All articles published by MDPI are made immediately available worldwide under an open access license. No special
permission is required to reuse all or part of the article published by MDPI, including figures and tables. For
articles published under an open access Creative Common CC BY license, any part of the article may be reused without
permission provided that the original article is clearly cited. For more information, please refer to
Feature papers represent the most advanced research with significant potential for high impact in the field. A Feature
Paper should be a substantial original Article that involves several techniques or approaches, provides an outlook for
future research directions and describes possible research applications.
Feature papers are submitted upon individual invitation or recommendation by the scientific editors and must receive
positive feedback from the reviewers.
Editor’s Choice articles are based on recommendations by the scientific editors of MDPI journals from around the world.
Editors select a small number of articles recently published in the journal that they believe will be particularly
interesting to readers, or important in the respective research area. The aim is to provide a snapshot of some of the
most exciting work published in the various research areas of the journal.
We show that an -algebra can be extended to a graded Hopf algebra with a codifferential. Then, we twist this extended -algebra with a Drinfel’d twist, simultaneously twisting its modules. Taking the -algebra as its own (Hopf) module, we obtain the recently proposed braided -algebra. The Hopf algebra morphisms are identified with the strict -morphisms, whereas the braided -morphisms define a more general -action of twisted -algebras.
-algebras or homotopy Lie algebras are generalizations of Lie algebras with infinitely many higher brackets, related to each other by higher homotopy versions of the Jacobi identity [1,2,3]. From a physical point of view, these higher algebras represent the geometrical structure providing a deeper understanding of the quantization of field theory and gravity. In particular, there exists a correspondence between the BV formalism used in the quantization of gauge theories and -algebras, as noted by Zwiebach in his seminal work on closed string field theory . Recently, the BV -algebra of the Yang–Mills theory was used in relating the squares of gluon amplitudes to those of ( super) gravity amplitudes and toall orders of perturbative quantum field theory . On the other hand, in the framework of deformation quantization , Kontsevich’s famous formality theorem on the existence and classification of star products on Poisson manifolds has been proven using the concept of -quasi-isomorphisms . Subsequently, Cattaneo and Felder  provided an interpretation of the Kontsevich quantization formula in terms of the perturbative expansion of the path integral for the Poisson sigma model [8,9].
More recently, attempts to improve the understanding of consistent non-commutative deformations of field theory using an -framework resulted in two interesting proposals. In , the homotopy relations defining an -algebra were used to bootstrap a consistent star-gauge-invariant theory starting from the star-deformed commutator of the symmetry algebra. However, there is another consistent way of introducing non-commutative deformations based on Drinfel’d twists of the symmetric Hopf algebra . The analysis of twisted gauge symmetries was initiated in [12,13,14,15], but only recently was it shown that these twisted symmetries can be understood in the framework of braided -algebras [16,17]. The construction of braided -algebras in  was based on a reinterpretation of all the defining relations of an -algebra in terms of morphisms in a suitable category. By twisting the enveloping Hopf algebra of vector fields on a manifold M to a non-cocommutative Hopf algebra and simultaneously deforming its category of modules, the -algebra was deformed into a braided -algebra. Here we show that an -algebra itself can be extended to a graded Hopf algebra with a compatible codifferential, and using this observation we construct the braided -algebra using a Drinfel’d twist of its underlying Hopf algebra structure.
In the next section we recall the coalgebraic structure of an -algebra and show that it can be extended to a graded Hopf algebra with a codifferential. In Section 3 we apply Drinfel’d twists to the Hopf algebras underlying -algebras and obtain twisted -algebras. In the spirit of deformation quantization, after twisting the algebra, one can further twist its modules. Taking the Hopf algebra as its own module, we obtain another Hopf algebra, , which is exactly the same as the braided -algebra defined in . Furthermore, we reinterpret the Hopf algebra (iso)morphism in terms of strict -(iso)morphisms, and define a more general class of braided -morphism. In the conclusion, we briefly discuss the relevance of our results for braided gauge field theory.
2. as Hopf Algebra
An -algebra can be defined in several different ways, depending on the context. In the standard approach one defines an -algebra on a graded vector space X as a generalization of a Lie algebra with possibly infinitely many higher brackets, related to each other by higher-homotopy versions of the Jacobi identity [1,2,3] Alternatively, one can describe an -structure as a degree 1 coderivation on the coalgebra generated by the suspension1 of X, as shown in [3,19]. In order to identify the Hopf algebra structure underlying an -algebra, this coalgebraic picture is more appropriate and we shall review it here2.
