3-dimensional $\Lambda$-BMS Symmetry and its Deformations

In this paper we study quantum group deformations of the infinite dimensional symmetry algebra of asymptotically AdS spacetimes in three dimensions. Building on previous results in the finite dimensional subalgebras we classify all possible Lie bialgebra structures and for selected examples, we explicitly construct the related Hopf algebras. Using cohomological arguments we show that this construction can always be performed by a so-called twist deformation. The resulting structures can be compared to the well-known $\kappa$-Poincar\'e Hopf algebras constructed on the finite dimensional Poincar\'e or (anti) de Sitter algebra. The dual $\kappa$ Minkowski spacetime is supposed to describe a specific non-commutative geometry. Importantly, we find that some incarnations of the $\kappa$-Poincar\'e can not be extended consistently to the infinite dimensional algebras. Furthermore, certain deformations can have potential physical applications if subalgebras are considered. Since the conserved charges associated with asymptotic symmetries in 3-dimensional form a centrally extended algebra we also discuss briefly deformations of such algebras. The presence of the full symmetry algebra might have observable consequences that could be used to rule out these deformations. }


Introduction
Gravity in 3 dimensions [1][2][3], see [4] for review, is a remarkably rich and interesting theory. It remained relatively obscured for many years, but from the seminal paper of Witten [5] (see also [6]) it became one of the most studied theory in theoretical physics.
There are many reasons for that. Gravity in 3 dimensions is a topological field theory with no local degrees of freedom, which makes the quantum theory exactly soluble, so it can serve as a toy model of quantum gravity. As it was shown in [7] in the case of negative cosmological constant this theory possesses an asymptotic Virasoro symmetry. This result was a precursor of AdS/CFT [8] and the AdS 3 /CFT 2 correspondence is actively and intensively investigated [9], [10]. Second, in spite of having no local dynamical degrees of freedom, 3 dimensional gravity with negative cosmological constant admits black hole solution [11,12], which makes 3-dimensional gravity a nice toy model for studying Hawking radiation. See [13] for a review of these aspects of the theory.
Another interesting property of 3-dimensional gravity is the fact that it provides a model of emergence of quantum group symmetries as physical symmetries of quantized gravitating systems. Envisioned in [14] and based on mathematical works [15][16][17] this idea was developed further, among others, in canonical formulation in [18][19][20][21], and for gravitational path integral in [22,23]. This aspect of gravity in 3 dimensions will be the central subject of the studies presented in this paper.
In any spacetime dimension Poincaré group and (Anti) de Sitter group are symmetries of classical vacuum spacetimes of gravity with zero, (negative) positive cosmological constant. More than half a century ago it was realized however that there are circumstances that the symmetry of particular configurations of gravitational field is much larger. It was shown that the symmetries of the asymptotically flat gravitational field near null infinity form an infinite dimensional group, called the Bondi, Metzner, Sachs (BMS) group [24][25][26], which contains the Poincaré group as its subgroup. A natural question then arises what are the quantum deformations of the BMS group. We addressed this problem in the case of zero cosmological constant in the recent papers [27,28].
In these papers we found a class of quantum deformations of BMS algebra 1 . Technically, we started with the twist deformation of Poincaré subalgebras of BMS algebra, extending them to the whole of BMS and obtaining in this way a Hopf-BMS algebra. Here we want to extend this analysis to the case of non-vanishing cosmological constant.
In this paper we consider deformations of BMS algebra of 3-dimensional gravity model of non-zero cosmological constant Λ-BMS algebra.
We are interested here in 3 dimensional models because, as mentioned above, it was shown using different approaches that deformed symmetries play an important role in 3 dimensional gravity. In fact, it was established in [30] using the non-perturbative methods of Loop Quantum Gravity that the symmetries of the Euclidean Anti de Sitter quantum spacetime in 3 spacetime dimensions is described by a quantum group. More precisely, that paper considers the algebra of the six conserved gravitational charges that form the Euclidean Anti de Sitter algebra. It is then shown by explicit computation that upon quantization these charges become operators, forming the SO q (3, 1) algebra, with the deformation parameter q = exp(i G |Λ|/2). In the Λ → 0 limit this algebra contracts to the 3 dimensional (Euclidean) version of κ-Poincaré algebra [31][32][33][34]. It is believed that an analogous result holds in the case of Lorentzian AdS. This result is a major motivation for investigating deformations BMS algebra with infinite number of charges in 3-dimensional asymptotically AdS spacetime. Indeed if the Anti de Sitter subalgebra of these charges is getting deformed by non-perturabative quantum gravity effects it can be expected that by the same token the whole algebra of asymptotic charges is deformed as well. It is of interest therefore to list and investigate in some details a class of quantum deformations of the 3-dimensional BMS algebra in the case of negative cosmological constant.
On the more technical side the undertaking to investigate a class of deformations of AdS-BMS algebras is further motivated by the recently obtained complete classification of deformations of (Anti) de Sitter algebras in three dimensions [35] (the discussion of the corresponding contractions Λ → 0 can be found in [40]). Using this classification in the present paper we find a class of deformations of 3-dimensional Λ-BMS algebra.
Another reason why we choose to investigate here the simpler 3-dimensional Λ−BMS algebra is that in 4 dimensions the non-vanishing cosmological constant extension of the BMS algebra cannot have a structure of Lie algebra. More precisely, it has been recently shown in [41], by applying cohomological arguments, that there does not exist a Λ−BMS 4 Lie algebra containing the 4 dimensional (Anti) de Sitter subalgebra that gives the BMS 4 algebra as a contraction limit Λ → 0. In fact, such extension of the BMS 4 algebra should presumably have the structure of Lie algebroids, with structure functions instead of structure constants [41], [43], [44]. As a consequence to deform it we would need a theory of quantized Lie algebroids, which is much less developed than the theory of quantized enveloping algebras we are dealing with.
Note that the deformations discussed in this work deform the coalgebra structure. Deformations and (central) extensions of the algebra sector have been addressed for example in [41] for the B 4 and in [42] for diffeomorphisms on the two-sphere.
The plan of the paper is as follows. In the next section we briefly recall the structure of asymptotic symmetries Λ−BMS 3 algebra in 3 dimensions and interpret the algebras for different signs of cosmological constant as two real forms of a complex algebra. In Section 3 we discuss Lie bialgebras and deformations, first in general terms and then in the specific case of interest of two copies of Witt algebra Λ−BMS 3 W ⊕ W. Section 4 is devoted to discussion of twist deformations, their classifications and contractions. A schematic overview is presented in Figure 1. We conclude our paper with some remarks on one-sided Witt algebra and specialization in Section 5.

