In this paper (arXiv:1403.4773) 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 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. The resulting non-commutative spacetime  is a nonlinear cosmological constant 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.

In this paper (arXiv:1403.1829) we present a new integrable generalization to the 2D sphere and to the hyperbolic space of the 2D Euclidean anisotropic oscillator Hamiltonian with Rosochatius (centrifugal) terms, and its curved integral of the motion is shown to be quadratic in the momenta. In order to construct such a new integrable Hamiltonian, we  make use of a group theoretical approach in which the curvature of the underlying space is treated as an additional (contraction) parameter, and we make extensive use of projective coordinates and their associated phase spaces. These findings supports the conjecture that for each commensurate (and thus superintegrable) m:n Euclidean oscillator there exists a two-parametric family of curved integrable (but not superintegrable) oscillators that turns out to be superintegrable only when the parameters are tuned to the m:n commensurability condition.

In this paper (arXiv:1402.2884) we show that the Drinfel’d double associated to the standard quantum deformation of sl(2,R) is isomorphic to the (2+1)-dimensional AdS algebra with the initial deformation parameter η related to the cosmological constant Λ. This gives rise to a generalisation of a non-commutative Minkowski spacetime that arises as a consequence of the quantum double symmetry of (2+1) gravity to non-vanishing cosmological constant. The properties of the AdS quantum double that generalises this symmetry to the case Λ≠0 are sketched, and it is shown that the new non-commutative AdS spacetime is a nonlinear Λ-deformation of the Minkowskian one.

In this paper (arXiv:1311.0792) we study Lie-Hamilton systems on the plane, i.e. systems of first-order differential equations describing the integral curves of a t-dependent vector field taking values in a finite-dimensional real Lie algebra of planar Hamiltonian vector fields with respect to a Poisson structure. We provide the complete local classification of Lie-Hamilton systems on the plane and we study new Lie-Hamilton systems of interest which are used to investigate relevant non-autonomous differential equations. In particular, the Milne-Pinney, second-order Kummer-Schwarz, complex Riccati and Buchdahl equations as well as some Lotka-Volterra and nonlinear biomathematical models are analysed from this Lie-Hamilton approach.

In this paper (arXiv:1310.6554) a new maximally superintegrable deformation of the N-dimensional Kepler–Coulomb Hamiltonian is presented. From a geometric viewpoint, this superintegrable Hamiltonian can be interpreted as a system on an N-dimensional Riemannian space with nonconstant curvature. The eigenvalues and eigenfunctions of the model are explicitly obtained, and the spectrum presents a hydrogen-like shape for positive values of the deformation parameter and of the corresponding coupling constant.

The Proceedings of the XXI IFWGP, that our group organized in Burgos last year, have been already published as the September 2013 issue of the International Journal of Geometric Methods in Modern Physics.