Let us start with a graded symmetric tensor algebra:
where X is a -graded vector space over the field and the degree of a homogeneous element is denoted as . The tensor product is graded-symmetric,
and we use ∨ to denote the product in . The algebra structure can be endowed with a unit map , where .
The coalgebra structure on is given by the coproduct:
where is the Koszul sign,
and denotes those permutations, ordered as and . We use the conventions that Sh(n,0) = Sh(0,n) equals ID and that an empty slot in the product equals the unit . Thus, we have
As a map , this reads:
where denotes the action of permutations , e.g., the non-identity permutation of two elements is:
and this includes the Koszul sign. Furthermore, the coalgebraic structure of includes counit , where and .
Next, we introduce a coderivation D that squares to zero and thus generates the appropriate homotopy relations. The coderivation is a map of degree 1, such that the co-Leibniz property is satisfied:
where the graded symmetric multilinear maps are of degree 1. When is non-vanishing, one talks about curved -algebras [20,21], whereas for we have flat -algebras3. The maps act on the full tensor algebra as a coderivation:
and can be written using the permutation map as:
Note that is a degree 1 element of . Now one can define an -algebra as a -graded vector space with multilinear graded symmetric maps of degree 1 such that the coderivation is nilpotent . As an example, we calculate the first few homotopy relations:
The vanishing of the above expression is equivalent to the following three identities:
Additionally, we can use the fact that is trivially zero due to the odd degree of , and is therefore not a constraint. The homotopy relations defining an -algebra can be written in the closed form :
Moreover, we have as .
So far we have identified an -structure with a counital coalgebra over a graded vector space with compatible coderivation that squares to zero. Now we wish to compare this structure with that of a Hopf algebra. In short, a Hopf algebra is a bialgebra that admits an antipode map with certain compatibility properties. Although the formal definition is given in Appendix B, we discuss here the prototypical example—a tensor algebra.
A tensor algebra , where V is a vector space over the field K, can be seen as a Hopf algebra . The coproduct , counit and antipode S are defined on as:
Since the coproduct and counit are algebra homomorphisms and the antipode is an algebra (and coalgebra) anti-homomorphism, we can extend the definition from the basis elements to the full tensor algebra:
where we use for the product in the tensor algebra. This example can be trivially extended to the symmetric-graded tensor algebra used in the construction of an -algebra, described above. In particular, the coproduct will be of the form (1), and the antipode will be extended to the graded antipode:
Using the axioms of a Hopf algebra, given in Appendix B, one can easily verify that the symmetric-graded tensor algebra is indeed a Hopf algebra. Thus, we arrive at the following theorem.
An extended -algebra is a bialgebra with coderivation D: of degree 1 s.t. the co-Leibniz property is satisfied
and . This naturally inherits the structure of a Hopf algebra from the graded symmetric tensor algebra, with:
where the codifferential
induces the same homotopy relations as D.
Note that the unit map is in general a morphism of graded coalgebras, and only for a flat -algebra, i.e., when , does it become a morphism of differential graded coalgebras with .
We need to show that the Hopf algebra structure of the symmetric-graded tensor algebra is compatible with the -algebra structure encoded in the nilpotent coderivation. The compatibility of the unit and counit with coderivation was already discussed; therefore, we need to check only the antipode compatibility relation. We apply the coderivations and on an element of using (5). The left-hand side is
where, in the second line, we introduced the sign
induced by the graded action of antipode (9) and in the third line we use . Similarly, the right-hand side gives
where . Equating the two sides gives . Inspecting the homotopy relations (7), it is easy to see that the relations induced by are the same. □
A coderivation of a graded Hopf algebra with similar properties was previously introduced in  in the context of the BRST formulation of quantum group gauge theory.
3. Braided -Algebra from a Drinfel’d Twist
The Hopf algebra underlying an -algebra that we have discussed so far is cocommutative and coassociative, as it is based on a (graded) symmetric tensor algebra. A systematic way to introduce a non-(co)commutative deformation is by applying the Drinfel’d twist approach . We twist a Hopf algebra H using a twist element , which is invertible and satisfies:
Relation (10) is known as the 2-cocycle condition, whereas condition (11) is known as (normalized) counitality. The 2-cocyle condition ensures that the deformed algebra remains coassociative. Using Sweedler’s summation notation we can write the twist element and its inverse as:
It was shown in [23,24] that a twist of a Hopf algebra H results in a new Hopf algebra, , which is given by . On the level of vector spaces , the product and counit are unchanged, whereas the coproduct transforms as follows:
In the case of an Abelian twist 4, which we assume in the following, the antipode is not deformed, . Thus, using the Drinfel’d twist, we obtain a twisted -algebra, namely, , where and are the same as vector spaces.