Asymptotic Symmetries of Spacetimes with Cosmological Constant
In this section we describe the structure of the Λ−BMS 3 algebra of asymptotic symmetries. An extensive discussion of this algebra can be found in [45] and [46], which contain also references to other works.

Asymptotic Symmetries in 3D
The study of asymptotic symmetries is usually carried out by starting from a general metric with given asymptotic structure (usually asymptotically Minkowski or (Anti) de Sitter) and imposing fall-off conditions on the expansion coefficients close to the asymptotic boundary. Then one looks for vector fields preserving the form of the asymptotic expansion. In the three dimensional asymptotically AdS spacetime such vector fields have the form ξ f,R = f ∂ u + R∂ z where R = R(z), f = T (z) + u∂ z R and their algebra reads [45,46] [ξ 1 , ξ 2 ] =ξ ≡f ∂ u +R∂ z (2.1) which in the contraction limit Λ → 0 gives the usual BMS 3 algebra (2.3). Depending on the sign of Λ (2.3)-(2.4) describes two different real algebras, into which one can embed the finite 3D (Anti) de Sitter algebra 2 generated by the generators K 2 , K ± , M +− , M ±2 satisfying the algebra in an infinitely many distinct ways by identifying (we rescale Λ → n 2 Λ) The isomorphism (2.9) is complex for positive Λ and therefore (2.10) has to be seen as a complex algebra with different real forms (cf. next section). As before, it is also easy to see from (2.9) that there are infinitely many embeddings of the (A)dS algebra into the Λ-BMS 3 , i.e. one shows that the o(4, C) ≡ sl(2, C) ⊕ sl(2, C) is multiply embedded in the two copies of the Witt algebra via L 0 , L ±n ,L 0 ,L ±n , (2.11) L n → L n n ,L n →L n n , n = 1, 2, . . . . (2.12) Using the isomorphism (2.9) this translates to the family of embeddings with the rescaling Alternatively one can rescale η µν instead of the generators l m , i.e. instead of (2.8) one would have the same relations with n = 1 and Λ is not rescaled.

Real Forms
A real Lie algebra is naturally defined as a real vector space with Lie bracket determined by real structure constants. However, for the purpose of quantum deformation one needs another, equivalent definition, which is based on the notion of a real form of a complex Lie algebra (see e.g. [35] and references therein). Thus real form is a pair (g, †) where g is a complex Lie algebra and † : g → g denotes antilinear involutive antiautomorphism mimicking Hermitian conjugation, see below. If the structure constants are real then the natural choice is X † a = −X a , X a ∈ gen(g). Favorite physicist convention is to establish imaginary structure constants and Hermitian generators Y † a = Y a , where Y a = iX a . For example, the simple sl(2, C) Lie algebra admits (up to an isomorphism) two real forms: noncompact sl(2, R) ∼ o(1, 2) ∼ su(1, 1) and compact o(3) ∼ su(2) 3 . Accordingly, the semisimple o(4, C) = sl(2, C) ⊕ sl(2, C) admits four non-isomorphic real forms: Euclidean, Lorentzian, Kleinian and quaternionic [37,38]. Each of them can be extended to the real form of the infinite-dimensional Λ-BMS algebra. However, in view of possible physical applications we are interested here in Lorentzian and Kleinian type. They correspond to de Sitter and anti de Sitter algebras of 3-dimensional Lorentzian spacetime R 1,2 .
Note that while (2.5)-(2.7) describes two different real algebras with Λ ≶ 0, in (2.10) there is only one complex algebra with two different reality conditions. If we consider the subalgebra o(4, C) spanned by L 0 , L ±1 ,L 0 ,L ±1 In (2.15) there are two real forms that correspond to the AdS and dS case resprectively. For negative Λ, i.e. the AdS case we have from (2.9) that and restrained to the subalgebra this defines two copies of the real form sl(2, R) o(2, 1). Thus this real form corresponds to the Kleinian algebra o(2, 2) o(2, 1) ⊕ō(2, 1).
3 Different notational coventions reference to different realisations or different system of generators. 4 It is worth noticing that Cartan-Weyl generators of sl(2, R) can be also considered as light-cone gen- The other case with positive Λ yields i.e. the Lorentzian real form when restricted to the o(4, C) subalgebra. This can be identified with the real structures listed in [35] in the last line of eq.(4.13) and eq.(4.14) with the automorphism E ± → −E ± ,Ē ± → −Ē ± . Note that this automorphism of the sl(2, C) can be extended uniquely to an automorphism of the Witt algebra. As mentioned above for the algebras (2.5)-(2.7) we have only one reality condition

