We derive finite boost transformations based on the Lorentz sector of the bicross-product-basis kappa -Poincare Hopf algebra. We emphasize the role of these boost transformations in a recently-proposed new relativistic theory, and their relevance for experimental studies presently being planned. We find that when the (dimensionful) deformation parameter is identified with the Planck length, which together with the speed-of-light constant has the status of observer-independent scale in the new relativistic theory, the deformed boosts saturate at the value of momentum that corresponds to the inverse of the Planck length. (C) 2001 Published by Elsevier Science B.V

A quantum deformation of the conformal algebra of the Minkowskian spacetime in $(3+1)$ dimensions is identified with a deformation of the $(4+1)$-dimensional AdS algebra. Both Minkowskian and AdS first-order non-commutative spaces are explicitly obtained, and the former coincides with the well known $\kappa$-Minkowski space. Next, by working in the conformal basis, a new non-commutative Minkowskian spacetime is constructed through the full (all orders) dual quantum group spanned by deformed Poincar\'e and dilation symmetries. Although Lorentz invariance is lost, the resulting non-commutative spacetime is quantum group covariant, preserves space isotropy and, furthermore, can be interpreted as a generalization of the $\kappa$-Minkowski space in which a variable fundamental scale (Planck length) appears.Comment: Revised version accepted for pubblication on PLB. Latex file, 9 page

Several recent studies have considered the implications for astrophysics and cosmology of some possible nonclassical properties of spacetime at the Planck scale. The new effects, such as a Planck-scale-modified energy-momentum (dispersion) relation, are often inferred from the analysis of some quantum versions of Minkowski spacetime, and therefore the relevant estimates depend heavily on the assumption that there could not be significant interplay between Planck-scale and curvature effects. We here scrutinize this assumption, using as guidance a quantum version of de Sitter spacetime with known Inönü-Wigner contraction to a quantum Minkowski spacetime. And we show that, contrary to common (but unsupported) beliefs, the interplay between Planckscale and curvature effects can be significant. Within our illustrative example, in the Minkowski limit the quantum-geometry deformation parameter is indeed given by the Planck scale, while in the de Sitter picture the parameter of quantization of geometry depends both on the Planck scale and the curvature scalar. For the much-studied case of Planck-scale effects that intervene in the observation of gamma-ray bursts we can estimate the implications of "quantum spacetime curvature" within robust simplifying assumptions. For cosmology at the present stage of the development of the relevant mathematics one cannot go beyond semiheuristic reasoning, and we here propose a candidate approximate description of a quantum FRW geometry, obtained by patching together pieces (with different spacetime curvature) of our quantum de Sitter. This semiheuristic picture, in spite of its limitations, provides rather robust evidence that in the early Universe the interplay between Planck-scale and curvature effects could have been particularly significant. * Electronic address: antonino.marciano@cpt.univ-mrs.fr † Electronic address: r.bruno@bnrenergia.it ‡ Electronic address: gianluca.mandanici@istruzione.it

The -deformation of the (2 + 1)D anti-de Sitter, Poincaré, 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 -deformation are explicitly given, and the three quantum kinematical groups are obtained as quantizations of such Poisson-Lie algebras. As a consequence, the noncommutative (2 + 1)D spacetimes that generalize the -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 noncommutative worldlines. Furthermore, quantum (anti-)de Sitter algebras are presented both in the known basis related to 2 + 1 quantum gravity and in a new one which generalizes the bicrossproduct one. In this framework, the quantum deformation parameter is related to the Planck length, and the existence of a kind of "duality" between the cosmological constant and the Planck scale is also envisaged.

A 'mass-like' quantum Weyl-Poincaré algebra is proposed to describe, after the identification of the deformation parameter with the Planck length, a new relativistic theory with two observer-independent scales (or DSR theory). Deformed momentum representation, finite boost transformations, range of rapidity, energy and momentum, as well as position and velocity operators are explicitly studied and compared with those of previous DSR theories based on κ-Poincaré algebra. The main novelties of the DSR theory here presented are the new features of momentum saturation and a new type of deformed position operators.

We analyze among all possible quantum deformations of the 3+1 (anti)de Sitter algebras, so(3, 2) and so(4, 1), which have two specific non-deformed or primitive commuting operators: the time translation/energy generator and a rotation. We prove that under these conditions there are only two families of two-parametric (anti)de Sitter Lie bialgebras. All the deformation parameters appearing in the bialgebras are dimensionful ones and they may be related to the Planck length. Some properties conveyed by the corresponding quantum deformations (zero-curvature and non-relativistic limits, space isotropy,. . . ) are studied and their dual (first-order) non-commutative spacetimes are also presented.

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The Poincaré sector of a recently deformed conformal algebra is proposed to describe, after the identification of the deformation parameter with the Planck length, the symmetries of a new relativistic theory with two observer-independent scales (or DSR theory). Also a new non-commutative space-time is proposed. It is found that momentum space exhibits the same features of the DSR proposals preserving Lorentz invariance in a deformed way. The space-time sector is a generalization of the well known non-commutative κ-Minkowski space-time which however does not preserve Lorentz invariance, not even in the deformed sense. It is shown that this behavior could be expected in some attempts to construct DSR theories starting from the Poincaré sector of a deformed symmetry larger than Poincaré symmetry, unless one takes a variable Planck length. It is also shown that the formalism can be useful in analyzing the role of quantum deformations in the "AdS-CFT correspondence".One of the most studied applications of quantum groups in physics is in the description of deformed spacetime symmetries that generalize classical Poincaré kinematics beyond the Lie-algebra level. Well-known examples are the κ-Poincaré 1 and the quantum null-plane (or light-cone) Poincaré 2 algebras. The deformation parameter has been interpreted as a fundamental scale which may be related with the Planck length. In fact, these results can be seen as different attempts to develop new approaches to physics at the Planck scale 3 . It has been also recently conjectured that the κ-Poincaré algebra may provide the basis for a so-called doubly special relativity (DSR) 4,5,6,7 in which the deformation parameter/Planck length is viewed as an observer-independent length scale completely analogous to the familiar observer-independent velocity scale c, in such a manner that (deformed) Lorentz invariance is preserved 8,9,10 . In this talk I want to review briefly some related results which have been presented in greater detail in [11,12]. These results provide an analysis of the Poincaré sector of a manageable deformation of so(4, 2) introduced in [13] together with its dual in order to extract some physical implications on the associated non-commutative Minkowskian spacetime.If {J i , P µ = (P 0 , P), K i , D} denote the generators of rotations, time and space translations, boosts and dilations, the non-vanishing deformed commutation rules 1