PLB30083S0370-2693(14)00192-010.1016/j.physletb.2014.03.036The AuthorsTheoryA (2+1) non-commutative Drinfel'd double spacetime with cosmological constantAngelBallesterosa⁎angelb@ubu.esFrancisco J.Herranzafjherranz@ubu.esCatherineMeusburgerbcatherine.meusburger@math.uni-erlangen.deaDepartamento de Física, Universidad de Burgos, E-09001 Burgos, SpainDepartamento de FísicaUniversidad de BurgosBurgosE-09001SpainbDepartment Mathematik, FAU Erlangen-Nürnberg, Cauerstr. 11, D-91058 Erlangen, GermanyDepartment MathematikFAU Erlangen-NürnbergCauerstr. 11ErlangenD-91058Germany⁎Corresponding author.Editor: M. CvetičAbstractWe show that the Drinfel'd double associated to the standard quantum deformation slη(2,R) is isomorphic to the (2+1)-dimensional AdS algebra with the initial deformation parameter η related to the cosmological constant Λ=−η2. This gives rise to a generalisation of a non-commutative Minkowski spacetime that arises as a consequence of the quantum double symmetry of (2+1) gravity to non-vanishing cosmological constant. The properties of the AdS quantum double that generalises this symmetry to the case Λ≠0 are sketched, and it is shown that the new non-commutative AdS spacetime is a nonlinear Λ-deformation of the Minkowskian one.Keywords(2+1) gravityNon-commutative spacetimeAnti-de SitterCosmological constantQuantum groupsPoisson–Lie groups1IntroductionFollowing the pioneering work by Snyder [1], non-commutative spacetimes have been pursued as an algebraic approach to model properties of spacetimes that should arise at the Planck scale (see, for instance, [2] and references therein). Essentially, if spacetime coordinates are converted into non-commuting operators in a consistent way (that usually requests invariance properties under certain spacetime transformations) then the discreteness of the spectrum of spacetime operators [1] or the non-trivial commutation relations between them [3] provide a useful description of the expected discretisation or fuzziness of spacetimes in the Planck regime.In this context, quantum groups [4,5] have provided a mathematically consistent and powerful approach to the rigorous definition of non-commutative spacetimes. In fact, any quantum (Hopf algebra) deformation of a given Lie algebra is associated in a canonical way with a Poisson–Lie (PL) structure on the associated Lie group [6,7], and the quantisation of this PL algebra defines a non-commutative algebra of local group coordinates. In particular, if the Lie group under consideration is a group of isometries of a certain spacetime (for instance, the Poincaré group in the case of Minkowski space), the associated non-commutative spacetime is defined by the commutation rules among the “quantum” space and time coordinate functions. In this way, different non-commutative algebras associated with Minkowski space have been considered in the literature (see [8–11] and references therein). However, explicit proposals concerning non-commutative spacetimes with non-vanishing cosmological constant are — to the best of our knowledge — still lacking, with the exception of the κ-AdS space introduced in [12] and further studied in [13]. In this respect, if astrophysical and cosmological tests of Planck-scale phenomena are devised (see [14] and references therein), the explicit introduction of the cosmological constant will be essential in order to model the interplay between Planck-scale effects and spacetime curvature.Moreover, for a given Lie group of isometries there are many possible quantum deformations, and a clear connection between specific deformations and any fundamental properties of (quantum) gravity on the corresponding spacetime remains as an important open problem. In (2+1) gravity, the question which quantum deformations are suitable quantum symmetries for gravity is easier to address and may provide important insights for higher dimensions.This is due to the fact that quantum group symmetries in (2+1)-gravity are not introduced ad hoc or from phenomenological considerations, but can be derived from the