthesis, and the symmetryproperties of the three terms give the last line, which vanishes by (4.23).

21

We have thus proven that the brackets (4.45)–(4.53) satisfy the higher Jacobi identities(4.2), hence defining a L∞-algebra. Note that this L∞-algebra associated to the (1, 0) su-perconformal model does not coincide with the one found in [45], which was obtained fromthe Bianchi identities, although the underlying tensor hierarchy is the same. We leave forfuture work a more general study on the relationship between those two L∞-algebras.

Acknowledgments

We would like to thank Roberto Bonezzi, Martin Cederwall, Olaf Hohm and Jim Stasheff forfor discussions and comments on the first version of this paper. In particular we are gratefulto Olaf Hohm for explaining the ideas behind the work [1]. We would also like to thank theanonymous referee for suggesting some well justified clarifications on the role of the degree-zero subspace of the differential graded Lie algebras. This work was initiated during a visitat Institut des Hautes Études Scientifiques (IHÉS), and we would like to thank the institutefor its hospitality. The work of JP is supported by the Swedish Research Council, project no.2015-02468. The work of SL is supported by the Agence Nationale de la Recherche, projectSINGSTAR.

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