Non-commutative relativistic spacetimes and worldlines from 2+1 quantum (anti-)de Sitter groups

The $\kappa$-deformation of the (2+1)D anti-de Sitter, Poincar\'e and de Sitter groups is presented through a unified approach in which the curvature of the spacetime (or the cosmological constant) is considered as an explicit parameter. The Drinfel'd-double and the Poisson-Lie structure underlying the $\kappa$-deformation are explicitly given, and the three quantum kinematical groups are obtained as quantizations of such Poisson-Lie algebras. As a consequence, the non-commutative (2+1)D spacetimes that generalize the $\kappa$-Minkowski space to the (anti-)de Sitter ones are obtained. Moreover, noncommutative 4D spaces of (time-like) geodesics can be defined, and they can be interpreted as a novel possibility to introduce non-commutative worldlines. Furthermore, quantum (anti-)de Sitter algebras are presented both in the known basis related with 2+1 quantum gravity and in a new one which generalizes the bicrossproduct one. In this framework, the quantum deformation parameter is related with the Planck length, and the existence of a kind of"duality"between the cosmological constant and the Planck scale is also envisaged.

These deformed Poincaré symmetries were later applied in the context of the so called doubly special relativity (DSR) theories [24,25,26,27,28,29,30,31,32] which introduced two fundamental scales: the usual observer-independent velocity scale c as well as an observer-independent length scale l p , which was related to the deformation parameter in the algebra. Since from all approaches to quantum gravity [33,34,35,36,37] the Planck scale is thought to play a fundamental role, DSR theories seem to establish a promising link between some Planck scale effects and quantum groups [38,39].
From a more general viewpoint, we recall that non-commutative spaces have been proposed as a suitable algebraic framework in order to describe the "quantum" structure of the geometry of spacetime at the Planck scale through a non-commutative algebra of quantum spacetime coordinates [40,41,42,43,44]. In this way the deformation parameter characterizes the non-commutativity of the spacetime algebra, thus generating uncertainty relations between non-commuting coordinates that can be thought to model a "fuzzy" or "discrete" nature of the spacetime at very small distances (or high energies) [45,46]. In particular, the non-commutative spacetime deduced from the κ-Poincaré algebra is the so-called κ-Minkowski spacetime [15,22], which is the algebra defined by the spacetime quantum group coordinates dual to the translation (momenta) generators.
In this framework, spacetime curvature (or non-zero cosmological constant) should play a relevant role concerning the possible cosmological consequences of a quantum spacetime (see, e.g., [38,47,48,49,50,51] and references therein). Therefore, it seems natural to consider the construction of the κ-deformation for the (anti-)de Sitter (hereafter (A)dS) groups, and to analyse their possible connections with quantum gravity theories with a non-zero cosmological constant. In this respect, we recall that the Hopf structure for the κ-deformation of the (2+1)D (A)dS and Poincaré (P) algebras were collectively obtained in [19], and their connection between their deformed commutation rules and 2+1 quantum gravity has been explored in [38]. The results obtained in [19] correspond to the l.h.s. of the commutative diagram: z = 1/κ (related to the Planck length l p ); reversed arrows correspond to the spacetime contraction ω → 0 and (classical) non-deformed limit z → 0. As a consequence, the construction of non-commutative (A)dS spacetimes in terms of intrinsic and ambient spacetime quantum group coordinates seems worth to be explored in detail and, moreover, the same framework could account for new proposals of non-commutative spaces of time-like geodesics (worldlines), which -to the best of our knowledge-have not been considered in the literature yet, even for the Poincaré case.
Here we present an enlarged and updated review version of our unpublished manuscript arXiv:hep-th/0401244, in which the above mentioned problems are faced for the three relativistic cases simultaneously, that is, by dealing explicitly with the spacetime curvature ω as a contraction parameter. Hence, we propose to explore the r.h.s. of the diagram (1) (dual to the l.h.s.) by computing the quantum deformation of the (2+1)D (A)dS groups (that is, Fun z ((A)dS)) which are obtained by quantizing the Poisson-Lie algebra of smooth functions on these groups (namely, Fun((A)dS)) coming from a suitable classical r-matrix. In this way, the (2+1)D non-commutative spaces (e.g. AdS 2+1 z ) can then be identified as certain subalgebras of the corresponding quantum groups. Moreover, we also construct and study in detail the corresponding 4D non-commutative spaces of worldlines. We stress that in our approach all the κ-Poincaré relations (including its non-commutative spaces) can be directly recovered from the general (A)dS expressions through the limit ω → 0. Moreover, all of the resulting non-commutative spaces are covariant under quantum group (co)actions (for the construction of Poisson and quantum homogeneous spaces we refer to [54,55,56,57,58] and references therein).