Algebra of Surface Charges
By the Noether theorem the new-found symmetries correspond to conserved quantities, although this relation is more involved in the case of gauge symmetries (see [45]). For example in the flat case one obtains infinitely many charges parametrized by supertranslations f (z,z) [39] where the integral is defined over a boundary of a spacelike slice and m B is the Bondi mass aspect. These charges generate the symmetry transformations and are thus tightly linked to the algebra derived above. In fact the Poisson bracket of the charges has to coincide with the Lie bracket of the algebra generators up to a constant, i.e. it is in general a central extension of the algebra [45]. For the Witt algebra it is well known that the central extension is uniquely given by the Virasoro algebra (Vir) satisfying the commutation relations Furthermore, two copies of the Virasoro algebra consitute the only possibility for a central extension of W ⊕ W and Brown and Henneaux calculated in their seminal paper [7] that It should be noticed that only the first embedding (L 0 , L ±1 ) of sl(2) algebra is not affected by the presence of central charge in (2.27). The higher order embeddings, e.g. (L 0 , L ±2 ), have to take into account the central charge.

Lie Bialgebras and Deformation
Recall that a Lie bialgebra is a Lie algebra g with a cobracket δ : g → g ⊗ g satisfying the cocycle condition [47] δ and the dual version of the Jacobi identity, the so-called co-Jacobi identity A coboundary Lie bialgebra has a cobracket defined by a classical r-matrix r ∈ 2 g via and δ r satisfies the co-Jacobi identity iff r = a ∧ b fulfills the modified classical Yang-Baxter equation where and Ω has to be ad-invariant in g. If the rhs of (3.4) vanishes the Lie bialgebra is called triangular. A * -Lie bialgebra over a real form of a complex algebra with an involution * is a Lie bialgebra that is a * vector space and bracket and cobracket are * homomorphisms. The latter condition implies for coboundary Lie bialgebras defined by an r-matrix r that r * ⊗ * = −r, (3.5) where (a ⊗ b) * ⊗ * := a * ⊗ b * . We recall that two r-matrices r 1 , r 2 ∈ g ∧ g ⊂ g ⊗ g are called equivalent if there exists a Lie algebra automorphism φ ∈ Aut(g) such that (φ ⊗ φ) (r 1 ) = r 2 . Equivalent rmatrices provide isomorphic Lie bialgebra structures on g. Choosing Lie subalgebras h ⊂ g and r 1 , r 2 ∈ h ∧ h ⊂ g ∧ g one can ask now whether h-equivalence implies g-equivalence.
The answer is not obvious since in general an automorphism of h does not extend to the automorphism of the full algebra g. Therefore, the classification problem depends on the choice of an algebra we are interested in, instead of just the minimal subalgebra generated by the r-matrix itself 5 . Similarly, if (g, ) is a real form of a complex Lie algebra g then Aut(g, ) ⊂ Aut(g). Therefore, equivalent complex r-matrices may not be equivalent as real ones.
Lie bialgebras can be considered as infinitesimal versions of Hopf algebras, i.e. unitary algebras with a compatible coproduct ∆, counit ε and an antipode S generalizing the inverse (cf. [47,48] for an extensive treatment of Hopf algebras/quantum groups). In particular the cobracket is related to the coproduct via where 1/κ is the deformation parameter (see below). Starting from any Lie algebra g one can generically construct a Hopf algebra H by considering the universal enveloping algebra U g with Non-trivial coalgebra structures can be obtained by a deformation, i.e. U g is first topologically extended to U g[[1/κ]] with the so-called h-adic topology to include formal power series in the deformation parameter 1/κ. If g admits a triangular Lie bialgebra structure such a deformation can be obtained by a twisting procedure, i.e. then a twist F ∈ H ⊗ H satisfying the 2-cocycle condition exists and defines a deformed coproduct via In the following we will find that all possible deformations are of this form in the W ⊕ W algebra with the help of Lie algebra cohomology.

