Twisted (2+1) $\kappa$-AdS Algebra, Drinfel'd Doubles and Non-Commutative Spacetimes

We construct the full quantum algebra, the corresponding Poisson-Lie structure and the associated quantum spacetime for a family of quantum deformations of the isometry algebras of the (2+1)-dimensional anti-de Sitter (AdS), de Sitter (dS) and Minkowski spaces. These deformations correspond to a Drinfel'd double structure on the isometry algebras that are motivated by their role in (2+1)-gravity. The construction includes the cosmological constant $\Lambda$ as a deformation parameter, which allows one to treat these cases in a common framework and to obtain a twisted version of both space- and time-like $\kappa$-AdS and dS quantum algebras; their flat limit $\Lambda\to 0$ leads to a twisted quantum Poincar\'e algebra. The resulting non-commutative spacetime is a nonlinear $\Lambda$-deformation of the $\kappa$-Minkowski one plus an additional contribution generated by the twist. For the AdS case, we relate this quantum deformation to two copies of the standard (Drinfel'd-Jimbo) quantum deformation of the Lorentz group in three dimensions, which allows one to determine the impact of the twist.


Introduction
Spacetime geometry is widely expected to exhibit discrete features at the Planck scale [1,2], which give rise to an interplay between algebra and geometry. A schematic approach to these phenomena is provided by non-commutative models, in which spacetime coordinates are replaced by (non-commuting) operators whose uncertainty relations encode the discrete nature of spacetime geometry.
Non-commutative models based on quantum groups [3,4] are well-established in this context and serve as a framework which allows one to construct non-commutative spacetimes together with an action of the corresponding generalisations of the classical spacetime kinematical groups [5]. Most of these models are based on q-deformed function algebras and the associated universal enveloping algebras, where the deformation parameter q is related to the Planck scale.
The structure of the paper is as follows. In the next section we review the basic geometrical properties of (2+1)-dimensional AdS, dS and Minkowski space, their isometry groups and the associated Lie algebras [45,46] and discuss how the latter can be grouped into a family AdS ω for which the cosmological constant plays the role of a deformation parameter ω. Section 3 exhibits the underlying Drinfel'd double structure that generates the twisted κ-AdS ω via its canonical classical r-matrix. We show that this Drinfel'd double structure leads to a spacelike κ-deformation [12,13] and that the passage to the usual time-like one (i.e., the proper κ-deformation [14]) involves a change of basis with complex coefficients. In Section 4 we compute the corresponding first-order structure of the full quantum deformation which is given by its Lie bialgebra. This Lie bialgebra depends on three deformation parameters (η, z, ϑ): the cosmological constant Λ = −ω = −η 2 , the usual quantum (Plank scale) deformation parameter κ = 1/z, and the additional twist parameter ϑ. Furthermore, we construct and analyse the first-order non-commutative quantum spacetime for AdS ω , which does not depend on the cosmological constant and hence is common to the three cases under consideration.
The construction of the full quantum twisted κ-AdS ω algebra U κ,ϑ (AdS ω ) is undertaken in Section 5. At this point, it is important to emphasise that the compatibility condition of the deformation with the underlying Drinfel'd double structure is satisfied only for very specific values of the deformation parameters z = 1/κ and ϑ, and this depends on whether one chooses the space-like or time-like deformation from Section 3. We construct the (flat) Poincaré limit ω → 0 of the quantum algebra and show that this leads to one of the known twisted κ-Poincaré algebras given in [23].
In Section 6, we construct the full Poisson-Lie group associated to this twisted deformation by making use of a simultaneous parametrisation of the (2+1) AdS, dS and Poincaré groups in terms of local coordinates. The Poisson subalgebra generated by the local spacetime coordinates {x 0 , x 1 , x 2 } provides the classical counterpart of the non-commutative twisted κ-AdS ω spacetime associated to this deformation. As expected, this Poisson AdS ω spacetime is a nonlinear algebra when the cosmological constant Λ is non-zero, and in the limit Λ → 0 reduces to linear non-commutative Minkowski spacetime that is consistent with the results in [23]. Due to the nonlinear nature of this new AdS ω spacetime, its quantisation is far from being trivial. Nevertheless, in Section 7 it is shown that by using the Lie algebra isomorphism between so(2, 2) and sl(2, R) ⊕ sl(2, R), the twisted κ-AdS Poisson-Lie group can be reconstructed and fully quantised. A final Section with comments and a discussion of open problems closes the paper.