The structure of the paper is the following. In the next section we recall the basics on the (A)dS groups in (2+1) dimensions and their associated homogeneous (2+1)D spacetimes and 4D spaces of worldlines (time-like lines). Both types of spaces are described in terms of intrinsic quantities (related to group parameters) as well as in ambient coordinates with one and two extra dimensions, respectively, which will be further used in their non-commutative versions. By starting from the classical r-matrix that generates the κ-deformation, we construct in section 3 the corresponding Drinfel'd-double and obtain some preliminary information on the first-order quantum deformation, from which first-order non-commutative spaces arise. On one hand, we find that at first-order in the deformation parameter the three non-commutative relativistic spacetimes are given by the same κ-Minkowski algebra. Moreover, we show that the deformation parameter can be interpreted as a curvature on a classical dS spacetime for the three cases, thus generalizing the results obtained in [29,30] for κ-Poincaré. On the other hand, we obtain that the first-order non-commutative spaces of worldlines are in fact non-deformed ones, and a relationship with the non-relativistic (Newtonian) kinematical groups is thus established.
As an intermediate stage in the search of the quantum (A)dS groups, we compute in section 4 the invariant (A)dS vector fields and next the Poisson-Lie structures coming from the classical r-matrix generating the κ-deformation. These results enable us to propose in section 5 the non-commutative (A)dS spaces, which are written in both intrinsic and ambient coordinates. The resulting non-commutative spacetimes show how the curvature modifies the underlying first-order κ-Minkowski space, while for the non-commutative spaces of worldlines we find that 2D velocity/rapidity space (spanned by the dual coordinates to the boost generators) remains non-deformed for κ-Poincaré but becomes deformed for the (A)dS cases. Hence Lorentz invariance seems to be lost (or somewhat "deformed") when a non-zero curvature/cosmological constant is considered. Section 6 is devoted to study the (dual) quantum (A)dS algebras and their deformed Casimirs in two different basis. In particular, starting from the expressions given in [19], a non-linear transformation involving the generators of the stabilizer subgroup of a time-like line allows us to obtain these quantum algebras in a new basis that generalizes for any ω the bicrossproduct basis of κ-Poincaré [16]. These results are analysed in connection with 2+1 quantum gravity [38] and a "duality" between curvature/cosmological constant and deformation parameter/Planck length is suggested along the same lines of the so-called "semidualization" approach for Hopf algebras in 2+1 quantum gravity [59] associated to the exchange of the cosmological length scale and the Planck mass (see also [60,61]). Finally, some remarks and comments concerning recent findings in this framework close the paper.
When the Lie group SO ω (2, 2) is considered, two relevant families of symmetric homogeneous spaces [65] can be constructed as follows.
• The (2+1)D spacetime. This is a rank 1 space, associated with the automorphism ΠΘ (4) and Cartan decomposition (5), which is defined through the quotient where SO(2, 1) is the Lorentz subgroup spanned by J and K. Thus momenta P µ characterize the tangent space at the origin. This space turns out to have constant curvature equal to the contraction parameter: ω = ±1/R 2 for (A)dS and ω = 0 (R → ∞) for Minkowski, where R is the universe radius.
• The 4D space of time-like lines (or wordlines). This is a rank 2 space, related to the automorphism Π (3) and Cartan decomposition (6), which is given by where SO(2) = J and SO ω (2) = P 0 . This space is of hyperbolic type as this has constant curvature equal to −1 (i.e., −1/c 2 in terms of the speed of light). The tangent space is determined by spatial momenta P and boosts K. In fact, S (2) can also be interpreted as a (2 × 2)D relativistic phase space [66] in which position and momentum coordinates are related to the group parameters dual to P and K, respectively.
We display in table 1 each of the above symmetrical homogeneous spaces for each of the three Lorentzian Lie groups.
On the other hand, the two Casimir invariants of the Lie algebra so ω (2, 2) are given by where P 2 = P 2 1 + P 2 2 and K 2 = K 2 1 + K 2 2 . Recall that C comes from the Killing-Cartan form, while W is the Pauli-Lubanski vector.