Drinfel’d twist deformation quantization consists of twisting the Hopf algebra as above, while simultaneously twisting all of its modules . Taking the Hopf algebra as a module itself, one obtains another Hopf algebra with the corresponding vector space once again being the same as before, namely, , with the following product:
This algebra is a Hopf algebra with:
The -matrix is an invertible matrix induced by the twist,
where . In the case of an Abelian twist, is triangular , and . The inverse -matrix controls the non-commutativity of the -product and provides a representation of the permutation group  and, in particular, the action of a non-identity permutation of two elements is:
As is triangular, squares to the identity. Now we can extend the coproduct (15) to the whole tensor algebra:
The coderivation is defined in terms of braided graded-symmetric maps :
with the condition reproducing the deformed homotopy relations. In particular, we have:
In passing to the last line we assume the equivariance of the maps , i.e., we assume that they commute with the action of the twist generators. Thus, one can show that the homotopy relations are the same5 as (7). The braided coproduct (18) and the compatible coderivation (19), equivariant under the action of the degree-zero twist element, reproduce, in the coalgebra picture, the braided -algebra constructed in , c.p., Definition 4.73 in . Formally, the homotopy relations have the same form as (8):
Moreover, we have as .
The Hopf algebras and are isomorphic and there exists an invertible map between the underlying vector spaces 
On the other hand, there exist maps between -algebras: an -morphism is a collection of graded symmetric maps of degree zero from to , such that they define a coalgebra morphism, i.e., they satisfy
and such that is compatible with the coderivations:
The first few components are:
When , we talk about curved -morphisms. From the compatibility of coderivations (26) one obtains the explicit relation between coderivation maps and 
When the map is invertible, we have an -isomorphism. Applying the -morphism to our case of interest, namely, finding a map , we need to define the component maps which are braided graded symmetric, i.e.,
and equivariant with respect to the action of twist generators. The expression for the morphism is then obtained from (28) by exchanging with and the action of the permutation with .
However, things are much simpler here; the Hopf algebra morphism is both an algebra morphism (21) and a coalgebra morphism (22) and (25); so we obtain that the only non-vanishing component of the morphism is :
The relation we established between Hopf and -algebras implies that the morphism between Hopf algebras (21)–(24) can be extended to a strict -morphism by demanding compatibility of the morphism with the coderivation (26)
Note that in [28,29] the authors discussed the special example of the twisting of an -algebra, where the -morphisms of twisted algebras went beyond strict -morphisms. The difference comes from the difference between Hopf algebra modules which we discuss here, and more general -algebra modules; see [19,30].
Finally, in complete analogy with Thm 1.,we can relate the braided -algebra with the Hopf algebra . The compatibility relation between the antipode and the codifferential follows from the equivariance of the coderivation maps and the fact that the antipode is a graded algebra anti-homomorphism.
4. Concluding Remarks
In this paper we have identified the cocommutative and coassociative Hopf algebra structure underlying -algebras. Thus, we were able to introduce a non-(co)commutative deformation by applying the Drinfel’d twist approach  and obtaining the braided -algebra of  as a module of the twisted one. In , the braided -algebra was used in the construction of a non-commutative deformation of the Chern–Simons and Einstein–Cartan–Palatini actions with a braided gauge symmetry. However, the physical interpretation of braided gauge symmetries encountered in these models was not well understood. One way to improve this situation is to construct an appropriate generalization of the BV formalism [17,31] that could help in identifying equivalent physical configurations. This is particularly natural in the coalgebra formulation, in which one can interpret the dual of the codifferential as locally being a cohomological vector field Q of degree 1 on a manifold M, i.e., or:
Here, the structure constants of the algebra are the components of the coderivation D on a basis of X:
and represents a basis of the dual6 vector space . In the BV formalism, Q becomes the BRST operator and denotes the physical fields.
Furthermore, in the -framework there exists a well-defined notion, at least for flat -algebras, of an -quasi-isomorphism that relates physically (gauge) equivalent configurations. Namely, when the 0-bracket vanishes, the 1-bracket is a differential (see (7)) and there is a cochain complex underlying the -algebra. In that case, one defines the -quasi-isomorphisms by the requirement that the linear morphism component induces an isomorphism of cohomologies of the respective -algebras; see detailed discussion in . For the case of a non-vanishing 0-bracket, a natural setting would be that of -models and -spaces, introduced by Costello . An -space includes target manifold data and a 0-bracket can be identified with the curvature of a connection on the target. Said differently, the connection is a degree-zero solution of Maurer–Cartan equation
Using this solution, one can define new -algebra on the same vector space, but with vanishing curvature [34,35].