classical theory. They arise as the quantum counterparts of certain PL symmetries that describe the Poisson structure on the phase space of the theory in its formulation as a Chern–Simons (CS) gauge theory. There is good evidence that the relevant quantum group symmetries are Drinfel'd doubles [15–18], and specific quantum deformations of the corresponding isometry groups were proposed in [19]. Following this approach, the full classification and explicit construction of all the possible Drinfel'd double quantum deformations of the de Sitter (dS) and anti-de Sitter (AdS) groups in (2+1) dimensions that are compatible with the CS formulation of (2+1) gravity was recently given in [20].In this article, we derive first results about the non-commutative spacetimes that arise from these Drinfel'd double symmetries in (2+1)-gravity. In particular, we summarise the main properties of the AdSξ2+1 quantum spacetime that results from a certain Drinfel'd double quantum deformation of the isometry group of the AdS space that was studied in [20]. By construction, this new AdS non-commutative spacetime should be connected with (2+1)-gravity, and we show that this is indeed the case: the AdSξ2+1 algebra turns out to be generalisation to non-vanishing cosmological constant of the so(2,1)-non-commutative (2+1) Minkowski spacetime Mξ2+1, which arises naturally in the quantisation of (2+1)-gravity when the Poincaré group in (2+1) dimensions is considered as the Drinfel'd double D(sl(2,R)) [21–23] (see also [24–26] for the Euclidean case based on D(su(2))). In particular, AdSξ2+1 is found to be a deformation of the Lie algebra so(2,1), in which the deformation parameter is directly related with Λ. Therefore, the quantum group framework here presented shows that a non-vanishing cosmological constant leads to a nonlinear generalisation of the previously considered Lie algebraic noncommutative spacetimes.The article is organised as follows. The next section summarises the construction of non-commutative spacetimes from Drinfel'd doubles. This approach is illustrated in Section 3, on the example of the Poincaré algebra in (2+1) dimensions, which is the Drinfel'd double D(sl(2,R)) associated to the trivial (i.e., non-deformed) Hopf algebra structure of the universal enveloping algebra U(sl(2,R)). Moreover, it is shown that the associated non-commutative Minkowski spacetime Mξ2+1 is the quantisation of the PL structure defined by the canonical classical r-matrix of D(sl(2,R)). In Section 4, we perform the same construction but taking instead as the starting point the standard or Drinfel'd–Jimbo quantum deformation Uη(sl(2,R)) [6,27]. In this case the Drinfel'd double associated to this η-deformed Hopf algebra turns out to be the (2+1) AdS algebra, in which the cosmological constant Λ that determines the curvature is given by the quantum sl(2,R) deformation parameter as Λ=−η2. Consequently, the associated non-commutative AdS spacetime AdSξ2+1 is obtained as the quantisation of the PL structure defined by the canonical classical r-matrix provided by D(slη(2,R)).Section 5 contains the explicit construction of the non-commutative spacetime AdSξ2+1. We first introduce the vector model of the classical isometry group of the (2+1) AdS space and explicitly compute the full PL structure determined by the canonical classical r-matrix. In particular, we determine the Poisson brackets of the AdS spacetime coordinates, which are the classical counterparts of the commutators of spacetime operators in the non-commutative AdS spacetime. The resulting quantum AdS spacetime turns out to be a deformation of the so(2,1) non-commutative spacetime Mξ2+1 with the cosmological constant as a deformation parameter. We emphasise that this possibility, that allows one to perform a “cosmological limit” Λ→0, is a general feature of our construction. In all expressions for AdSξ2+1 and the corresponding quantum algebra, the cosmological constant is contained explicitly as a parameter, and the corresponding expressions for Mξ2+1 and the corresponding deformation