Vector model of the (2+1)D relativistic spacetimes
The action of the (A)dS groups on the homogeneous spaces that we have just described is not linear. As it is well known, this problem can be circumvented by considering the vector representation of the Lie group SO ω (2, 2) which makes use of an ambient space with an "extra" dimension. In particular, the 4D real matrix representation of so ω (2, 2) given by fulfils (Y T is the transpose of Y ). The exponential of (8) leads to the vector representation of SO ω (2, 2) as a Lie group of matrices which acts linearly in a 4D space with ambient (or Weierstrass) coordinates (s 3 , s µ ). The one-parameter subgroups of SO ω (2, 2) obtained from (8) turn out to be where hereafter we also express the curvature as ω = ρ 2 . Hence, ρ = 1/R and ρ = i/R for AdS 2+1 and dS 2+1 , while the (contraction) limit ρ → 0 gives rise to M 2+1 .
is identified with the orbit of the origin of the spacetime O = (s 3 , s µ ) = (1, 0, 0, 0) which is contained in the pseudosphere provided by I (1) : where s 2 = s 2 1 + s 2 2 . The metric on S (1) comes from the flat ambient metric divided by the curvature and restricted to the above constraint: Ambient coordinates can be parametrized in terms of three intrinsic spacetime coordinates in different ways. We shall introduce the geodesic parallel coordinates x µ [67] through the following action of the momenta subgroups (9) on the origin: The role of the coordinates x µ that parametrize a generic point Q under (12) in the (2+1)D spacetime is the following. Let l 0 a time-like geodesic and l 1 , l 2 two space-like geodesics such that these three basic geodesics are orthogonal at O. Then x 0 is the geodesic distance from O up to a point Q 1 measured along l 0 ; x 1 is the 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 distance between Q 2 and Q along a space-like geodesic l ′ 2 orthogonal to l ′ 1 and parallel to l 2 . Recall that time-like geodesics (as l 0 ) are compact in AdS 2+1 and noncompact in dS 2+1 , while space-like ones (as l i , l ′ i ) 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 (ρ = 1/R) and hyperbolic in dS 2+1 (ρ = i/R) and, conversely, those depending on x i are circular in dS 2+1 but hyperbolic in AdS 2+1 .

Bivector model of the 4D spaces of worldlines
The action of SO ω (2, 2) on the space of time-like lines S (2) is also a non-linear one. As in the previous case, this problem can be solved by introducing an ambient space, now 6D with two "extra" dimensions, on which the group acts linearly and where S (2) is embedded. Let us consider the so called bivector representation of (2) given by [66]: where e ab is the 6 × 6 matrix with entries (e ab ) ij = δ ai δ bj . Under this representation any generator Y ∈ so ω (2, 2) fulfils By exponentiation of (14) we obtain the bivector representation of SO ω (2, 2), that is, a group of matrices which acts linearly in a 6D space with ambient (or Plücker) coordinates (η 3 , η 1 , η 2 , y 1 , y 2 , y 3 ). The origin of S (2) is O = (1, 0, 0, 0, 0, 0) and this space is identified with the intersection of the pseudosphere Σ (2) determined by I (2) with a quadratic cone P known as Plücker or Grassmann relation (invariant under the group action); these constraints are given by [66]: where η 2 = η 2 1 + η 2 2 and y 2 = y 2 1 + y 2 2 . The metric on S (2) follows from the 6D flat ambient metric divided by the negative curvature of S (2) and subjected to both conditions (15): Plücker coordinates can be expressed through four intrinsic quantities of S (2) . We shall consider the space x and rapidity (boost) ξ group coordinates. The action of the following sequence of one-parameter subgroups on O (those defining the tangent space to S (2) ) under the representation (14), gives rise to Under the contraction ρ → 0 (ω = 0), this parametrization reduces to that of the 4D Minkowskian space of worldlines LM 2×2 : Such expressions indicate that the Plücker coordinates (y, y 3 ) and (η, η 3 ) can be interpreted as "position-like" and "momentum-like" ones, respectively, within the phase space (x, ξ) [66]. In LM 2×2 the metric (16) is degenerate and reads which corresponds to a 2D space of rank 1 with negative curvature, that is, the 2D velocity Minkowskian space (so with coordinates ξ) is hyperbolic. Nevertheless, we stress that this is no longer true when ω = 0 (in both LAdS 2×2 and LdS 2×2 ) where the complete (2 × 2)D space structure is required (so with the four coordinates (x, ξ)), thus precluding the possibility of using a "reduced" 2D velocity space.