Cohomology
It is well known that the relation is the condition that δ is a 1-cocycle of the Chevalley-Eilenberg cohomology [49]. Recall that this cohomology is constructed on the vector spaces of cochains C n = Hom(Λ n g, V ) where V is a module of the Lie algebra g (in our case V = 2 g). The coboundary operators ∂ n : C n → C n+1 are given by wherex i means that the i-th tensor leg is dropped and denotes the right action on the module. We denote the cohomology groups by Ker ∂ n /Im ∂ n−1 ≡ H n (g, V ). If the first cohomology H 1 (W ⊕ W, 2 (W ⊕ W)) vanishes it would follow that all Lie bialgebras in W ⊕ W are coboundary. The result is even a bit stronger as not all cocycles of the cohomology define Lie bialgebras but only those which additionally fulfill the co-Jacobi identity.
In the theory of finite dimensional Lie algebras a fundamental result is the Whitehead lemma which states that all cohomology groups H n (g, V ) for finite-dimensional semi-simple g and V vanish. However, since W is not finite dimensional the lemma is not applicable here. Thus we prove the following theorem in Appendix A Theorem 1 The first cohomology H 1 (W, 2 W) of the Witt algebra with values in the exterior product of the adjoint module is zero.
Using this theorem one can also prove the following In [56] the authors independently prove a slightly more general result than our first theorem at the cost of a longer proof. The proofs presented here conceptually follow the proof of H 1 (W, W) = {0} in [50].
As stated above the second theorem establishes that all Lie bialgebra structures are coboundary and we now just need to show that the corresponding deformations are all given by a twist.

Twist Deformation and Classification
In the recent paper [28] several twists were considered for the BMS algebra in three and four dimensions (denoted by B 3 and B 4 , respectively). It was noticed that all deformations from coboundary Lie bialgebras have to be triangular since there is no ad-invariant element in ∧ 3 B 3 and ∧ 3 B 4 . The same observation also holds for W ⊕ W. Since in four dimensions the Λ-BMS is a Lie algebroid it is not known how a suitable concept of quantum group can be defined on it. In three dimensions, however, we can investigate the possible twists in a similar way.
As motivated earlier we will focus on Λ < 0 in the following, if not stated otherwise. In the contraction limit one can identify the generators K i with the momenta of the B 3 , i.e.
Let us first consider the three dimensional Poincaré (P 3 ). The abelian twist and the Jordanian twist, corresponding to the r-matrices are then also viable if P i is replaced with K i , i.e. the r-matrices are triangular and the twists satisfy the 2-cocycle condition. Also the r-matrix associated with the light-cone κ-Poincaré is triangular in three dimensions and when expressed in terms of L m ,L m and it is apparent that it coincides with r II (ζ = 0) from [35] and [40] where all classes of available twists of o(4) were obtained. Demanding triangularity we are left with the following r-matrices from the classification The abelian twist corresponds to r III and the Jordanian twist to r I .
In general there are also other classical r-matrices in W ⊕ W and the full classification is not known even for the Witt algebra [36]. For example it is easy to see that r-matrices of the form are triangular. However, one has to take into account that the asymptotic symmetry is spontaneously broken in the bulk in the sense that the vacua related by supertranslations and superrotations are physically distinguishable [39]. There is a correspondence between these vacua and the embeddings of Poincaré subalgebras which leave the associated vacuum invariant. Therefore we require that the restriction of the Hopf algebra deformed by the twist associated with a given r-matrix to an embedding is a sub Hopf algebra and we are interested in r-matrices of the form (4.5)-(4.8) where {H, E ± ,H,Ē ± } is replaced with the embedding. Note that while in the case of positive Λ the involution mixes left and right-handed elements this is not happening for negative Λ, leaving the potential possibility to use different embeddings for them.
The classification (4.5)-(4.8) is defined up to automorphisms of the o(4) but there might be inequivalent r-matrices that are related by Aut(o(4)) that do not extend to Aut(W⊕W).
Therefore, in Appendix B the classification of triangular r-matrices on o(4) is revised along the lines of [37,38] but using only the W ⊕ W automorphisms (Aut(W ⊕ W)) where 0 = γ,γ ∈ C, ,¯ = ±1. As a result we obtain the following classes of r-matrices 14) where , ,¯ ∈ {0, 1} and In the case of complex W⊕W all the parameters in 4.12-4.19 can take values in C but for the real forms associated with the involutions †, ‡ the condition (3.5) constrains the choice of parameters. For † in the classes r 1 to r 5 and r 7 , r 8 this enforces β, β 1 , β 2 , α,ᾱ ∈ iR and in r 6 one can restrict β, β 0 ∈ iR andβ,β 0 ∈ R without loss of generality.
For positive Λ, i.e. the involution ‡, the reality condition is more restrictive. In particular As the r-matrices from (4.12)-(4.19) that are not included in (4.5)-(4.8) are at least in o(4) automorphism orbits containing them one can use the inverse of the automorphisms to obtain the full twists. For example r = ( From the abelian twist for r III one then gets the twist

Twisting of the Coalgebra Sector
In this section we will explicitly construct the Hopf algebras from an abelian and a Jordanian twist. The abelian twist here has the peculiarity that it consists only of elements that are contained in all embeddings so it does not single out any specific. The Jordanian twist can already be constructed in a very basic example namely the only non-abelian two dimensional algebra where F = exp(X ⊗log(1+Y )) satisfies the 2-cocycle condition (3.8). Because of the semisimplicity of the relevant algebras here there is always a Cartan element that diagonalizes the adjoint action and thus a subalgebra of the form (4.25). Indeed many of the possible twists are of this form or have it as a building block, making it an ideal example to study.