of the Poincaré algebra can be recovered as a limit Λ→0. The final section contains some remarks and open problems for future research.2Non-commutative spacetimes from Drinfel'd doublesLet G be a finite-dimensional Lie group G with Lie algebra g and {Yi} a basis of g. Consider the natural Hopf algebra structure of the universal enveloping algebra U(g) given by the primitive coproduct map Δ0:U(g)→U(g)⊗U(g) defined by(1)Δ0(Y)=Y⊗1+1⊗Y,∀Y∈g. A quantum algebra (Uη(g),Δη) is a Hopf algebra deformation of (U(g),Δ0) in which the new “quantum” coassociative coproduct map Δη is constructed as a formal power series in the quantum deformation parameter η as(2)Δη=∑k=0∞ηkΔk=Δ0+ηΔ1+o[η2], and the product of U(g) is modified in such a way that the deformed coproduct Δη becomes an algebra homomorphism Δη:Uη(g)→Uη(g)⊗Uη(g). As the structures are defined as a formal power series in the deformation parameter η, each quantum deformation (Uη(g),Δη) defines a unique Lie bialgebra structure (g,δ) obtained as the linearisation of (Uη(g),Δη) in the parameter η. Then, the skew-symmetric part of the first-order deformation Δ1 of the quantum coproduct Δη (2) defines the cocommutator map δ:g→g⊗g(3)δ(Yn)=fnlmYl∧Ym and the skew-symmetrisation of the multiplication map gives rise to the usual Lie bracket on g.In this framework, the Hopf algebra dual to (Uη(g),Δη) can be interpreted as the non-commutative Hopf algebra of functions on the quantum group Funη(G). The algebra of quantum coordinate operators is obtained via Hopf algebra duality from the deformed coproduct (2), and its non-commutativity is a consequence of the fact that the deformed coproduct (2) is in general non-cocommutative. Moreover, the resulting quantum group is the quantisation of the unique PL structure on G that is associated to the cocommutator map δ (3). In particular, if G is the group of isometries of a spacetime, a given quantum deformation of its Lie algebra will induce a specific non-commutative spacetime.From this, it could be concluded that non-deformed Hopf algebras would be trivial from the point of view of non-commutative spacetimes. However, this is not true: the Hopf algebra (U(g),Δ0) with a trivial coproduct induces interesting non-commutative geometry structures on the double Lie group D(G), whose Lie algebra is the so-called Drinfel'd double Lie algebra D(g) (see [21,22,24]). This statement can be made explicit by considering the “trivial” Lie bialgebra structure (g,δ0) that corresponds to the “non-deformed” quantum universal enveloping algebra (U(g),Δ0) and is given by(4)[Yi,Yj]=cijkYk,δ0(Y)=0, where cijk denotes the structure constants of g with respect to the basis {Yi}. If we fix a basis {yi} of the vector space g⁎ dual to {Yi}, this yields a pairing(5)〈Yi,Yj〉=0,〈yi,yj〉=0,〈yi,Yj〉=δji,∀i,j and the “double” vector space a=g⊕g⁎ can be endowed with a Lie algebra structure, the so-called Drinfel'd double [6], given by(6)[Yi,Yj]=cijkYk,[yi,yj]=0,[yi,Yj]=cjkiyk. The Lie group D(G) with tangent Lie bialgebra a=Lie(D(G)) is the Drinfel'd double Lie group associated to the trivial Lie bialgebra (g,δ0). By construction, it is a semidirect product Lie group G⋉g⁎. Moreover, the double Lie algebra D(g)≡a can be endowed with a (quasi-triangular) Lie bialgebra structure (D(g),δD) that is defined by the “canonical” classical r-matrix(7)r=∑iyi⊗Yi or, equivalently, by its skew-symmetric counterpart r′=12∑iyi∧Yi via the coboundary relation(8)δD(X)=[X⊗1+1⊗X,r′],∀X∈D(g). The cocommutator δD derived from (8) is then given by(9)δD(yi)=12cjkiyj∧yk,δD(Yi)=0. This means that the trivial Lie bialgebra structure (g,δ0) (4) associated to the non-deformed Hopf algebra (U(g),Δ0) induces a unique non-trivial quantum deformation of the double Lie algebra D(g) whose first-order deformation in the coproduct is given by (9). Therefore, in this deformation of U(D(g)) the subalgebra of D(g) generated by g will be primitive and the coproduct of the subalgebra generated by g⁎ (which is abelian) contains all information about the deformation.In the corresponding quantum group with