Conceptualization, L.J.; formal analysis, C.J.G., L.J., T.K. and G.M.; investigation, C.J.G., L.J., T.K. and G.M.; methodology, L.J.; Writing—original draft, L.J.; Writing—review and editing, C.J.G., T.K. and G.M. All authors have read and agreed to the published version of the manuscript.
This research was funded by Croatian Science Foundation, grant number: IP-2019-04-4168.
Data Availability Statement
We thank Marija Dimitrijević Ćirić and Peter Schupp for extensive discussions and Paolo Aschieri, Athanasios Chatzistavrakidis, and Richard Szabo for useful comments.
Conflicts of Interest
The authors declare no conflict of interest.
Appendix A. On L∞-Algebras
(-algebra ). An -algebra is a graded vector space X equipped with a collection of multilinear maps that are graded totally antisymmetric:
of degree , where and satisfy the homotopy Jacobi identities:
for all , . Here, indicates the graded Koszul sign, including the sign from the parity of the permutation of that is ordered as: and .
We use the convention that totally graded antisymmetric means:
with indicatingthe degree of homogeneous element . When , this algebra is called a curved -algebra, whereas the name flat -algebra refers to the case .
The homotopy Jacobi identities defining the structure exists for any given level n, and there can be, in principle, an infinite number of these. The first few homotopy relations are:
These homotopy relations can be related to (7) using a degree map s between the algebra and coalgebra pictures, called a suspension or shift isomorphism:
which induces an isomorphism of the graded tensor algebras,
and décalage isomorphism of the brackets:
Appendix B. On Hopf Algebras
A bialgebra over K is a vector space which is both an algebra and a coalgebra in a compatible way:
The comultiplication and counit map are both algebra homomorphisms, whereas the multiplication and unit map are coalgebra homomorphisms.
DefinitionA3 (Hopf algebra).
A Hopf algebra over K is a bialgebra over K equipped with an antipode map satisfying the following:
The existence of an inverse antipode map is not assumed, but if , the inverse is equivalent to the antipode map itself. A consequence of the antipode’s uniqueness is that it obeys the following relations :
The first two relations state that the antipode is an antialgebra map, whereas the second two state that it is an anticoalgebra map.
The suspension map is also called a shift isomorphism ; see Appendix A for more details.
In the rest of the paper, we work in the coalgebraic picture and denote with X the underlying graded vector space in order to simplify the notation.
We shall use the term -algebra for both cases when the distinction is not relevant.
The twist generators commute in the case of an Abelian twist.
Only when acting on explicit elements of the tensor algebra does one have to take into account the braided transposition map. In that case, the first difference with respect to the untwisted algebra appears when acting on three or more elements, as shown in .
In the infinite-dimensional case, one either restricts to the space spanned by , or considers continuous duals in infinite-dimensional topological vector spaces; see discussion in .
Stasheff, J. Differential graded Lie algebras, quasi-Hopf algebras and higher homotopy algebras. In Quantum Groups; Springer: Berlin/Heidelberg, Germany, 1992; pp. 120–137. [Google Scholar]
Borsten, L.; Jurčo, B.; Kim, H.; Macrelli, T.; Saemann, C.; Wolf, M. Double Copy from Homotopy Algebras. Fortsch. Phys.2021, 69, 2100075. [Google Scholar] [CrossRef]
Bayen, F.; Flato, M.; Fronsdal, C.; Lichnerowicz, A.; Sternheimer, D. Deformation Theory and Quantization. 1. Deformations of Symplectic Structures. Ann. Phys.1978, 111, 61. [Google Scholar] [CrossRef]
Dimitrijević Ćirić, M.; Giotopoulos, G.; Radovanović, V.; Szabo, R.J. Braided L∞-Algebras, Braided Field Theory and Noncommutative Gravity. Lett. Math. Phys.2021, 111, 148. [Google Scholar] [CrossRef]
Giotopoulos, G.; Szabo, R.J. Braided Symmetries in Noncommutative Field Theory. arXiv2022, arXiv:2112.00541. [Google Scholar] [CrossRef]
Jurčo, B.; Raspollini, L.; Sämann, C.; Wolf, M. L∞-Algebras of Classical Field Theories and the Batalin-Vilkovisky Formalism. Fortsch. Phys.2019, 67, 1900025. [Google Scholar] [CrossRef][Green Version]
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely
those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or
the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas,
methods, instructions or products referred to in the content.