In this new paper (**arXiv:1902.09132**), the space of time-like geodesics on Minkowski spacetime is constructed as a coset space of the Poincaré group in (3+1) dimensions with respect to the stabilizer of a worldline. When this homogeneous space is endowed with a Poisson homogeneous structure compatible with a given Poisson-Lie Poincaré group, the quantization of this Poisson bracket gives rise to a noncommutative space of worldlines with quantum group invariance. As an oustanding example, the Poisson homogeneous space of worldlines coming from the κappa-Poincaré deformation is explicitly constructed, and shown to define a symplectic structure on the space of worldlines. Therefore, the quantum space of κappa-Poincaré worldlines is just the direct product of three Heisenberg-Weyl algebras in which the inverse of the kappa parameter plays the very same role as the Planck constant ℏ in quantum mechanics. In this way, noncommutative spaces of worldlines are shown to provide a new suitable and fully explicit arena for the description of quantum observers with quantum group symmetry.

# Publications

# Drinfel’d double structures for Poincaré and Euclidean groups

In this contribution (**arXiv:1812.02075**) all non-isomorphic three-dimensional Poisson homogeneous Euclidean spaces are constructed and analyzed, based on the classification of coboundary Lie bialgebra structures of the Euclidean group in 3-dimensions, and the only Drinfel’d double structure for this group is explicitly given. The similar construction for the Poincaré case is reviewed and the striking differences between the Lorentzian and Euclidean cases are underlined. Finally, the contraction scheme starting from Drinfel’d double structures of the so(3,1) Lie algebra is presented.

# The Poincaré group as a Drinfel’d double

In this new paper (**arXiv:1809.09207**) the eight nonisomorphic Drinfel’d double (DD) structures for the Poincaré Lie group in (2+1) dimensions are explicitly constructed in the kinematical basis. Also, the two existing DD structures for a non-trivial central extension of the (1+1) Poincaré group are also identified and constructed, while in (3+1) dimensions no Poincaré DD structure does exist. Each of the DD structures here presented has an associated canonical quasitriangular Poincaré r-matrix whose properties are analysed. Some of these r-matrices give rise to coisotropic Poisson homogeneous spaces with respect to the Lorentz subgroup, and their associated Poisson Minkowski spacetimes are constructed. Two of these (2+1) noncommutative DD Minkowski spacetimes turn out to be quotients by a Lorentz Poisson subgroup: the first one corresponds to the double of 𝔰𝔩(2) with trivial Lie bialgebra structure, and the second one gives rise to a quadratic noncommutative Poisson Minkowski spacetime. With these results, the explicit construction of DD structures for all Lorentzian kinematical groups in (1+1) and (2+1) dimensions is completed, and the connection between (anti-)de Sitter and Poincaré r-matrices through the vanishing cosmological constant limit is also analysed.

# Global versus local superintegrability of nonlinear oscillators

Liouville (super)integrability of a Hamiltonian system of differential equations is based on the existence of globally well-defined constants of the motion, while Lie point symmetries provide a local approach to conserved integrals. In this new paper (**arXiv:1809.02248**) we investigate in which sense Lie point symmetries can be used to provide information concerning the superintegrability of a given Hamiltonian system. The two-dimensional oscillator and the central force problem are used as benchmark examples to show that the relationship between standard Lie point symmetries and superintegrability is neither straightforward nor universal. In general, it turns out that superintegrability is not related to either the size or the structure of the algebra of variational dynamical symmetries. Nevertheless, all of the first integrals for a given Hamiltonian system can be obtained through an extension of the standard point symmetry method, which is applied to a superintegrable nonlinear oscillator describing the motion of a particle on a space with non-constant curvature and spherical symmetry.

# Extended noncommutative Minkowski spacetimes and hybrid gauge symmetries

In this new paper (**arXiv:1805.07099**) we study the Lie bialgebra structures that can be built on the one-dimensional central extension of the Poincaré and (A)dS algebras in (1+1) dimensions. These central extensions admit more than one interpretation, but the simplest one is that they describe the symmetries of (the noncommutative deformation of) an Abelian gauge theory, U(1) or SO(2) on the (1+1) dimensional Minkowski or (A)dS spacetime. We show that this highlights the possibility that the algebra of functions on the gauge bundle becomes noncommutative. This is a new way in which the Coleman-Mandula theorem could be circumvented by noncommutative structures, and it is related to a mixing of spacetime and gauge symmetry generators when they act on tensor-product states. We obtain all Lie bialgebra structures on centrally-extended Poincaré and (A)dS which are coisotropic w.r.t. the Lorentz algebra, and therefore all of them admit the construction of a noncommutative principal gauge bundle on a quantum homogeneous Minkowski spacetime. It is shown that several different types of hybrid noncommutativity between the spacetime and gauge coordinates are allowed by introducing quantum extended Poincaré symmetries. In one of these cases, an alternative interpretation of the central extension leads to a new description of the well-known canonical noncommutative spacetime as the quantum homogeneous space of a quantum Poincaré algebra of symmetries.