generators {yˆi,Yˆj}, the first-order relations for the quantum coordinates on D(G) would be given by the dual of δD (9). This means that the only non-vanishing relations for the coordinates operators will be given — up to higher-order terms — by(10)[yˆi,yˆj]=12cijkyˆk. This is a general construction that yields a non-commutative subset of local coordinates on the quantum double group whose commutation rules are just isomorphic to the ones given by the initial Lie algebra g. In other words, any finite-dimensional Lie algebra g induces a quantum deformation on the semidirect product Lie group D(G)=G⋉g⁎ in which a subset of non-commutative coordinates have commutation rules isomorphic to g. This is a canonical way to construct non-commutative spaces of Lie algebraic type with prescribed commutation rules.Now it is worth stressing that for a quantum deformation (Uη(g),Δη), the cocommutator δη is no longer trivial (f≠0 in (3)) and the Drinfel'd double Lie algebra is given by(11)[Yi,Yj]=cijkYk,[yi,yj]=fkijyk,[yi,Yj]=cjkiyk−fjikYk, which means that the semidirect product structure is lost. As a consequence, the corresponding Drinfel'd double non-commutative spacetime will be a deformation of (10) with a deformation parameter related to η.3The (2+1) Poincaré Lie algebra as a Drinfel'd doubleWe will now illustrate the construction in the preceding section with the example of the Lie algebra sl(2,R). Consider a basis of sl(2,R) in which the Lie bracket takes the form(12)[Y0,Y1]=2Y1,[Y0,Y2]=−2Y2,[Y1,Y2]=Y0, and the universal enveloping algebra with its primitive (non-deformed) coproduct (1). This corresponds to a vanishing cocommutator map δ0(Y)=0, and the Drinfel'd double Lie algebra D(sl(2,R)) is given by the relations (6), namely(13)[Y0,Y1]=2Y1,[Y0,Y2]=−2Y2,[Y1,Y2]=Y0,[y0,y1]=0,[y0,y2]=0,[y1,y2]=0,[y0,Y0]=0,[y0,Y1]=y2,[y0,Y2]=−y1,[y1,Y0]=2y1,[y1,Y1]=−2y0,[y1,Y2]=0,[y2,Y0]=−2y2,[y2,Y1]=0,[y2,Y2]=2y0. This is essentially the Drinfel'd double proposed in [21] as the algebraic structure providing the non-commutative geometry for (2+1) Lorentzian quantum gravity with vanishing cosmological constant. Now, following the approach in [19,20] we can identify this Lie algebra with the (2+1) Poincaré algebra through the following change of basis(14)J0=−12(Y1−Y2),J1=12Y0,J2=12(Y1+Y2),P0=y1−y2,P1=2y0,P2=y1+y2. It is immediate to check that the resulting Lie bracket is the one of the Poincaré algebra p(2+1)≡iso(2,1) in (2+1) dimensions(15)[J0,J1]=J2,[J0,J2]=−J1,[J1,J2]=−J0,[J0,P0]=0,[J0,P1]=P2,[J0,P2]=−P1,[J1,P0]=−P2,[J1,P1]=0,[J1,P2]=−P0,[J2,P0]=P1,[J2,P1]=P0,[J2,P2]=0,[P0,P1]=0,[P0,P2]=0,[P1,P2]=0, where Pa and Ja (a=0,1,2) are, respectively, the generators of translations and Lorentz transformations. The inverse change of basis reads(16)Y0=2J1,Y1=−J0+J2,Y2=J0+J2,y0=12P1,y1=12(P0+P2),y2=12(−P0+P2), and in terms of the new basis the pairing (5) is given by(17)〈Ja,Pb〉=gab,〈Ja,Jb〉=〈Pa,Pb〉=0, with gab=(−1,1,1). The fact that p(2+1) can be interpreted as the double Lie algebra D(sl(2,R)) yields a canonical quasi-triangular Lie bialgebra structure on p(2+1) that is generated by the classical r-matrixr=∑yi⊗Yi=−P0⊗J0+P1⊗J1+P2⊗J2. By taking into account that the two quadratic Casimirs of p(2+1) are given by(18)C1=−P02+P12+P22,C2=−J0P0+J1P1+J2P2, we see that the symmetric part of r is one half the tensorised Casimir C2. This allows one to fully skew-symmetrise the r-matrix and yields(19)r′=12(J0∧P0+P1∧J1+P2∧J2). It is directly apparent from the structure of this r-matrix that it generates a quantum deformation of p(2+1), which is a superposition of three non-commuting twists. In order to analyse this deformation in more depth, we introduce a new quantum deformation parameter ξ as a global multiplicative factor for the classical r-matrix that generates the deformation:(20)rξ≡ξr′=ξ2(J0∧P0+P1∧J1+P2∧J2). The cocommutator induced by rξ is given by (8) and reads(21)δξ(J0)=δξ(J1)=δξ(J2)=0,δξ(P0)=ξP1∧P2,δξ(P1)=ξP0∧P2,δξ(P2)=ξP1∧P0, and defines the first-order in ξ of the full quantum coproduct. Therefore, Eq. (21) implies that the