Abelian Twist
The abelian twist can be expressed as which factorizes into the twist F 3 from [35] and a factor that only produces symmetric deformations of the coproduct. Explicitly and The antipodes can also be inferred easily from and turn out to be

Jordanian Twist
Considering the Jordanian twist Using (4.32) and the previous equations we find For general generators we find where in the last line we iteratively commuted the s terms in front of L m+kn to the right. From this formula it can be seen that in general there is an infinite number of terms in the coproduct involving a tower of infinitely many different generators. However, when restricting ourself to (two copies of) the one-sided Witt algebra spanned by {L m ,L m ; m ≤ 1} the situation is different. In that case there are only two possible embeddings with n = ±1 and by choosing n = 1 the sum over k in (4.42) terminates after min{1 − m, j} terms instead of when k = j which is not finite as we sum j to infinity. As a consequence also the sum over l terminates after 1 − m terms and there appear only a finite number of generators and as we will see also only a finite number of terms.
In that case we proceed with the identity j−1 It is straightforward to see that it holds for k = 1. Now assume that it holds for k = k , 1 ≤ k < j and it follows that the lhs for k = k + 1 is where in the last step the hockey stick identity was used. Thus we proved (4.44) by induction. Therefore and comparing with one finds that the summands in (4.46) and (1 − (m + p))  it follows that (4.42) is indeed given by (4.48) if the one-sided Witt algebra with n = 1 is considered. Plugging the result into (4.41) yields where and all sums are finite for m ≤ 1. Similarly forL m one finds (1 − (m + k 1 + p 2 )) × ...

Contraction Limit and Uniqueness of Deformations
So far we obtained general Hopf algebras on the symmetry algebras for asymptotically AdS spacetimes which algebraically also carry over to the dS case easily. There are several reasons why the asymptotically flat case is of special interest though. One motivation is the possibility of deformed dispersion relations that is associated with non-trivial Hopf algebra structures but in (A)dS there are no true momenta. However, by performing the contraction limit Λ → 0, one can obtain information about quantum groups in the three dimensional BMS from the Λ−BMS. For all r-matrices of the o(4) the contraction limit was obtained in [40]. The resulting r-matrices were compared to the full classification of r-matrices of the Poincaré algebra in three dimensions from [51] and it was claimed that all of them could be derived by an appropriate contraction limit. It should be noted that the contraction is ambiguous, i.e. the contraction of a class of r-matrices can be performed in different non-equivalent ways (cf. below) and not injective, i.e. there are r-matrices in the 3D Poincaré that can be obtained as a contraction from distinct o(4) r-matrices. A general scheme describing corresponding classes of r-matrices and their contractions is depicted in Figure 1. Let us first explicitly perform the contraction limit of r 1 from (4.12). To this end it is expressed in terms of l m , T m with the help of (2.9) and subsequently expanded in powers In order to obtain a finite result we have to rescale (α+ᾱ) → (α+α)(−Λ) and β →β √ −Λ. Then taking the limit Λ → 0 yieldŝ This is not the only possibility to abtain a finite limit, in the case α = −ᾱ one can also rescale α →α √ −Λ to get Similarly the contraction limit can be performed for all r-matrices in (4.12)-(4.19). Comparing to (the triangular part of) the classification of r-matrices on P 3 by [51] shows that the contractions of r 1 are in general not automorphic to it via a B 3 automorphism. Instead the set of r-matrices of the P 3 up to Aut(B 3 ) is strictly larger than up to Aut(P 3 ) similar to the case of non-vanishing cosmological constant.
It is also not clear a priori if the contraction from (4.12)- (4.19) to this set is surjective and one has to take into account that the contraction limit can be performed along different axis as explained in the following. The (anti) de Sitter algebra (2.5)-(2.7) is isomorphic to i.e. the fourth axis is chosen for contraction. With the isomorphism one can express the r-matrices in terms of M AB . Depending on which axis is chosen the result of the contraction differs, e.g. when choosing the second instead of the fourth axis and can thus be related to the three dimensional Lorentz sector via Note that this r-matrix of the P 3 can not be obtained from a contraction of (4.12)- (4.19) which are associated with the fourth axis. But even after taking into account the possibility to contract along different axis it turns out that there are triangular r-matrices in P 3 , e.g. that can not be obtained in its general from a contraction of a triangular r-matrix. This is important insofar it would enable a constructive method to obtain a twist for all deformations as we will see now.
Namely there is the possibility in performing the contraction limit on the level of the full twist. As an example let us consider the light-cone twist F LC = e L 0 ⊗log(1+aLn) eL 0 ⊗log(1−aLn) (4.73) corresponding also to (4.4). We express the L m ,L m in terms of l m , T m In order to obtain a finite contraction limit we have to rescale a → a = a/ √ −Λ. Expanding the first exponent of the twist in powers of √ −Λ and taking the limit then results in After repeating this procedure for the second exponent the final twist is given by F LC = exp l 0 ⊗ log(1 + a T n ) + a T 0 ⊗ l n 1 + a T n . Similarly from the contraction of the abelian twist discussed in section 4.1.1 we obtain the abelian twist discussed in [28], where its physical interpretation is discussed and compared to the Jordanian twist deformation.