The AdS ω Lie algebra
In this section we describe, in a unified setting, the realisation of the (2+1)-dimensional AdS, dS and Minkowski spaces as symmetric homogeneous spaces, their isometry groups, their Lie algebras and the associated left-and right-invariant vector fields. We start by considering their Lie algebras, which form a family of three six-dimensional real Lie algebras which are related by a real contraction parameter ω and will be denoted AdS ω ≡ so ω (2, 2) in the following.
In terms of a basis {P 0 , P i , J, K i }, i = 1, 2, consisting of the infinitesimal generators of, respectively, a time translation, spatial translations, a spatial rotation and boosts, the Lie brackets of AdS ω take the form where i, j = 1, 2 and ǫ ij is a skew-symmetric tensor normalised such that ǫ 12 = 1. For positive, zero and negative values of ω, this Lie bracket defines a Lie algebra isomorphic to so(2, 2), iso(2, 1) = so(2, 1) ⋉ R 3 and so(3, 1), respectively. Note that when ω = 0, this parameter can always be transformed to ω = ±1 by a rescaling of the basis. Moreover, the case ω = 0 can be understood as an Inönü-Wigner contraction [47]: so(2, 2) → iso(2, 1) ← so(3, 1). The two quadratic Casimir invariants of the Lie algebra AdS ω are given by where here and in the following we denote P = (P 1 , P 2 ), P 2 = P 2 1 + P 2 2 and similarly for any other vector object with two components. Recall that C corresponds to the Killing-Cartan form and it is related to the energy of a point particle, while W is the Pauli-Lubanski vector.

Vector model: ambient and geodesic parallel coordinates
The action of the isometry groups SO ω (2, 2) on their (2+1)-dimensional homogeneous spaces is not linear. However, this problem can be circumvented by considering the vector representation of AdS ω which makes use of an ambient space with an "extra" dimension (called s 3 below). This leads to the vector representation of AdS ω , in which the basis {P 0 , P, J, K} is represented by the following 4 × 4 real matrices [12]: The corresponding one-parameter subgroups of SO ω (2, 2) are obtained by exponentiation where hereafter the parameter η is related to the curvature by ω = η 2 = −Λ. This means that η is either a real number (η = 1/R) for AdS 2+1 or a purely imaginary one (η = i/R) for dS 2+1 .
The matrix representation of the isometry groups SO ω (2, 2) and their Lie algebras AdS ω can be characterised in terms of the bilinear form represented by the matrix which identifies them with the isometry groups of the four-dimensional linear space (R 4 , I ω ) with ambient or Weierstrass coordinates (s 3 , s 0 , s 1 , s 2 ): SO ω (2, 2) = G ∈ GL(4, R) : G T I ω G = I ω , where Y T denotes the transpose of Y . The origin of the ambient space has ambient coordinates O = (1, 0, 0, 0) and is invariant under the Lorentz subgroup SO(2, 1) ⊂ SO ω (2, 2) given by (4). The orbit passing through O corresponds to the (2+1)-dimensional homogeneous spacetime which is contained in the pseudosphere Σ ω ≡ s 2 3 + ω(s 2 0 − s 2 ) = 1, determined by I ω (5). Note that the in the limit ω → 0 (R → ∞), which corresponds to the contraction to the Minkowski space, the pseudosphere Σ ω gives rise to two hyperplanes, which are characterised by the condition s 3 = ±1 in Cartesian coordinates (s 0 , s). From now on we will identify the Minkowski space with the hyperplane given by s 3 = +1. The metric on the homogeneous spacetime is obtained from the flat ambient metric given by I ω by dividing by the curvature and restricting the resulting metric to the pseudosphere Σ ω : Now let us introduce three intrinsic spacetime coordinates that will be helpful in the sequel: these are the so-called geodesic parallel coordinates (x 0 , x 1 , x 2 ) [45], which can be regarded as a generalisation of the flat Cartesian coordinates to non-vanishing curvature. They are defined in terms of the action of the one-parameter subgroups (4) for P 0 , P on the origin O = (1, 0, 0, 0): The geometrical meaning of the coordinates (x 0 , x) that parametrize a generic point Q in the spacetime via (7) is as follows. Let l 0 a time-like geodesic and l 1 , l 2 two space-like geodesics such that these three basis geodesics are orthogonal at O. Then x 0 is the geodesic distance from O up to a point Q 1 measured along the time-like geodesic l 0 ; x 1 is the geodesic distance between Q 1 and another point Q 2 along a space-like geodesic l ′ 1 orthogonal to l 0 through Q 1 and parallel to l 1 ; and x 2 is the geodesic distance between Q 2 and Q along a space-like geodesic l ′ 2 orthogonal to l ′ 1 through Q 2 and parallel to l 2 . Recall that time-like geodesics (as l 0 ) are compact in AdS 2+1 and non-compact in dS 2+1 , while space-like ones (as l i , l ′ i ; i = 1, 2) are compact in dS 2+1 but non-compact in AdS 2+1 . Thus the trigonometric functions depending on x 0 are circular in AdS 2+1 (η = 1/R) and hyperbolic in dS 2+1 (η = i/R) and, conversely, those depending on x i are circular in dS 2+1 but hyperbolic in AdS 2+1 . By inserting the parametrisation (7) into the metric (6) we obtain the corresponding expression in terms of geodesic parallel coordinates For ω ∈ {±1, 0} this expression reduces to