corresponding full quantum deformation Uξ(D(sl(2,R)))≃Uξ(p(2+1)) (the quantum double) has a non-deformed Lorentz sector(22)Δξ(Ja)=Δ0(Ja)=Ja⊗1+1⊗Ja,a=0,1,2 and the full deformation is concentrated in the subalgebra of translations, which is an Abelian subalgebra before deformation. Consequently, both the Lorentz and the translation sectors will be Hopf subalgebras and no modification of the commutation rules (15) is expected.On the other hand, if we introduce the coordinate functions (xˆa,θˆa) that are dual to the generators (Pa,Ja) (a=0,1,2) by setting(23)〈xˆa,Pb〉=δab,〈xˆa,Jb〉=0,〈θˆa,Pb〉=0,〈θˆa,Jb〉=δab, the quantum group dual to Uξ(D(sl(2,R))) would be characterised in first-order only by non-vanishing relations obtained from dualising (21), namely(24)[xˆ0,xˆ1]=−ξxˆ2,[xˆ0,xˆ2]=ξxˆ1,[xˆ1,xˆ2]=ξxˆ0, which means that, up to higher-order terms in the quantum coordinates, the non-commutative spacetime linked to this quantum double is just the (1+1) AdS Lie algebra so(2,1), as proposed in [21].In principle, when all orders in the quantum coordinates are considered, the spacetime (24) could exhibit further non-linear contributions. The easiest way to address this question is to construct the Poisson bracket that defines the unique PL structure on the Poincaré group P(2+1)=ISO(2,1) induced by the classical r-matrix rξ (20). The quantisation of this PL structure will then provide the full non-commutative spacetime associated to this quantum double. As we will show in Section 5, in this case the PL structure is given by the Poisson analogues of the above relations, which just coincide with (24), namely(25){x0,x1}=−ξx2,{x0,x2}=ξx1,{x1,x2}=ξx0, and the remaining Poisson brackets vanish. The relations (24) therefore define the non-commutative Minkowski spacetime Mξ2+1, and they are compatible with the coproduct Δξ. The latter is given by the multiplication law of the corresponding quantum Poincaré group which, in this case, has non-commutative group parameters only in the quantum translations sector.This construction is the one underlying all previous investigations of such quantum double spacetimes (see [23–26]). All of them are Lie algebraic spacetimes, and the representation theory of the corresponding algebra (so(2,1) in the Mξ2+1 case) characterises their physics properties. On the other hand, we recall that such Lie algebraic deformation of (2+1) Minkowski space was obtained by twisting the (2+1) Poincaré algebra in [28], without making use of the underlying Drinfel'd double structure.4The (2+1) AdS Lie algebra as a Drinfel'd doubleIn the remainder of the paper, we show how the cosmological constant can be introduced into the previous construction by considering the standard quantum deformation of sl(2,R) as the starting point of the construction. In fact, as shown in [19,20], the Drinfel'd double group associated to this deformation is just the isometry group of AdS in (2+1) dimensions.Recall first that the so-called standard (or Drinfel'd–Jimbo [6,27]) quantum deformation of sl(2,R) is the Hopf algebra defined by(26)[Y0,Y1]=2Y1,[Y0,Y2]=−2Y2,[Y1,Y2]=sinh(ηY0)η,Δη(Y0)=Y0⊗1+1⊗Y0,Δη(Y1)=Y1⊗eη2Y0+e−η2Y0⊗Y1,(27)Δη(Y2)=Y2⊗eη2Y0+e−η2Y0⊗Y2. In the following, we denote this Hopf algebra by slη(2,R), where initially η is a real deformation parameter (and q=eη). The non-trivial Lie bialgebra structure associated to this deformation is given by(28)δη(Y0)=0,δη(Y1)=η2Y1∧Y0,δη(Y2)=η2Y2∧Y0. This Lie bialgebra is generated by the classical r-matrix r=η2Y1∧Y2 via the coboundary condition (8). In this case, the double Lie algebra D(slη(2,R)) is obtained from (11):(29)[Y0,Y1]=2Y1,[Y0,Y2]=−2Y2,[Y1,Y2]=Y0,[y0,y1]=−η2y1,[y0,y2]=−η2y2,[y1,y2]=0,[y0,Y0]=0,[y0,Y1]=y2+η2Y1,[y0,Y2]=−y1+η2Y2,[y1,Y0]=2y1,[y1,Y1]=−2y0−η2Y0,[y1,Y2]=0,[y2,Y0]=−2y2,[y2,Y1]=0,[y2,Y2]=2y0−η2Y0.As shown in [19,20], this Lie algebra is isomorphic to the isometry algebra of the (2+1) AdS space. In terms of the alternative basis(30)J0=−12(Y1−Y2),J1=12Y0,J2=12(Y1+Y2),P0=−η2(Y1+Y2)+(y1−y2),P1=2y0,P2=η2(Y1−Y2)+(y1+y2), the Lie bracket