(Non-)uniqueness of the Twist
From [28] we also know that the extended Jordanian twist of the form exists for the B 3 r-matrix r = l 0 ∧ T n + l n ∧ T 0 which is the contraction limit of (4.3).
Comparing the two twists reveals that they are related by a flip in first order and differ in higher orders. However, the inequivalence is only superficial as we have to take into account automorphisms on the universal envelope. We find there exist invertible elements by a similarity transformation. In general for every twist deformed Hopf algebra with F one can obtain a gauge equivalent twist via F ω = ω −1 ⊗ ω −1 F∆(ω). The new twist then satisifies the 2-cocycle condition because and f (x) = ωxω −1 establishes the isomorphism between the twisted Hopf algebras If the untwisted Hopf algbebra admits a * -structure the twist has to satisfy i.e. be unitary in order to preserve the * -structure. On the invertible element ω this enforces the unitarity condition as well In our particular example we find that in first order of 1/κ the isomorphism induced from the element ω = e − a 4 (lnT 0 +T 0 ln) (4.84) relates the extended Jordanian and the contraction limit of the light-cone twist for a = −a . It is easy to see that it is hermitian with respect to the reality condition l * m = −l m , T * m = −T m and a ∈ iR.

Deformations of the Surface Charge Algebra
As we noted in section 2.3 the algebra of the surface charges in an asymptotically AdS spacetime differs from the previously examined W ⊕ W by a central extension. Therefore we want to investigate whether this has any impact on the possible deformations. First let us note without proof that the first cohomology group H 1 (Vir ⊕ Vir, Vir ⊕ W) vanishes and there are again no ad-invariant elements in the exterior product. As a consequence all LBA are coboundary and we will now investigate how the classification of the r-matrices is affected. Note that there is no automorphism on Vir ⊕ Vir that mixes elements of W with central elements so it is enough to consider (4.12)- (4.19). More exactly, the formulas (4.10) are still valid if we assume φ(c L ) = c L . By a straightforward computation one finds that all of these r-matrices are still triangular except for those containing a 1 if we choose a different embedding. In particular for a 1 with L ±1 → L ±n we have But also Lie bialgebras from r-matrices that contain no central element can contribute extra terms, e.g. r = χL 0 ∧ L n yields For all r-matrices from (4.86)-(4.88) we can write down the twist. This is easy to see in the first two cases as everything is abelian. In the last case note that we can obtain the twist from the Jordanian twist by simply redefining L 0 → L 0 − αc L which leaves [L 0 , L p ] invariant. From the twists we can directly compute the coalgebra structures. In general these will contain infinite expressions which, as we will explore in the next section, could be remedied by considering one-sided algebras. Even though this works in the case of the Witt algebra, with a central extension the one-sided algebra is pointless to consider as the central elements do not appear in the algebra sector. Thus out of (4.86)-(4.88) only r = c L ∧ (χL 0 +χL 0 ) leads to a finite coalgebra sector. In particular,

One-sided Witt Algebra and Specialization
So far the Hopf algebras we considered were defined with the h-adic topology and thus allowed for infinite power series in the formal parameter 1/κ. While this is mathematically consistent it is ultimately problematic when interpreting the formalism in a physical context where 1/κ is to be identified with an energy scale of the order of the Planck mass.
The problem of finding a Hopf algebra (the so-called q-analog) with the same (co)algebra structure where the formal parameter can be specialized to a complex (or real) parameter is known as specialization [48,52]. Most importantly, all the structures in the q-analog need to be finite power series in the generators. It is also easy to compute and reexpressing (4.29)-(4.31) gives Endowed with this algebra and coalgebra structures the set of polynomials in the generators {L m ,L m , K, K −1 ,K,K −1 } does indeed form a q-analog of the twisted Hopf algebra and it can be defined for any q ∈ C. In particular the classical limit κ → ∞ ↔ q → 1 gives simply the Lie algebra W ⊕ W but extended by the central elements K,K.
In the case of the Jordanian twist the situation is different. We discovered in (4.41) that for L m , m ∈ Z the coproduct contains infinitely many different generators and thus it would be impossible to define a q-analog. However, by restricting to two copies of the one-sided Witt algebra W − containing L m , m ≤ 1 it was shown that all coproducts contain finitely many terms. Similarly one could use the embedding corresponding to n = −1 and restrict to W + containing L m , m ≥ −1. In order to express all algebra and coalgebra relations involving only finite powers of 1/κ the elements Π + , defined in (4.38) and its inverse Π −1 + are used. The additional commutation relations then read and similarly 14) From (4.50) one has in particular All these formulas are well defined forã ∈ C and for κ → ∞ the elements Π + , Π −1 + become central. Thus, similar to the abelian twist, the classical limit is the centrally extended Lie algebra W + ⊕ W + .
Additionally, if we consider the centrally extended algebra of surface charges, the rmatrix r = c L ∧(χL 0 +χL 0 ) with the twist F = exp(c L ⊗(χL 0 +χL 0 )) induces a Hopf algebra which admits a specialization by simply adding the elements Π = exp(χc L ),Π = exp(χc L ). These elements are thus just redefinitions of the Brown-Henneaux central charge.
It turns out that all twist deformations except for the abelian twist do not have a q-analog on the full Witt algebras. But those (and only those) which do not contain both L 1 and L −1 orL 1 andL −1 simultaneously can be shown to permit specialization on the one-sided Witt algebras in a similar way as for the Jordanian twist. Therefore we will investigate what physical implications the restriction of the generators has.
To this end we consider an asymptotically flat spacetime in four dimensions, described by B 4 (see also [28]). Recall that the superrotation Killing vectors are parametrized by functions R A on the sphere. For m ≥ −1 these functions do not contain negative powers of z,z and are thus holomorphic on the whole sphere except for z = ∞. Note that only the ordinary rotations with m = 0, 1, 2 are globally defined as the vectorfields R z m ≡ z m ∂ z , m < 0 and, after redefining ω = z −1 ,R z m = ω 2−m ∂ ω , m > 2 have a singularity at the origin [46]. In contrast, consider the following construction due to Penrose where Minkowski space is cut along the light-cone u = 0 [53], [54]. Then, after performing a diffeomeorphism on the u > 0 patch, it is glued together such that the metric is continuous at u = 0. That procedure introduces singularities and was later linked to cosmic strings [55]. A cosmic string is a topological defect with dimension one and is conjectured to exist if in the early universe the topology was not simply connected. The geometry containing a cosmic string is not exactly asymptotically flat because of the singularities but it satisfies a weaker requirement and is said to be asymptotically locally flat. A snapping string with ends at z = 0, ∞ that starts to snap from u = 0 is indeed described by Penrose' construction and furthermore one can show that certain superrotations of flat space yield cosmic strings. In other words a superrotation that is only meromorphic, i.e. isolated singularities are allowed, maps a flat geometry to a flat geometry except at the singularities [54].
If the results we obtained in three dimensions carry over qualitatively to the four dimensional case, i.e. that the consistent specialization of particular twist deformations requires the restriction to the one-sided B 4+ , then we could conclude that the remaining superrotations do not allow for the formation or decay of cosmic strings. Thus phenomenological evidence for the existence of cosmic strings, e.g. from observing gravitational wave signatures of their decay, could be used to constrain theories of quantum groups and noncommutative geometry.