(Anti-)de Sitter Drinfel'd-doubles and first-order non-commutative spaces
The first-order deformation terms in the coproduct of the κ-Poincaré algebra [9,13,14,15,16,17,18,22] are known to be generated by the following classical r-matrix: where ∧ denotes the skewsymmetric tensor product. Recall that r is a solution of the modified classical Yang-Baxter equation and that z is related to the usual κ and q deformation parameters by z = 1/κ = ln q.
Such a classical r-matrix also holds for the (A)dS algebras [68], so that we shall consider (19) for the whole family so ω (2, 2). Hence this element gives rise to the cocommutator δ of any generator Y i through the relation δ( Next if we denote byŷ i the quantum group coordinate dual to Y i , such that ŷ i |Y j = δ i j , and write the cocommutators as δ( Lie bialgebra duality provides the so called Drinfel'd-double Lie algebra [7,8] formed by three sets of brackets: the initial Lie algebra, the dual relations and the crossed commutation rules, namely The cocycle condition for the cocommutator δ implies the following compatibility equations among the structure constants c k ij and f ij k : In our case, we denote by {θ,x µ ,ξ i } the dual non-commutative coordinates of the generators {J, P µ , K i }, respectively. Thus the (A)dS and Poincaré Drinfel'd-doubles are collectively given in terms of the curvature ω and deformation parameter z by the initial Lie algebra so ω (2, 2) (2), the dual commutators together with the crossed relations Parity and time-reversal automorphisms (3) can be generalized to the full Drinfel'ddouble as follows Π z : (P 0 , P, K, J;x 0 ,x,ξ,θ; z) → (P 0 , −P, −K, J;x 0 , −x, −ξ,θ; z), Θ z : (P 0 , P, K, J;x 0 ,x,ξ,θ; z) → (−P 0 , P, −K, J; −x 0 ,x, −ξ,θ; −z).
Since the first-order structure of the complete quantum deformation of so ω (2, 2) is described by the corresponding Drinfel'd-double, some preliminary information concerning the physical properties of the associated non-commutative spaces can be extracted from it. Notice that, in this first-order approach, all the expressions will be linear both on the generators and on the dual quantum group coordinates.

Non-commutative spacetimes: linear relations
The usual way to propose a non-commutative spacetime is to consider the commutation rules involving the quantum coordinatesx µ . Therefore, from (21) we find that the three (A)dS and Minkowskian non-commutative spacetimes are simultaneously defined by the same first-order relations: which coincide with the κ-Minkowski space, M 2+1 z , [15,16,17,18] for any value of the curvature ω. As we shall see in section 5, further corrections of (23) depending on ω will appear when the full quantum (A)dS groups are considered.
As it was already studied in [29,30], the action of the generators on the non-commutative spacetime follows by replacing formally P µ →x µ , which requires to consider the commutators involving {J,x µ , K i } within the Drinfel'd-double. Next the change of basis given by [30]:p 0 = −x 0 ,p i =x i + zK i , provides the following commutation relations that can directly be related to the initial Lie algebra (2). Consequently, whenever z is a real deformation parameter, the commutators (24), that do not depend on ω, close the Lie algebra so(3, 1) for the three cases. Hence we obtain the dS spacetime as the homogeneous space dS 2+1 ≡ J, K,p µ / J, K = SO(3, 1)/SO(2, 1), such that the deformation parameter now plays the role of the (negative) curvature equal to −z 2 .
We stress that the connection between M 2+1 z and the dS space was so established in [29] and further developed in [30], so that the expressions (24) generalize such a link for the non-commutative (A)dS cases as well.

Non-commutative spaces of worldlines: linear relations
A similar procedure suggests that the corresponding non-commutative spaces of worldlines arise within the Drinfel'd-double through the commutators ofx andξ (dual to P and K); which are trivially independent of z and ω.
The adjoint action on the quantum coordinatesx,ξ of the isotropy subgroup of a worldline spanned by J and P 0 (6) gives the following non-deformed commutation rules: where the deformation parameter z does not appear. To unveil this structure we rename the former generators as (26) we obtain the commutation relations Surprisingly enough, these relations define just the Newtonian Lie algebras coming from the non-relativistic limit c → ∞ of the three Lorentzian Lie algebras (2), keeping the constant curvature ω. Namely, the relations (27) close the oscillating Newton-Hooke, Galilei and expanding Newton-Hooke algebras [19,53,63,69] according to ω >, =, < 0, respectively. This fact is consistent with the known result that establishes that each of the non-relativistic Newtonian spaces of constant curvature ω can be obtained from the corresponding relativistic one through a contraction around a time-like line. Therefore the classical (non-deformed) picture is preserved for worldlines.
Summing up, the first-order deformation of so ω (2, 2) characterized by the chosen rmatrix (19) conveys non-commutativity on the spacetime (23) but commutativity on the space of worldlines (25). Furthermore, space isotropy is ensured in both cases as the corresponding commutation relations do not involve the quantum rotation coordinateθ.