reads(31)[J0,J1]=J2,[J0,J2]=−J1,[J1,J2]=−J0,[J0,P0]=0,[J0,P1]=P2,[J0,P2]=−P1,[J1,P0]=−P2,[J1,P1]=0,[J1,P2]=−P0,[J2,P0]=P1,[J2,P1]=P0,[J2,P2]=0,[P0,P1]=η2J2,[P0,P2]=−η2J1,[P1,P2]=−η2J0. Following [20], we realise that (31) is the Lie bracket of so(2,2), and the deformation parameter η is directly related to the (negative) cosmological constant Λ through(32)Λ=−η2. In fact, the Lie brackets (15) and (31) are precisely the Lie brackets from [29], which allow one to express the Lie algebras p(2+1)≡iso(2,1), so(2,2) and sl(2,C)≃so(3,1) of the isometry groups of (2+1)-dimensional Minkowski, AdS and dS spaces in terms of a common basis, such that the cosmological constant appears as a structure constant.The AdS quadratic Casimirs are given by(33)C1=−P02+P12+P22+η2(−J02+J12+J22),C2=−J0P0+J1P1+J2P2, and the canonical pairing of the Drinfel'd double reads(34)〈J0,P0〉=−1,〈J1,P1〉=1,〈J2,P2〉=1,〈Ja,Jb〉=〈Pa,Pb〉=0,〈Ja,Pb〉=0fora≠b,a,b=0,1,2, which is exactly the appropriate pairing for the CS formulation of (2+1) gravity on the constant curvature space whose isometries are given by (31).By inverting (30) one finds that the canonical classical r-matrix (7) inherited from the Drinfel'd double structure reads(35)r=ηJ0∧J2+(−P0⊗J0+P1⊗J1+P2⊗J2), and its fully skew-symmetric counterpart is obtained by subtracting the tensorised Casimir C2 (see [20] for details)(36)r′=ηJ0∧J2+12(−P0∧J0+P1∧J1+P2∧J2). Again, we will multiply r′ by the quantum double deformation parameter ξ, in such a way that the classical r-matrix rξ≡ξr′ defines a quantum deformation Uξ(D(slη(2,R)))≃Uξ(AdS). Moreover, rξ defines the unique PL structure on the AdS group manifold that is associated to the previous double structure. As we will see in the sequel, once this Poisson–Hopf algebra is obtained in appropriate coordinates, its quantisation provides the quantum AdS group dual to Uξ(D(slη(2,R))), and the non-commutative AdS spacetime AdSξ2+1 will arise as the quantisation of the PL brackets among the space and time coordinates.Explicitly, the cocommutator generated by rξ reads(37)δξ(J0)=ηξJ1∧J0,δξ(J1)=0,δξ(J2)=ηξJ1∧J2,δξ(P0)=ξ(P1∧P2+ηP1∧J0+η2J2∧J1),δξ(P1)=ξ(P0∧P2+ηP0∧J0−ηP2∧J2+η2J2∧J0),δξ(P2)=ξ(P1∧P0+ηP1∧J2+η2J0∧J1), which gives the first-order term Δ1 of the full quantum coproduct in Uξ(D(slη(2,R))). Note that the zero cosmological constant limit is obtained by taking η→0 in all the above expressions, and leads to the (2+1)-Poincaré quantum double with classical r-matrix (20).In terms of the dual basis (xˆa,θˆa) (a=0,1,2) defined by (23), we find from (37) that the first-order dual Lie brackets among the spacetime coordinates are given by(38)[xˆ0,xˆ1]=−ξxˆ2,[xˆ0,xˆ2]=ξxˆ1,[xˆ1,xˆ2]=ξxˆ0. However, the additional terms in (37) that appear due to the non-vanishing cosmological constant η give rise to the first-order non-commutative relations between the quantum spacetime and Lorentz parameters:(39)[θˆ0,θˆ1]=−ηξ(θˆ0−ηxˆ2),[θˆ0,θˆ2]=−η2ξxˆ1,[θˆ1,θˆ2]=ηξ(θˆ2−ηxˆ0),[θˆ0,xˆ0]=−ηξxˆ1,[θˆ0,xˆ1]=−ηξxˆ0,[θˆ0,xˆ2]=0,[θˆ1,xˆ0]=0,[θˆ1,xˆ1]=0,[θˆ1,xˆ2]=0,[θˆ2,xˆ0]=0,[θˆ2,xˆ1]=−ηξxˆ2,[θˆ2,xˆ2]=ηξxˆ1.Therefore, up to higher-order corrections in the quantum spacetime coordinates, the non-commutative space AdSξ2+1 is again isomorphic to sl(2,R)≃so(2,1), and coincides with the quantum Minkowski space (24) obtained in the previous section. Nevertheless, modifications are expected to arise in (38) when higher-orders in terms of the quantum spacetime coordinates are considered, since in this case η≠0. To obtain such higher-order terms explicitly, one must construct the full AdS quantum algebra Uξ(D(slη(2,R))) (recall that (37) gives only the first-order deformation of the coproduct) and, afterwards, compute its dual Hopf algebra. However, this lengthy procedure can be circumvented by computing directly the PL brackets associated to the r-matrix rξ≡ξr′ (36) in terms of the classical AdS coordinates (xa,θa) (a=0,1,2), since the quantisation of this PL algebra will provide the all-orders AdS non-commutative spacetime in terms of the quantum coordinates (xˆa,θˆa).5The (2+1) AdS PL group and non-commutative spacetimeIt