Invariant vector fields
Left-and right-invariant vector fields, Y L and Y R of the group SO ω (2, 2) can be described in terms of the matrix representation (4). For this, one parametrises the group elements T ∈ SO ω (2, 2) in terms of the matrices (4) as This yields a matrix of the form where the entries A αβ and B µν depend on all the group coordinates (x 0 , x, ξ, θ) and on the parameter η. In the limit η → 0, these expressions reduce to the well known matrix representation of the Poincaré group ISO(2, 1) where the entries Λ µν parametrise an element of SO(2, 1) and depend only on (ξ, θ). From the group action of SO ω (2, 2) on itself via right-and left-multiplication, one obtains expressions for the left-and right-invariant vector fields in terms of the coordinates (x 0 , x, ξ, θ), which are displayed in table 1. We stress that the limit η → 0 of these expressions is always well defined and gives the left-and right-invariant vector fields on the Poincaré group ISO(2, 1). Table 1: Left-and right-invariant vector fields on the isometry groups of the (2+1)-dimensional (anti-)de Sitter and Minkowski spaces in terms of the sectional curvature ω = η 2 = −Λ.
Twisted κ-AdS ω algebra as a Drinfel'd double We now consider the Lie bialgebra structures underlying the quantum deformations of the isometry groups SO ω (2, 2). It is well-known that the Lie bialgebra structures underlying quantum deformations of semisimple Lie algebras are always coboundary ones [49] and hence characterised by classical r-matrices. This will be the case for all possible quantum deformations of the isometry groups of AdS and dS. Moreover, it is well known that all quantum deformations of the (2+1)-dimensional Poincaré algebra are also coboundaries [50]. These quantum deformations can therefore be classified by considering the classical r-matrices of AdS ω and relating them via the cosmological constant. The first steps towards such a classification for the Lie algebras AdS ω were recently presented in [51], where it became evident that there is a plethora of possible quantum deformations. Therefore, criteria to select the physically relevant cases are required. In this respect, the Chern-Simons formulation of (2+1)-gravity can be helpful since Poisson-Lie structures and Lie bialgebra structures arise naturally in the description of its classical phase space. The compatibility of a given classical r-matrix for AdS ω with the Chern-Simons formulation imposes restrictions on the possible r-matrices [38,52]. We have recently shown in [39] that these constraints are always fulfilled if the classical r-matrix that defines the deformation is the canonical r-matrix of certain Drinfel'd double structures of the Lie algebra AdS ω . In this Section we show that the twisted κ-deformation is one of these compatible structures and thus appears to be the appropriate generalisation of the κ-deformation in the context of (2+1)-gravity.