# A unified approach to Poisson-Hopf deformations of Lie-Hamilton systems based on sl(2)

Based on a recently developed procedure to construct Poisson-Hopf deformations of Lie-Hamilton systems, a novel unified approach to nonequivalent deformations of Lie-Hamilton systems on the real plane with a Vessiot-Guldberg Lie algebra isomorphic to 𝔰𝔩(2) is proposed in the paper **arXiv:1803.07404**. This, in particular, allows us to define a notion of Poisson-Hopf systems in dependence of a parameterized family of Poisson algebra representations. Such an approach is explicitly illustrated by applying it to the three non-diffeomorphic classes of 𝔰𝔩(2) Lie-Hamilton systems. Our results cover deformations of the Ermakov system, Milne-Pinney, Kummer-Schwarz and several Riccati equations as well as of the harmonic oscillator (all of them with t-dependent coefficients). Furthermore t-independent constants of motion are given as well. Our methods can be employed to generate other Lie-Hamilton systems and their deformations for other Vessiot-Guldberg Lie algebras and their deformations.

# Curved momentum spaces from quantum (Anti-)de Sitter groups in (3+1) dimensions

In this new paper (**arXiv:1711.05050**), curved momentum spaces associated to the κ-deformation of the (3+1) de Sitter and Anti-de Sitter algebras are constructed as orbits of suitable actions of the dual Poisson-Lie group associated to the κ-deformation with non-vanishing cosmological constant. The κ-de Sitter and κ-Anti-de Sitter curved momentum spaces are separately analysed, and they turn out to be, respectively, half of the (6+1)-dimensional de Sitter space and half of a space with SO(4,4) invariance. Such spaces are made of the momenta associated to spacetime translations and the “hyperbolic” momenta associated to boost transformations. The known κ-Poincaré curved momentum space is smoothly recovered as the vanishing cosmological constant limit from both of the constructions.

# Poisson-Hopf algebra deformations of Lie-Hamilton systems

In this paper (**arXiv:1708.08185**) Hopf algebra deformations are merged with a class of Lie systems of Hamiltonian type, the so-called Lie-Hamilton systems, to devise a novel formalism: the Poisson-Hopf algebra deformations of Lie-Hamilton systems. This approach applies to any Hopf algebra deformation of any Lie-Hamilton system. Remarkably, a Hopf algebra deformation transforms a Lie-Hamilton system, whose dynamic is governed by a finite-dimensional Lie algebra of functions, into a non-Lie-Hamilton system associated with a Poisson-Hopf algebra of functions that allows for the explicit description of its t-independent constants of the motion from deformed Casimir functions. We illustrate our approach by considering the Poisson-Hopf algebra analogue of the non-standard quantum deformation of sl(2) and its applications to deform well-known Lie-Hamilton systems describing oscillator systems, Milne-Pinney equations, and several types of Riccati equations. In particular, we obtain a new position-dependent mass oscillator system with a time-dependent frequency.

# Curved momentum spaces from quantum groups with cosmological constant

In this paper (**arXiv:1707.09600**) we bring the concept that quantum symmetries describe theories with nontrivial momentum space properties one step further, looking at quantum symmetries of spacetime in presence of a nonvanishing cosmological constant Λ. In particular, the momentum space associated to the κ-deformation of the de Sitter algebra in (1+1) and (2+1) dimensions is explicitly constructed as a dual Poisson-Lie group manifold parametrized by Λ. Such momentum space includes both the momenta associated to spacetime translations and the `hyperbolic’ momenta associated to boost transformations, and has the geometry of (half of) a de Sitter manifold. Known results for the momentum space of the κ-Poincaré algebra are smoothly recovered in the limit Λ→0, where hyperbolic momenta decouple from translational momenta. The approach here presented is general and can be applied to other quantum deformations of kinematical symmetries, including (3+1)-dimensional ones.

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

The κ-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 κ-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 κ-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. This paper (**arXiv:hep-th/0401244**) is an updated review version of a 2004 manuscript with the same title and authors.