The quantum duality principle is used to obtain explicitly the Poisson analogue of the κ-(A)dS quantum algebra in (3+1) dimensions as the corresponding Poisson-Lie structure on the dual solvable Lie group. The construction is fully performed in a kinematical basis and deformed Casimir functions are also explicitly obtained. The cosmological constant Λ is included as a Poisson-Lie group contraction parameter, and the limit Λ → 0 leads to the well-known κ-Poincaré algebra in the bicrossproduct basis. A twisted version with Drinfel'd double structure of this κ-(A)dS deformation is sketched. PACS: 02.20.Uw 04.60.-m KEYWORDS: anti-de Sitter, cosmological constant, quantum groups, Poisson-Lie groups, Lie bialgebras, quantum duality principle.techniques [17,18]. One of the main features of the κ-Poincaré quantum algebra (which is the Hopf algebra dual to the quantum Poincaré group, and is defined as a deformation of the Poincaré algebra in terms of the dimensionful parameter κ) consists in its associated deformed second-order Casimir, which leads to a modified energy-momentum dispersion relation. From the phenomenological side, this type of deformed dispersion relations have been proposed as possible experimentally testable footprints of quantum gravity effects in very different contexts (see [19,20,21] and references therein).Moreover, if the interplay between quantum spacetime and gravity at cosmological distances is to be modeled, then the curvature of spacetime cannot be neglected and models with nonvanishing cosmological constant have to be considered [22,23,24,25]. Thus, the relevant kinematical groups (and spacetimes) would be the (anti-)de Sitter ones, hereafter (A)dS, and the construction of quantum (A)dS groups should be faced. In (1+1) and (2+1) dimensions, the corresponding κ-deformations have been constructed [26,27] (see also [28,29,30] for classification approaches). In fact, it is worth stressing that the κ-(A)dS deformation introduced in [27] was proposed in [31] as the algebra of symmetries for (2+1) quantum gravity (see also [32]), and compatibility conditions imposed by the Chern-Simons approach to (2+1) gravity have been recently used [33] in order to identify certain privileged (A)dS quantum deformations [34] (among them, the twisted κ-(A)dS algebra [35,36,37,38]).Concerning (3+1) dimensions, we recall that in the papers [13,14,15,16] the κ-Poincaré algebra was obtained as a contraction of the Drinfel'd-Jimbo quantum deformation [6,39] of the so(3, 2) and so(4, 1) Lie algebras, by starting from the latter written in the Cartan-Weyl or Cartan-Chevalley basis, and then obtaining suitable real forms of the corresponding quantum complex simple Lie algebras. However, to the best of our knowledge, no explicit expression of the (3+1) κ-(A)dS algebras in a kinematical basis (rotations J, boosts K, translations P ) and including the cosmological constant Λ has been presented so far, thus preventing the appropriate physical analysis of the interplay between Λ and the quantum deformation. Moreover, explicit e...

The correspondence between Poisson homogeneous spaces over a Poisson-Lie group G and Lagrangian Lie subalgebras of the classical double D(g) is revisited and explored in detail for the case in which g = D(a) is a classical double itself. We apply these results to give an explicit description of some coisotropic 2d Poisson homogeneous spaces over the group SL(2, R) ∼ = SO(2, 1), namely 2d anti de Sitter space, 2d hyperbolic space and the lightcone in 3d Minkowski space. We show how each of these spaces is obtained as a quotient with respect to a Poisson-subgroup for one of the three inequivalent Lie bialgebra structures on sl(2, R) and as a coisotropic one for the others.We then construct families of coisotropic Poisson homogeneous structures for 3d anti de Sitter space AdS 3 and show that the ones that are quotients by a Poisson subgroup are determined by a three-parameter family of classical r-matrices for so(2, 2), while the non Poisson-subgroup cases are much more numerous. In particular, we present the two Poisson homogeneous structures on AdS 3 that arise from two Drinfel'd double structures on SO(2, 2). The first one realises AdS 3 as a quotient of SO(2, 2) by the Poisson-subgroup SL(2, R), while the second one, the non-commutative spacetime of the twisted κ-AdS deformation, realises AdS 3 as a coisotropic Poisson homogeneous space.

Abstract. 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 Λ 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 κ-AdS and dS quantum algebras; their flat limit Λ → 0 leads to a twisted quantum Poincaré algebra. The resulting non-commutative spacetime is a nonlinear Λ-deformation of the κ-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.

Abstract. Noncommutative spacetimes are widely believed to model some properties of the quantum structure of spacetime at the Planck regime. In this contribution the construction of (anti-)de Sitter noncommutative spacetimes obtained through quantum groups is reviewed. In this approach the quantum deformation parameter z is related to a Planck scale, and the cosmological constant Λ plays the role of a second deformation parameter of geometric nature, whose limit Λ → 0 provides the corresponding noncommutative Minkowski spacetimes.

We present the generalisation to (3 + 1) dimensions of a quantum deformation of the (2 + 1) (Anti)-de Sitter and Poincaré Lie algebras that is compatible with the conditions imposed by the Chern-Simons formulation of (2 + 1) gravity. Since such compatibility is automatically fulfilled by deformations coming from Drinfel'd double structures, we believe said structures are worth being analysed also in the (3 + 1) scenario as a possible guiding principle towards the description of (3 + 1) gravity. To this aim, a canonical classical r-matrix arising from a Drinfel'd double structure for the three (3 + 1) Lorentzian algebras is obtained. This r-matrix turns out to be a twisted version of the one corresponding to the (3 + 1) κ-deformation, and the main properties of its associated noncommutative spacetime are analysed. In particular, it is shown that this new quantum spacetime is not isomorphic to the κ-Minkowski one, and that the isotropy of the quantum space coordinates can be preserved through a suitable change of basis of the quantum algebra generators. Throughout the paper the cosmological constant appears as an explicit parameter, thus allowing the (flat) Poincaré limit to be straightforwardly obtained.