Drinfel'd double Lie algebras
A 2d-dimensional Lie algebra a has the structure of a Drinfel'd double (DD) [53] (see also [54,55,56]) if there exists a basis {Y 1 , . . . , Y d , y 1 , . . . , y d } of a in which the Lie bracket takes the form Note that this implies that {Y 1 , . . . , Y d } and {y 1 , . . . , y d } span two Lie subalgebras with structure constants c k ij and f ij k , respectively. From the form of the "crossed" brackets [y i , Y j ] it follows that there is an Ad-invariant quadratic form on a given by and a quadratic Casimir for a is given by A Lie algebra a with a DD structure can therefore be regarded as a pair of Lie algebras, g with basis {Y 1 , . . . , Y d } and g * with basis {y 1 , . . . , y d }, together with a specific set of crossed commutation rules (8) that ensures the existence of the Ad-invariant symmetric bilinear form (9). We refer to Lie algebras with a DD structure as DD Lie algebras in the following. A DD Lie algebra D(g) ≡ a is always endowed with a (quasi-triangular) Lie bialgebra structure (D(g), δ D ) that is generated by the canonical classical r-matrix through the coboundary relation Thus the cocommutator δ D takes the form Note that the cocommutator δ D only depends on the skew-symmetric component of the rmatrix (11), namely while the symmetric component of the r-matrix defines a canonical quadratic Casimir element of D(g) in the form (10). This implies that the associated element of D(g) ⊗ D(g) given by To summarise, if a Lie algebra a has a DD structure (8), then this implies that (a, δ D ) is a Lie bialgebra with canonical r-matrix (11). Therefore, there exists a quantum algebra (U z (a), ∆ z ) whose first-order coproduct is given by δ D (12), and this quantum deformation can be viewed as the quantum symmetry corresponding to the given DD structure for a.

The AdS Drinfel'd double
As shown in [42], the Lie algebras so(3, 1), iso(2, 1) = so(2, 1) ⋉ R 3 and so(2, 2) of the isometry groups of Lorentzian (2+1)-gravity can be described in terms of a common basis in which the cosmological constant Λ plays the role of a structure constant. In terms of the generators T a (a = 0, 1, 2) of the translations and the generators J a (a = 0, 1, 2) of the Lorentz transformations, the Lie bracket then takes the form As these are six-dimensional real Lie algebras, they can carry DD structures. In that case, the DD structure provides a canonical r-matrix and, therefore, an associated quantum deformation compatible with the constraints imposed by (2+1)-gravity. The compatible DD structures on these Lie algebras were investigated in [39]. The twisted κ-AdS r-matrix arises as the case F in the classification of admissible r-matrices, which corresponds to the DD (6 0 |5.iii|λ) in [57] and case (11) in [58]. This DD depends on an essential deformation parameter η = 0 and is explicitly given by together with the crossed relations The change of basis that transforms these expressions into (14) is Hence we obtain the AdS Lie algebra so(2, 2) with negative cosmological constant Λ = −η 2 , which is directly related to the deformation parameter η. The canonical pairing (9) takes the form where g = diag(−1, 1, 1) denotes the Minkowski metric in three dimensions. We stress that (16) was shown in [42] to be the appropriate pairing for the Chern-Simons formulation of (2+1)-gravity, while other choices of pairing lead to a different symplectic structure. By inserting the inverse of (15) into the canonical classical r-matrix (11) and by using the Casimir operator (10) in order to get a fully skew-symmetric expression as in (13), we find that the AdS deformation induced by this DD structure is generated by Since the Lie algebra elements J a are the generators of the Lorentz group and the generators T a generate the (non-commutative) AdS translation sector (a = 0, 1, 2), we conclude that the classical r-matrix (17) is just a superposition of the standard deformation of so(2, 2) [12] generated by (J 1 ∧ T 0 − J 0 ∧ T 1 ) and a Reshetikhin twist generated by J 2 ∧ T 2 (note that J 2 and T 2 commute). Therefore, we have obtained a twisted κ-AdS algebra which is a DD structure.

The dS Drinfel'd double
The analogous DD deformation of so(3, 1) is given by case (9) in [58] and case (7 0 |5.ii|λ) in [57], and corresponds to case C in [39]. It depends again on one essential deformation parameter η = 0, and the Lie bracket is given by To obtain a Lie algebra isomorphism between this DD Lie algebra and the isometry algebra (14) of the dS space, we consider the change of basis This yields the Lie algebra so(3, 1) with bracket (14) and positive cosmological constant Λ = η 2 , together with the same canonical pairing (16). Using again the Casimir operator (10), one obtains the skew-symmetric classical r-matrix from the canonical one (11) As the r-matrix r ′ F = r ′ C does not depend on the deformation parameter η and, consequently, is independent of the cosmological constant, it is a common classical r-matrix for the three DD structures so(2, 2), so(3, 1) and iso(2, 1) on the (2+1)-dimensional Lorentzian spacetimes AdS 2+1 , dS 2+1 and M 2+1 , the latter being obtained from the limit η → 0.

