A Lie system is a nonautonomous system of first-order differential equations admitting a superposition rule, i.e., a map expressing its general solution in terms of a generic family of particular solutions and some constants. In this paper (arXiv:1404.2740), by using that a Lie system can be considered as a curve in a finite-dimensional Lie algebra of vector fields, a so-called Vessiot-Guldberg Lie algebra, we associate every Lie system with a Lie algebra of Lie point symmetries induced by the Vessiot-Guldberg Lie algebra. This enables us to derive Lie symmetries of relevant physical systems described by first- and higher-order systems of differential equations by means of Lie systems in an easier way than by standard methods. A generalization of our results to partial differential equations is introduced. Among other applications, Lie symmetries for several new and known generalizations of the real Riccati equation are studied.

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.

A Lie system is a nonautonomous system of first-order differential equations possessing a superposition rule, i.e. a map expressing its general solution in terms of a generic finite family of particular solutions and some constants. Lie-Hamilton systems form a subclass of Lie systems whose dynamics is governed by a curve in a finite-dimensional real Lie algebra of functions on a Poisson manifold. In this new paper (arXiv:1305.6272) is shown that Lie-Hamilton systems are naturally endowed with a Poisson coalgebra structure. This allows us to devise methods to derive in an algebraic way their constants of motion and superposition rules. We illustrate our methods by studying Kummer-Schwarz equations, Riccati equations, Ermakov systems and Smorodinsky-Winternitz systems with time-dependent frequency.

We present an overview of maximally superintegrable classical Hamitonians on spherically symmetric spaces (arXiv:1304.4544). It turns out that each of these systems can be considered either as an oscillator or as a Kepler-Coulomb Hamiltonian. We show that two possible quantization prescriptions for all these curved systems arise if we impose that superintegrability is preserved after quantization, and we prove that both possibilities are gauge equivalent.

In this new paper (arXiv:1303.3080) in collaboration with C. Meusburger, all possible Drinfel’d double structures for the anti-de Sitter Lie algebra so(2,2) and de Sitter Lie algebra so(3,1) in (2+1)-dimensions are explicitly constructed and analysed in terms of a kinematical basis adapted to (2+1)-gravity. Each of these structures provides in a canonical way a pairing among the (anti-)de Sitter generators, as well as a specific classical r-matrix, and the cosmological constant is included in them as a deformation parameter. It is shown that four of these structures give rise to a Drinfel’d double structure for the Poincaré algebra iso(2,1) in the limit where the cosmological constant tends to zero. We explain how these Drinfel’d double structures are adapted to (2+1)-gravity, and we show that the associated quantum groups are natural candidates for the quantum group symmetries of quantised (2+1)-gravity models and their associated non-commutative spacetimes.