A Poisson-Lie structure on the (anti-)de Sitter groups
So far we have studied the first-order quantum (A)dS deformation. However, the obtention of the complete (in all orders in z and in the generators) deformation of a semisimple group in terms of local coordinates is, in general, a very involved task. A way to study the noncommutative structures is to compute the Poisson-Lie brackets (derived from (19)) for the commutative coordinates and next to analysing their possible non-commutative version.
In particular, let us consider the 4 × 4 matrix element of the group SO ω (2, 2) obtained through the following product written under the representation (9): where the group coordinates are commutative ones. Left-and right-invariant vector fields, Y L and Y R , of SO ω (2, 2) deduced from (28) are displayed in table 2. Notice that such expressions hold for any value of the curvature ω. In the Poincaré case with ω = 0, the vector fields, coming from the smooth contraction limit ρ → 0, are rather simplified as shown in table 3.
The Poisson-Lie brackets that close the algebra of smooth functions on the (A)dS groups, Fun(SO ω (2, 2)) (r.h.s. of the diagram (1)), associated to an r-matrix r = r ij Y i ⊗Y j come from the Sklyanin bracket defined by [70]: Thus by substituting the vector fields of table 2 and the classical r-matrix (19) in (29) we obtain the Poisson-Lie brackets between the six commutative group coordinates {θ, x µ , ξ i } which are splitted in the following three sets.

Non-commutative (anti-)de Sitter spaces
In general, the simplest way to quantize Poisson-Lie structures [5,70] consists in the usual Weyl substitution of the initial Poisson brackets between commutative coordinates by commutators between non-commutative coordinates. Several quantum deformations of non-semisimple groups have been constructed by applying this procedure; amongst them we underline the κ-Poincaré group [15,16,17,18], for which the full set of commutation rules are linear in the deformation parameter. Nevertheless, we stress that a quantum group does not always coincide with the Weyl quantization of their underlying Poisson-Lie brackets, specially when dealing with semisimple groups as the (A)dS ones, since ordering problems often appear during the quantization procedure.
In our case, the Poisson-Lie brackets (30)- (33) are the commutative counterpart of the full non-commutative quantum (A)dS groups Fun z (SO ω (2, 2)). Thus if we write a generic Poisson-Lie bracket as {y i , y j } = zf (y 1 , . . . , y 6 ), its non-commutative version would read [ŷ i ,ŷ j ] = zf (ŷ 1 , . . . ,ŷ 6 ) + o(z 2 ), whose first-order in allŷ k and z is given by (21), while the o(z 2 ) terms come from the reordering of the quantum coordinatesŷ k . By following this point of view, we will analyse the generalization of the first-order non-commutative spaces presented in section 3 to higher orders in the quantum coordinates and up to second-order in z.

Non-commutative spacetimes
The first set of Poisson-Lie brackets (30) allows us to introduce the defining commutation relations of the (2+1)D non-commutative (A)dS spacetimes, namely These expressions demonstrate how the underlying first-order M 2+1 z (23) for the (A)dS groups is now generalized with an explicit dependence on the curvature ω, in such a way that three different cases appear. Furthermore, space isotropy is preserved in the quantum case sinceθ is again absent (as well as the quantum boost coordinatesξ).
It is worth mentioning that the asymmetric form of (34) could be expected fom the beginning (see, e.g., the classical metric (13)) as we are dealing with local quantum coordinates. However, if we consider non-commutative ambient (Weierstrass) coordinates (ŝ 3 ,ŝ µ ) defined in terms of the former onesx µ by the same formal relations (12) and subjected to the constraint (10), we obtain the non-commutative spacetimes written in a fully symmetric way as a quadratic algebra: These expressions are clearly invariant under the permutationŝ 1 ↔ŝ 2 . The two first relations in (35) are just M 2+1 z , sinceŝ 3 → 1 when ω → 0.