The AdS ω Drinfel'd double in the kinematical basis
We now analyse the DD structures of the Lie algebra AdS ω in the kinematical basis {P 0 , P, J, K} with commutation relations (1). Recall that AdS ω gives a unified description of the three Lie algebras so(2, 2), so(3, 1) and iso(2, 1), which are parametrised by the constant sectional curvature ω of their corresponding homogeneous spacetime AdS 2+1 , dS 2+1 and M 2+1 , correspondingly. In particular, we show that the classical r-matrix (18) defines two different quantum DD deformations that, according to [12,13], we shall call space-like and time-like deformations.

The space-like r-matrix
It is easy to see that the change of basis transforms (14) into (1) provided that ω = −Λ. Consequently, the deformation parameter η above coincides with the one introduced in Section 2 in the form ω = η 2 , assuming that η is a real number for AdS but a purely imaginary one for dS. Using (19), one finds that the classical r-matrix (18) is given by To construct the associated quantum deformation, we scale this r-matrix by a quantum deformation parameter z (different from η) as This allows us to relate (21) to the results in [12]. It becomes apparent that the first term in (21) is just a space-like r-matrix of type (a) for AdS ω , while the second one is a twist. From (12) it follows that the primitive (non-deformed) generators are P 2 and K 1 . Note that for the space-like r-matrix the physical dimension of z is determined by the generator P 2 of spatial translations [z] = [P 2 ] −1 , which implies that z has the dimension of a length. It is related to the usual deformation parameters κ and q by The classical limit of the quantum deformation therefore corresponds to z → 0 (κ → ∞, q → 1). In the following we therefore call (21) the twisted κ-space-like r-matrix. Moreover, the Lie algebra isomorphism so(2, 2) ≃ sl(2, R) ⊕ sl(2, R) suggests that a (real) change of basis should exist between the AdS ω basis (1) and that of two copies of sl(2, R) when ω = η 2 > 0 (η ∈ R). A direct computation shows that such an isomorphism is given by and all other Lie brackets vanish. In this basis, the classical r-matrix (20) reads It becomes apparent that it is the superposition of two standard (Drinfel'd-Jimbo [53,59]) deformations, J l + ∧ J l − , on each of the two copies of sl(2, R) but with opposite sign, and of a twist generated by J 1 3 ∧ J 2 3 . This identification U z (so(2, 2)) ≃ U z (sl(2, R)) ⊕ U −z (sl(2, R)) was first introduced in [60,61] and further investigated in [62].