Conclusion
It was shown in this work that all Lie bialgebra structures on the symmetry algebra of asymptotically (A)dS spacetime in 3 dimensions are coboundary and triangular and can thus be quantized with the help of the Drinfeld twist technique. Physically viable rmatrices, that is those which are compatible with singling out an embedding representing a vacuum choice, are all classified. Also the triangularity condition constrains the possible Lie bialgebras and in particular some of the structures that are defined on the 3-dimensional Poincaré algebra related to κ-Poincaré quantum groups are eliminated due to this. With the help of the quantization of the Lie bialgebra structures on (real forms of) o(4, C) there is a constructive way to obtain the associated Hopf algebras in all orders of the deformation parameter also for the revised classes of r-matrices in the infinite-dimensional W ⊕ W algebra and its BMS 3 contraction limits, cf. a schematic overview presented in Figure 1. When performing this twist procedure it becomes apparent that the specialization of the formal deformation parameter to real values can not be done for all Hopf algebras. Rather, in these cases, this is only possible when a subalgebra of the asymptotic symmetry algebra is considered. We propose that this would have testable consequences when transferred to a realistic setting, namely the existence of cosmic strings would be inconsistent with the quantum group deformations. Further phenomenological consequences were already studied for the flat case in [28] and we make contact with this work by performing a contraction limit.
There is a number of problems that we hope to be able to address in a future. First of all it would be of great interest to directly derive the deformation of 3 dimensional BMS algebra by using the non-perturbative methods similar to those that made it possible to derive the deformation of quantum AdS algebra of charges in [30]. Given that by AdS 3 /CFT 2 asymptotic symmetries of gravity with negative cosmological constant should correspond to the symmetries of the conformal field theory on the asymptotic boundary of spacetime it is natural to investigate CFTs with deformed conformal symmetries, to understand, among others, what would be the origin of the deformation, corresponding to the non-perturbative quantum gravity effect leading to the deformation of symmetries in the bulk.
In this paper we considered only deformations of the Λ-BMS 3 algebra (2.3)-(2.4). Recently, various generalizations of this algebra were proposed, which result from boundary conditions different from the Brown-Henneaux ones [7] adopted here. In the paper [57] the authors consider chiral boundary conditions and the resulting algebra of charges differs from (2.3)-(2.4). It would be of interest to look for possible deformations of this algebra, however since the translational sector of it differs from the one we consider, to do so one has to adopt a new class of twists. Another generalization of the Λ-BMS 3 algebra considered here was proposed recently in [58] where the conformal extension of BMS algebra were considered. Since o(3, 1) is a subalgebra of the three dimensional conformal algebra o (3,2), to deform the latter one can readily use the twists constructed here. In the case of the conformal BMS 3 there are also other twist deformations, which it would be of interest to investigate in some details One should notice however that in contrast to o(4, C) case not all quantum deformations of o(5, C) algebra are yet known. Other proposals leading to different algebras that can be, in principle, deformed using the method presented in this paper have been reported in the papers [59], [60], [61], [62].
In this paper we considered only the 3-dimensional model of non-zero cosmological constant Λ-BMS algebra. The reason why we choose to investigate here the simpler 3dimensional Λ-BMS algebra is that in 4 dimensions the non-vanishing cosmological constant extension of the BMS algebra has the structure of a Lie algebroid, with structure functions instead of structure constants [41], [43], [44], [63]. Despite some attempts to generalize the notions of quantum groups/Lie bialgebras to bialgebroids/Hopf algebroids [64], [65], [66] there is no established concept for deformations of Lie algebroids.
From the cocycle condition for m = 0, we infer which concludes the proof for cocycles of degree d = 0.
Next, let us consider cocycles of degree d = 0 which can be written in the form Note that without loss of generality we can restrict the indices i m to be smaller than m/2 since otherwise, i.e. if there is an index i m > m/2, we simply substitute i m = m − i m and γ m i m = γ m im − γ m im−m to describe the same cocycle. We will make repeated use of this in the rest of the proof.