Non-commutative spaces of worldlines
Likewise, a first insight into the non-commutative (A)dS and Minkowskian spaces of timelike lines can be performed by starting with the Poisson-Lie brackets (31) among space x and boost ξ group coordinates.
Let us consider firstly the quantum Poincaré group with ω = 0. In this case, the expressions (31) are rather simplified. Their corresponding quantum deformation reads which determine the complete (in all orders in z,x andξ) non-commutative Minkowskian space of worldlines LM 2×2 z . Hence the commutativity of the first-order relations [x i ,ξ j ] is lost (see (25)). It is worth mentioning that [ξ 1 ,ξ 2 ] = 0 ensures the self-consistency of the remaining commutators as no ordering problems appear. Moreover, this condition means that the 2D velocity space remains non-deformed (commutative).
On the contrary, in the (A)dS cases with ω = 0 the brackets (31) are rather involved, again asymmetric, andξ 1 ,ξ 2 do not longer commute. As in the classical case, we can consider the non-commutative ambient (Plücker) coordinates (η 3 ,η,ŷ,ŷ 3 ) formally defined by (17), but now with non-commutative entriesx andξ. Then the non-commutative (A)dS spaces of worldlines are given by the following quadratic relations which are invariant under the map defined by The flat contraction ω = 0 of (36) to LM 2×2 The commutativity of the 2D velocity space in LM 2×2 z now comes from the fact that the commutators involving the "momentum-like" coordinates (η 3 ,η) vanish; note that they only depend onξ, which commute in this case. On the contrary, the "position-like" ones (ŷ 3 ,ŷ) do not commute; they depend on bothξ andx (see (18)). We stress that this quantum Poincaré group property is in full agreement with the study developed in [71,72] by working with the (dual) quantum algebra. In these works it is shown how the κdeformed Poincaré boost transformations close a group as in the non-deformed case, and the additivity of the boost parameter for transformations along a same direction is also preserved. Therefore, the relations (36) indicate that such properties may be either lost or somewhat modified in the quantum (A)dS groups.

Quantum (anti-)de Sitter algebras
In this section we firstly review the Hopf algebra structure and the associated invariants of the quantum (A)dS algebras that quantize the cocommutators (20), commutators (2) and Casimirs (7). These results are presented in the kinematical basis in which they were formerly obtained [19], and the contraction ω = 0 gives the κ-Poincaré written in the form deduced in [13]. We also point out some remarks concerning the connection between these structures and quantum gravity that has been introduced in [38]. Secondly, we obtain a new basis through a non-linear map for these quantum algebras in such a manner that the κ-Poincaré algebra is recovered in the so called bicrossproduct basis [8,16]. This change of basis, at the level of the quantum algebra, is the dual counterpart of a change of non-commutative coordinates on the quantum group.

"Symmetrical" basis
The Drinfel'd-Jimbo quantum deformation of so(4, C) was obtained in [11] by considering two copies of the quantum sl(2) algebra [3] and applying the prescription U z (so(4, C)) = U z (sl(2)) ⊕ U −z (sl (2)). From this result, a further analysis of real forms together with a contraction scheme led to quantum deformations of the family of (2+1)D kinematical algebras [19,68] which included, amongst others, the κ-deformation of the three relativistic algebras (2) with underlying Lie bialgebra (20), that we denote here by U z (so ω (2, 2)). The Hopf structure of U z (so ω (2, 2)) is characterized by the following coproduct and commutation relations [19]: [J, Counit and antipode maps can directly be derived from the Hopf algebra axioms. The deformation of the two Casimir invariants (7) turns out to be This deformation is governed by the generators spanning the isotropy subgroup of a worldline, P 0 and J, which remain non-deformed at the level of the coproduct (38). Thus deformed functions of P 0 and J arise for the coproduct of P and K in both spaces in the tensor product, in such a manner that their coproduct is invariant under the composition σ • T of the flip operator, σ(x ⊗ y) = y ⊗ x, and a "parity" operator T acting on the deformation parameter as T (z) = −z. Hence we say that U z (so ω (2, 2)) is written in a "symmetrical" basis.
The physical dimension of z is inherited from P 0 , [z] = [P 0 ] −1 , so that this can be interpreted as a fundamental length (provided that c = 1), which in the usual DSR theories is considered to be of the order of the Planck length l p .
Expressions (38)- (40) show the commutativity character of the l.h.s. of the diagram (1); the limit z → 0 in each of the three particular quantum algebras contained in the family U z (so ω (2, 2)) leads to the corresponding Lie algebra so ω (2, 2) and its invariants, while the contraction ω = 0 (ρ → 0) in U z (so ω (2, 2)) gives rise to the κ-Poincaré algebra and its deformed invariants in the basis formerly worked out in [13]. Both types of limits can be applied separately. Such a viewpoint suggests some kind of "duality" between quantum deformations induced by z and "curvature deformations" parametrized by ω; recall that (A)dS symmetries can be considered as a "classical" deformation of Poincaré invariance [52,53,65].
The dual relationship between quantum deformation parameters and classical deformation ones (z ↔ ω) was already analysed for the (1 + 1)D case in [73] and, in fact, the role of z as a curvature also arises within the Drinfel'd-double approach, as commented in section 3.1 (see [29,30]). Moreover, the "semidualization" approach in 2+1 quantum gravity introduced in [59] (see also [60,61]) provides a more complete Hopf algebraic understanding of this duality between the Planck scale l p and the cosmological constant ω = ρ 2 = −Λ, and shows its direct connection with the quantum version of the so-called Born reciprocity principle [74] between (now non-commutative) coordinates and (curved) momenta. This framework is also helpful in order to understand the existing constraints on contraction limits involving these two parameters, that we will discuss in the sequel.