is well known that the action of the isometry group SO(2,2) on its homogeneous spaceAdS2+1=SO(2,2)/SO(2,1),SO(2,1)=〈J0,J1,J2〉 is nonlinear. However, a linear SO(2,2) action can be obtained by considering the vector representation of the Lie group which makes use of an ambient space with an “extra” dimension. In particular, the 4×4 real matrix representation of so(2,2) with Lie brackets (31) is given by(40)P0=(0−η200100000000000),P1=(00η20000010000000),P2=(000η2000000001000),J0=(00000000000−10010),J1=(0000000−100000−100),J2=(0000001001000000), and fulfils(41)YTIη+IηY=0,Y∈so(2,2),Iη=diag(1,η2,−η2,−η2). The exponential of (40) leads to the vector representation of SO(2,2) as a Lie group of matrices which acts linearly, via matrix multiplication, on a 4-dimensional space with ambient or Weierstrass coordinates (w3,w0,w1,w2). By definition, any element G∈SO(2,2) satisfies the relation GTIηG=Iη. Note that this realisation includes explicitly the cosmological constant parameter η, and the one-parameter subgroups of SO(2,2) obtained from (40) are, for instance,ex0P0=(cosηx0−ηsinηx000sinηx0ηcosηx00000100001),ex1P1=(coshηx10ηsinhηx100100sinhηx1η0coshηx100001). In this vector model, the 3-dimensional space AdS2+1 is identified with the orbit containing the origin of the 4-dimensional space O=(1,0,0,0), which is contained in the pseudosphere Ση provided by Iη:(42)Ση:w32+η2(w02−w12−w22)=1. Note that any element of the Lorentz subgroup SO(2,1)=〈J0,J1,J2〉 leaves the origin O invariant. The metric on AdS2+1 comes from the flat ambient metric divided by η2 (the sectional curvature) and restricted to the above pseudosphere constraint:(43)ds2=1η2(dw32+η2(dw02−dw12−dw22))|Ση. In particular, let us consider a generalisation of the Cartesian coordinates to curved spaces known as “geodesic parallel coordinates” xa [30] which are the classical counterpart of the quantum coordinates xˆa (23). They are defined through the following action of the translation subgroups on the origin O(44)(w3,w0,w1,w2)(x0,x1,x2)=exp(x0P0)exp(x1P1)exp(x2P2)O, which yields(45)w3=cosηx0coshηx1coshηx2,w0=sinηx0ηcoshηx1coshηx2,w1=sinhηx1ηcoshηx2,w2=sinhηx2η. In terms of these coordinates, the metric (43) reads(46)ds2=cosh2(ηx1)cosh2(ηx2)dx02−cosh2(ηx2)dx12−dx22. If η→0, the parametrisation (45) gives the flat Cartesian coordinates w3=1, wa=xa, and the metric (46) reduces to ds2=dx02−dx12−dx22, which is the metric of the classical Minkowski space M2+1.Consider now the 4×4 matrix element of the group SO(2,2) obtained through(47)T=exp(x0P0)exp(x1P1)exp(x2P2)exp(θ2J2)exp(θ1J1)exp(θ0J0) where the group coordinates are the ones defined above. The PL brackets associated to a given classical r-matrix r′=rijXi∧Xj and defined on the algebra of smooth functions C∞(SO(2,2)), are obtained from the Sklyanin bracket [7]:(48){f,g}=rij(XiLfXjLg−XiRfXjRg),f,g∈C∞(SO(2,2)). Thus, after computing from (47) the SO(2,2) left- and right-invariant vector fields, XL and XR, one obtains the PL brackets between the six commutative group coordinates (xa,θa) (a=0,1,2) associated to the classical r-matrix rξ≡ξr′ (36). The main point of interest are the brackets defined by the xa group coordinates, which read(49){x0,x1}=−ξtanhηx2ηϒ,{x0,x2}=ξtanhηx1ηϒ,{x1,x2}=ξtanηx0ηϒ,whereϒ(x0,x1)=cosηx0(cosηx0coshηx1+sinhηx1). The expressions (49) for the Poisson brackets are surprisingly elegant and simple, and they involve the deformation parameter η related to the cosmological constant in a symmetric way. Note also that this Poisson structure is not symplectic. Its symplectic leaves are the level surfaces of the functionC=cos(ηx0)cosh(ηx1)cosh(ηx2) which Poisson commutes with all coordinate functions. By comparing this expression with (45), one finds that this Casimir function coincides with the ambient coordinate w3. In the limit η→0, this Casimir function becomes constant. Note, however, that the associated Casimir function C′=2(1−C)/η2 can be defined in such a way that it satisfieslimη→02(1−C)η2=x02−x12−x22 which is the quadratic Casimir function for the corresponding Poisson bracket in