The time-like r-matrix
Alternatively, the classical r-matrix (18) can be expressed as a superposition of the time-like r-matrix for AdS ω [12] with a twist. This requires the complex change of basis which again transforms (14) into (1) with ω = −Λ. The classical r-matrix (18) then takes the form r ′ F = r ′ C = i 2 (K 1 ∧ P 1 + K 2 ∧ P 2 ) + 1 2 J ∧ P 0 . Introducing again the quantum deformation parameter (22) via a rescaling we can write In this case, the first term in (25) is the time-like r-matrix of type (b) for AdS ω [12], which coincides with the κ-Poincaré r-matrix [8,14] when ω = Λ = η = 0, and the second one is the twist. The primitive generators are now P 0 and J and the dimensions of z are given by those of the generator of time translations P 0 as [z] = [P 0 ] −1 . This implies that z has the dimension of a time. Accordingly, we shall call (25) the twisted κ-time-like r-matrix. Similarly, we can also consider the isomorphism so(2, 2) ≃ sl(2, R) ⊕ sl(2, R), and apply the (complex) change of basis which gives rise to the same r-matrix (23).
4 Lie bialgebra of the twisted κ-AdS ω algebra So far, we have obtained a common classical r-matrix (18) from DD structures for the three Lie algebras that form the family AdS ω . Moreover, we have identified two physically distinct quantum deformations: the twisted κ-space-like deformation and the twisted κ-time-like one.
As the latter (without the twist motivated by (2+1)-gravity) has been widely studied in the literature due to the role of the deformation parameter z = 1/κ as a fundamental time or energy scale possibly related to the Planck length, we construct its full quantisation in the following. Note, however, that in all the expressions to be presented in the sequel, it will be always possible to obtain the corresponding κ-space-like counterparts by simply applying the map between both basis provided by (19) and (24), namely Time-like → Space-like: Moreover, in order to highlight the effect of J ∧ P 0 in the twisted κ-time-like deformation of AdS ω with classical r-matrix (25), we shall consider the two-parameter classical r-matrix given by where ϑ is a generic deformation parameter associated to the twist, that for ϑ = −iz yields the underlying DD structure. We also recall that the r-matrix (26) arises as a particular case within the classification of deformations of AdS ω given in [51] for which is a solution of the modified classical Yang-Baxter equation with Schouten bracket The first term is just the Schouten bracket for the κ-Poincaré r-matrix, while the second one includes the effect of the curvature or cosmological constant in the AdS and dS cases with ω = 0. As expected, the twist does not affect the Schouten bracket. The Lie bialgebra (AdS ω , δ) generated by (26) can be computed via (12), which yields the cocommutator δ(P 0 ) = δ(J) = 0, Denoting by {x 0 , x, θ, ξ} the dual non-commutative coordinates of the generators {P 0 , P, J, K}, respectively, one obtains from the cocommutators (27) the following dual Lie brackets between the non-commutative spacetime coordinates as well as Note that the expressions (28) are just the first-order of the non-commutative twisted κ-AdS ω spacetimes, that will be constructed in the following sections. Nevertheless, these expressions are sufficient to analyse the role of both the cosmological constant Λ = −ω and the twist with quantum parameter ϑ and to compare them to the well-known κ-Minkowski spacetime (see [8,10,11,14,22] and references therein) that is given by It was already shown in [36,51] that, at the first-order in the quantum coordinates, both non-commutative AdS and dS spacetimes coincide with the κ-Minkowski one (30) and that (see [36]) the parameter ω = η 2 only enters in the higher-order terms of the corresponding quantum deformation.

The twisted κ-AdS ω quantum algebra
In this section we construct the twisted κ-AdS ω algebra with underlying Lie bialgebra given by (1) and (27). We first present this algebra in the so-called symmetrical kinematical basis [36,51], in which the corresponding algebra without the ϑ-twist was obtained in [12]. We then present the twisted κ-AdS ω quantum deformation in a bicrossproduct-type basis, which is characterised by the fact that its limit ω → 0 gives rise to the κ-Poincaré algebra endowed with a proper bicrossproduct structure [4,9] (see also [17,18,20]). Finally, we compare these results to quantum deformations investigated in the literature [23].

"Symmetrical" basis
We start by recalling the expressions for the quantum κ-AdS ω algebra in terms of the kinematical basis (1), as obtained in [12]. The corresponding coproduct and compatible deformed commutation rules for the κ-AdS ω algebra read with ω = η 2 = −Λ. The quantum deformation of the two Casimir invariants (2) reads With these results, we construct the twisted (ϑ, κ)-quantum AdS ω algebra with general twist parameter ϑ by applying the well-known twisting procedure [63]. This preserves the deformed commutation relations (32) and hence the Casimir invariants (33). The new coproduct ∆ ϑ,z is obtained by twisting (31) with an element F ϑ ∈ κ-AdS ω ⊗ κ-AdS ω given by It satisfies the so-called twisting co-cycle and normalisation conditions where F ϑ,12 = F ϑ ⊗ id, F ϑ,23 = id ⊗ F ϑ and ǫ is the co-unit map ǫ(Y ) = 0, ∀Y ∈ AdS ω . The explicit form of the twisted coproduct is obtained through cumbersome computations and reads for i, j = 1, 2. Note that the limit ϑ → 0 is always well-defined and gives the "untwisted" coproduct ∆ z in (31).