Contractions and the Planck length
It has been shown (see [38] and references therein) that there exists a natural link bewteen the deformed commutation relations (39) for the (A)dS algebras with ω = 0 and 2+1 quantum gravity. In the latter framework, the curvature and deformation parameters can be identified with the cosmological constant Λ ≡ −ω and the fundamental Planck length l p ≡ z. Hence, by considering these identifications, the above results show that both limits Λ → 0 and l p → 0 can be taken independently (if Λ → 0 we obtain κ-Poincaré with l p ≡ z), and that a "duality" l p ↔ Λ might exist. Nevertheless, in [38] it is emphasized that the contraction to κ-Poincaré should be taken as the simultaneous limits z → 0 and Λl 2 p → 0, due to the coupling z = √ Λl p existing in 2+1 quantum gravity. This fact can be explained from a Lie bialgebra contraction approach [68] as follows.
Alternatively to the expressions (38) and (39) defining U z (so ω (2, 2)), that unify the Hopf structure for the quantum (A)dS and Poincaré algebras parametrized by ω, one could have started from the quantum AdS algebra U z (so(2, 2)) with ω = 1 (or from U z (so(3, 1)) with ω = −1). Next the Lie bialgebra contraction analysis of the r-matrix (19) and Lie bialgebra (20) shows that there exists a unique quantum Inönü-Wigner contraction (a coboundary Lie bialgebra contraction) that ensures the convergency of both (19) and (20). This is defined through the new generators Y ′ and deformation parameter z ′ given by [19,68]: ( √ ω = ρ plays the role of the usual Inönü-Wigner contraction parameter). This map is associated to the Z 2 -grading of U z (so(2, 2)) given by the composition of the deformed parity and time-reversal (22): By computing the coproduct and commutation relations for the new generators Y ′ from U z (so(2, 2)) and then by applying the limit √ ω → 0, one finds the κ-Poincaré algebra U z ′ (iso(2, 1)) in terms of the transformed deformation parameter z ′ . Therefore if z ′ is now taken as the Planck length l p and ω ≡ −Λ > 0, the map (41) means that z = √ −Λl p , so that the contraction √ −Λ → 0 conveys the limit z → 0 as well, leaving z ′ ≡ l p as the only fundamental scale in κ-Poincaré, in agreement with [38].

"Bicrossproduct-type" basis
So far, most of the applications of the κ-Poincaré algebra have been developed by working in the so called bicrossproduct basis [16,22], in which the Lorentz sector has non-deformed commutation rules. Thus it is natural to wonder about the existence of a non-zero curvature counterpart of such a basis. This can be achieved by means of the ω-generalization of the invertible non-linear map introduced for κ-Poincaré in [16]: This transformation can be written as the following matrix transformation of P and K depending on functions of the generators of the isotropy subgroup of a worldline: The transformed coproduct and commutation rules of U z (so ω (2, 2)) in this new basis turn out to be ∆(P 0 ) = 1 ⊗P 0 +P 0 ⊗ 1, ∆(J ) = 1 ⊗J +J ⊗ 1, Therefore the Lorentz sector remains deformed in the quantum (A)dS algebras with ω = 0, while the contraction ω = 0 (ρ → 0) produces the κ-Poincaré algebra in the bicrossproduct basis, in which the only deformed commutation rules are [P i ,K j ]. However, although the Lorentz sector in κ-Poincaré is non-deformed, the quantum deformation is still kept in the coproduct for the boost generators (43). Note also that this new coproduct is not invariant under the map σ • T .
The corresponding deformed Casimirs are obtained from (40) and read We remark that for both bases the full Hopf structure U z (so ω (2, 2)) is invariant under the quantum involutions (22) (as it should be) and also under the "classical" symmetry (P 0 , P 1 , P 2 , K 1 , K 2 , J) → (P 0 , P 2 , P 1 , K 2 , K 1 , −J), which shows that the equivalences P 1 ↔ P 2 and K 1 ↔ K 2 are preserved by this quantum deformation. This, in turn, means that there is no privileged space/boost direction. The analogous (dual) maps to (46) in the non-commutative spacetimes and spaces of worldlines are given by the interchangeŝ 1 ↔ŝ 2 and (37), respectively.