the Minkowski case, see Eq. (38).The quantisation of the Poisson algebra (49) would be the quantum non-commutative spacetime with cosmological constant AdSξ2+1 that we are looking for. It becomes clear that the complete quantum spacetime AdSξ2+1 for η≠0 is different from Mξ2+1 (24), which is obtained from it in the limit η→0, in full agreement with the first-order spacetime Lie brackets (38). In fact, if we consider the power series expansion of (49) in terms of the cosmological constant η we obtain(50){x0,x1}=−ξx2−ηξx1x2+η2ξ(x02x2−12x12x2+13x23)+o[η3],{x0,x2}=ξx1+ηξx12−η2ξ(x02x1−16x13)+o[η3],{x1,x2}=ξx0+ηξx0x1−η2ξ(23x03−12x12x0)+o[η3]. The remaining Poisson brackets between the classical coordinates can be straightforwardly computed from (48). They are, in general, non-vanishing, which means that this quantum double is much more complicated than the expressions obtained in the limit η→0. The latter are just given by the Poisson analogues of the relations (38)–(39) that include the quantum Minkowski spacetime Mξ2+1. As a consequence, the quantisation of (49) seems to be quite complicated, since many ordering problems appear. Nevertheless, the ambient space variables (w3,wa) (45) are more accessible in this respect, since their PL brackets turn out to be homogeneous quadratic brackets, namely(51){w0,w1}=−ξw2(w3+ηw1),{w0,w2}=ξw1(w3+ηw1),{w1,w2}=ξw0(w3+ηw1),{w3,wa}=0,a=0,1,2. In agreement with the remark after Eq. (49), the coordinate w3 turns out to be a Casimir function for the PL bracket. Note also that in the limit of zero cosmological constant η→0 we again have w3→1 and wa→xa, such that the expressions in (51) reduce to the classical counterpart of (38).6Remarks and open problemsThe results of this article strongly suggest that the Lie algebraic non-commutative Minkowskian spacetimes that were proposed in the literature have to be transformed into non-linear algebras when the cosmological constant does not vanish. The algebra (49) makes this assertion explicit, and the representation theory of its quantum version is a challenging open problem. Also, it is well known that non-commutative spacetimes with zero cosmological constant of the type in (24) are associated with curved momentum spaces for point particles that are related to certain group manifolds [1,23,31–33]. The generalisation of this construction to the case with non-vanishing Λ would have as a prerequisite the knowledge of the full quantum group generated by the r-matrix rξ, in order to apply the Heisenberg double construction [34,35]. Nevertheless, some insight into the complexity of the outcoming structure can be inferred from the results here presented: such a momentum space structure with non-vanishing cosmological constant would have non-commuting momenta (31), the dispersion relation coming from the deformed analogue of the Casimir C1 in (33) would include the Lorentz sector – which would also become quantum deformed, as it can be deduced from the first-order deformation given by the cocommutator (37) – and the associated noncommutative spacetime (24) is not of Lie algebraic type. All these facts seem to indicate that this problem goes far beyond the framework on which both standard and novel approaches [36–38] to curved momentum spaces are based on, and it deserves a separate study.Moreover, besides the PL structure presented here, the classical r-matrix rξ can also be used to construct the so-called dual PL and the Heisenberg double Poisson structure that are essential in the description of point particles on compact surfaces in the CS formulation of (2+1)-gravity [39–43]. The effect of the deformation by the cosmological constant should therefore also be studied in these contexts, see also [44]. Finally, as it was explicitly shown in [20], there exist two other, non-equivalent, realisations of the isometry algebra of the (2+1) AdS algebra as a Drinfel'd double. The corresponding non-commutative spacetimes can be constructed by following the same approach as in this article. 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