"Bicrossproduct-type" basis
Similarly to the previous subsection we now consider the κ-AdS ω algebra expressed in the "bicrossproduct-type" basis introduced in [36]. In this basis, the coproduct is given by and the deformed commutation rules read while the deformed Casimir invariants are given by By applying the twist (34) to (35) we obtain the two-parameter coproduct Again, the untwisted coproduct (35) is recovered straightforwardly by setting ϑ = 0. As expected, there exists a nonlinear mapping between the "symmetrical" and the "bicrossproduct" bases. This is just the (z, ϑ)-generalisation of the invertible nonlinear map introduced for κ-Poincaré in [9] and coincides with the one given in [36] for the quantum κ-AdS ω algebra without the ϑ-twist. If we denote byỸ i the generators of the quantum twisted κ-AdS ω algebra expressed in the above "bicrossproduct-type" basis and by Y i the corresponding generators in the "symmetrical" basis of Section 5.1, then the nonlinear map between both bases is given byP
Note that in lowest order, these PL brackets reduce to the first-order commutators (28) and (29). On the other hand, if the ϑ-twist vanishes, we recover the PL brackets on Fun(AdS ω ) presented in [36].

Non-commutative (anti-)de Sitter and Minkowski spacetimes
The first naive possibility for the quantisation of a PL structure is given by the Weyl prescription and consists of replacing the initial PL brackets between commutative group coordinates y i by Lie brackets between the corresponding non-commutative coordinatesŷ i . This indeed works for linear PL structures, such as the κ-Poincaré group [8,9,10,11], in which the full set of PL brackets for the local spacetime coordinates is linear in the deformation parameter z = 1/κ. However, for a generic nonlinear PL bracket, this strategy will not work in general due to ordering ambiguities.
In the case of AdS ω , the PL brackets (38) and (39) should be quantised to give rise to the non-commutative quantum group Fun z,ϑ (AdS ω ). In particular, the non-commutative spacetime will arise as the quantization of the Poisson algebra (38). In this case, although the brackets are nonlinear, we have {x 1 , x 2 } = 0, and if we assume that the two corresponding quantum coordinates commute [x 1 ,x 2 ] = 0, the quantisation of the full PL bracket can be achieved via the Weyl prescription. As the only potential ordering ambiguities involve the coordinatesx 1 andx 2 , it follows that the quantum twisted AdS ω spacetime is given by Thus, we have shown how the first-order non-commutative space-time (28), which is common to the three quantum twisted κ-AdS ω algebras, is generalised with an explicit dependence on the curvature or cosmological constant ω = −Λ.
The asymmetric form of (40) with respect to thex 1 andx 2 quantum coordinates could be expected from the beginning, as we are dealing with local coordinates. However, if we consider non-commutative ambient (Weierstrass) coordinates (s 3 , s 0 , s), defined in terms of (x 0 , x) through (7), we obtain that the PL twisted κ-AdS ω spacetime can be written as a quadratic (and much more symmetric) Poisson algebra: {s 1 , s 2 } = 0, whose limit ω → 0 is given by (s 3 , s 0 , s) → (1, x 0 , x). In fact, the quantization of this Poisson algebra in a way consistent with the relations (7) will provide an alternative description for the non-commutative spacetime (40).

Twisted κ-AdS quantum group
In this section, we relate the quantum deformation of AdS ω to the standard quantum deformations of sl(2, R) via the Lie algebra isomorphism so(2, 2) ≃ sl(2, R) ⊕ sl(2, R). As in Section 3.4.1, we therefore consider two copies of the Lie algebra sl(2, R) with bases {J l + , J l − , J l 3 }, group coordinates (a +,l , a −,l , χ l ) and (a l , b l , c l , d l ) (l = 1, 2). To exhibit the relevant structures more clearly, we consider the three-parametric r-matrix which coincides with the classical r-matrix r ′ F from (23) for α = δ = − η 2 and β = η 2 . This allows one to determine the role of each term in the construction of the corresponding quantum AdS group.