Concluding remarks
We have presented a unified and global study of the κ-deformation of the (A)dS and Poincaré algebras and groups in 2+1 dimensions by making use of an explicit contraction parameter ω that corresponds to the curvature/cosmological constant of the underlying classical spacetimes. We remark that the limit ω → 0 is always well defined in all the expressions, thus providing a straightforward Poincaré/Minkowskian counterpart of all the results here presented.
At the quantum algebra level, we have introduced a new basis (42) for the quantum (A)dS algebras, which is the analogous of the κ-Poincaré bicrossproduct basis. As far as the (dual) quantum groups are concerned, the results cover the non-commutative (A)dS spacetimes (34) up to second-order in the deformation parameter, thus generalizing the κ-Minkowskian spacetime (23) which turns out to be their common first-order seed. Furthermore, we have presented the first approach, to the best of our knowledge, to noncommutative spaces of worldlines. For the three relativistic cases they are related to phase spaces (x,ξ) ↔ (P, K), which are different from the κ-Poincaré phase spaces proposed in [29,30] (see also [75]). We also remark that an appropriate treatment of the classical (non-deformed) structures have allowed us to write both non-commutative spacetimes and spaces of worldlines in terms of ambient ("global") coordinates, by starting with the particular expressions (12) and (17) for the ("local") parametrizations of such spaces. In such ambient coordinates, the non-commutative spaces turn out to be determined through quadratic commutation relations (35) and (36). Morevover, they are shown to be symmetric with respect to some maps that generalize the P 1 ↔ P 2 and K 1 ↔ K 2 invariance of the dual quantum algebras.
It is worth stressing that the connection between the Poisson-Lie group approach here presented and the role that classical r-matrices and Drinfel'd-doubles play in the context of 2+1 quantum gravity [76,77,78,79,80,81,82,83,84] has been studied in detail in the works [85,86,87,88,89,90]. Also, the deformed Casimir operators (40) (or (45)) can be used to provide modified dispersion relations, which should be related to those appearing in several phenomenological approaches to quantum gravity (see [91,92,93,94]). Some comments on other possible quantum deformations of the (A)dS algebras are in order. Since any possible quantum deformation of the (A)dS algebras has to come from a classical r-matrix, a complete classification of the latter would be certainly useful in order to obtain and analyse other physically interesting quantum (A)dS groups (for each of them, the vector fields displayed in table 2 would give rise to the associated Poisson non-commutative spaces). In this respect, the full classification of classical r-matrices for the (3+1)D Poincaré Lie algebra can be found in [95]. For the (2+1)D (A)dS algebras a similar result has been recently given in [96] (see also [97]).
On the other hand, the generalization of the (2+1)D κ-(A)dS algebras and groups to the (3+1)D case can be obtained by generalizing the classical r-matrix (19) to (see [68]) which include a term J 1 ∧ J 2 coming from the rotation sector. This term does not appear either in the (3+1)D κ-Poincaré algebra (with ω = 0) or in the (2+1)D κ-(A)dS algebras. A twisted version of (47) with a second deformation parameter ϑ was considered in [98] by imposing some physical requirements, and the very same classical r-matrix has been derived in [99] from a Drinfel'd-double approach, namely r = z(K 1 ∧ P 1 + K 2 ∧ P 2 + K 3 ∧ P 3 ) + z √ ω J 1 ∧ J 2 + ϑJ 3 ∧ P 0 .
The corresponding Poisson-Hopf algebra has been recently constructed in [100] so obtaining the (3+1)D Poisson version of the coproduct (43), commutation relations (44) and