The conditions implie that all degree 0 cocycles vanish on L 0 because it has to hold for all m and there is no ad-invariant element in 2 (W ⊕ W).
As a next step we show that all cocycles are cohomolog to 0 on L ±1 . Let us assume without loss of generality that the indices of are given by i 1 ∈ I 1 = {−p 1 , −p 2 , ..., −p n |p 1 > p 2 > ... > p n > 1, n ∈ N}. From the cocycle condition we get Lets focus on the first term in the second line of (A.13) with p 1 ; it can only be cancelled by any of the other p j terms if p 2 = p 1 − 1 which we discuss below. In the case p 2 = p 1 − 1 there are two terms that can contribute, one from the first and the second summand in the first line in (A.13) which we will call type I and type II terms respectively 6 . The type II term would correspond to i −1 = −1 − p. If it existed with non-zero γ −1 −1−p it would imply the existence of a type I term of the form γ −1 −1−p (1 − p 1 )L 1+p 1 ∧ L −1−p 1 which in turn can only be cancelled by a type II term with i −1 = −2 − p 1 . Since also none of the prefactors (2 + i −1 ) and (i −1 − 1) vanishes if p 1 = 1 this would go on forever so that we need infinitely many terms in δ(L −1 ) which is not possible. Thus γ −1 −1−p = 0 and we need a type I term which implies a type II term with the same i −1 This term can be cancelled only by a type I term with 16) and the corresponding type II term ending the sequence. The cancellation of (A.17) with (A.18) implies the following ratio of the coefficients and when considering the 0-cochain we find the same ratio between the two summands. Thus setting γ 1−p 1 (p 1 ) = γ −1 1−p 1 in the cocycle both coefficients γ −1 −p 1 , γ −1 1−p 1 vanish and therefore also γ 1 p 1 has to be zero.
Next, we have to consider the case p 2 = p 1 − 1. In (A.13) the term can be cancelled by a type I term with i −1 = 1 − p 1 or a type II term with i −1 = −2 − p 1 . If the second term does not vanish it implies the existence of a type I term with the same i −1 which can only be eliminated by a type II term with i −1 = −3 − p 1 and so on, so that infinitely many terms are necessary, ruling out this option. Using the same cochain as above in (A.20) with the same choice for s and γ s we can eliminate the coefficient γ −1 and thus the possibility to cancel (A.23) with a type I term is not possible which means that γ 1 −p 1 has to vanish. For the rest of the p j , j > 1 we can iteratively use the same argumentation. In particular the arguments with the infinite number of terms in δ(L −1 ) can be extended to the higher j as the sequence would stop at the i −1 = −p j−1 terms which already has to vanish. Furthermore, one has to add coboundaries from the cochains 25) iteratively so that the required terms in δ ... (L −1 ) are eliminated. Finally, let us explicitely consider the case p 1 = 1 that was excluded in the argumentation above. In that case and from the cocycle condition we infer that On L ±1 δ then coincides with δ r , where r = γ 1 −1 /2L 1 ∧ L −1 and thus δ = δ − δ r is zero on these elements. This concludes the proof that δ is cohomolog to 0 on L 1 .
In the next step it will be shown that δ(L 1 ) = 0 implies that δ(L m ) = 0 for m > 1. . Starting from one explicitely obtains by using (3.10) with m = 1, n = 2, m = 1, n = 3 and m = 1, n = 4 Using the same argumentation as above we can restrict i 2 to be bigger than 1 and we consider the largest index i 2 . Then, (3.10) with m = 2, n = 3 yields = 0 for the largest index i −1 and thus δ(L −1 ) = 0.
Finally, one shows explicitely that (3.10) with m = 1, n = −2 results in δ(L −2 ) = 0 and, similarly to the case of positive m that can be used to show that δ(L m ) = 0 for all m < −2, completing the proof of the first theorem.

A.2 Proof of Theorem 2
Note that a 1-cocycle δ applied to an element of W can be split into three parts δ I , δ II , δ III , mapping to W∧W, W∧W or W∧W respectively, which have to satisfy the cocycle condition separately. From the previous theorem it follows that δ I is cohomolog to zero and from (3.10) one can easily see that δ II has to vanish. Thus we only need to consider the part δ III which again can be separated by the degree d, which we define such that A general cocycle of homogenous degree d = 0 is given by we then have Using this in it follows that it follows that and it follows that and thus γ 0 i 0 = 0. Because of (A.47) γ m im = γ 0 i 0 for m < 0 and for m > 0 all coefficients are given by γ m im = γ 2 i 2 . However, from (3.10) with m = 2, n = 3 we find and thus γ 2 i 2 = 0, concluding the proof.