Quantum standard SL(2, R) group
To quantise the PL bracket defined by the r-matrix (41), it is worth recalling the well-known construction of the standard quantum SL(2, R) group (see, for instance, [64]). For this, consider the Lie algebra sl(2, R) with Lie bracket and its fundamental representation given by This allows one to parametrise elements of the group SL(2, R) near the unit element according to where (a + , a − , χ) are local coordinates. The corresponding left-and right-invariant vector fields of SL(2, R) are given by The Sklyanin bracket (37) is induced by the standard (Drinfel'd-Jimbo) classical r-matrix [53,59] given by then this reads in terms of the local coordinates (a + , a − , χ) as Passing to the coordinates given by the matrix entries (a, b, c, d) from (42), one finds that this PL structure is homogeneous quadratic and C = ad − bc is a Casimir function. The quantisation of the PL algebra (44) in terms of the non-commutative coordinates (â,b,ĉ,d) takes the form and is compatible with the coproduct for the quantum SL(2, R) group that is induced by the group multiplication ∆(T ) =T⊗T , namely ∆(â) =â ⊗â +b ⊗ĉ, ∆(b) =â ⊗b +b ⊗d, Moreover, the deformed Casimir operator is just the "quantum determinant" ofT Note that the commutation relations (45)  7.2 Quantisation of the AdS group in the sl(2, R) ⊕ sl(2, R) basis The PL brackets for the r-matrix (41) are readily obtained by inserting the vector fields (43) into the Sklyanin bracket (37). A straightforward computation shows that, in terms of the SO(2, 2) coordinates (a l , b l , c l , d l ) (l = 1, 2), this yields It is immediate to check that C 1 = a 1 d 1 − b 1 c 1 and C 2 = a 2 d 2 − b 2 c 2 are Casimir functions for this multi-parametric Sklyanin bracket. The quantisation of this PL algebra is the quantum group SO qα,q β ,q δ (2, 2), where the (nonintertwined) deformation parameters are q α = e α , q β = e β and q δ = e δ . Evidently, the quantum group coproduct is given by a copy of (46) for each of the two non-commutative sets of coordinates (â l ,b l ,ĉ l ,d l ) (l = 1, 2). The associated q-commutation rules are the ones for two copies of the quantum SL(2, R) group with independent parameterŝ b 1â1 − q αâ1b1 = 0,ĉ 1â1 − q αâ1ĉ1 = 0, Additionally, the quantum algebra exhibits "crossed relations" that are governed by the twist parameter: To conclude the discussion, we consider the structure dual to the quantum group SO qα,q β ,q δ (2, 2) above, namely the Hopf algebra structure of the associated quantum algebra. Its coproduct is given by a formal series in the deformation parameters, and its first-order is the cocommutators obtained via (12) for the classical r-matrix (41): δ(J 1 3 ) = 0, δ(J 2 3 ) = 0, δ(J 1 + ) = J 1 + ∧ αJ 1 3 − δJ 2 3 , δ(J 2 + ) = J 2 + ∧ βJ 2 3 + δJ 1 3 , δ(J 1 − ) = J 1 − ∧ αJ 1 3 + δJ 2 3 , δ(J 2 − ) = J 2 − ∧ βJ 2 3 − δJ 1 3 .
Thus, we are effectively dealing with two almost-disjoint copies of sl(2, R) (not truly independent due to the δ-mixed terms in the cocommutators), and one readily obtains the full coproduct:

Concluding remarks
In this article, we have constructed the full quantum algebra as well as the associated noncommutative spacetimes for the family of Lie algebras AdS ω with associated DD structures. In these quantum algebras, the cosmological constant Λ plays the role of a deformation parameter in addition to the energy scale given by κ, and a further deformation parameter ϑ parametrises a twist that is motivated by the compatibility of this quantum deformation with (2+1)-gravity. It would be interesting to investigate the impact of this twist in more detail by considering multi-particle models in which the momenta are added via the coproduct of the quantum algebra. While a twist does not affect the commutation relations of the quantum algebra, it manifests itself in the coproduct, and, consequently, different values of the twist parameter ϑ lead to different momentum addition laws for point particles. It would be interesting to see if the precise value of this parameter that ensures the compatibility with (2+1)-gravity is also motivated by physical considerations in the context of multi-particle models and whether it has a geometrical interpretation.
It would also be desirable to investigate in more depth the role of the cosmological constant or curvature in these models and the spectra of the associated quantum operators. At least in the AdS case, where the quantum algebra is obtained via a twist from two commuting copies of the standard quantum deformation of SL(2, R), known results about the representation theory of this standard deformation [3,53,59] should permit one to work out in detail the spectrum of the associated quantum operators.
Finally, the introduction of another graded contraction parameter associated to the involution Π in (3) would allow one to compute the non-relativistic limits of all the quantum twisted κ-AdS ω algebras presented in the paper. This would lead to twisted κ-deformations of both the Newton-Hooke algebra and the Galilean one [23,24,65].