A Lie-Hamilton system is a nonautonomous system of first-order ordinary differential equations describing the integral curves of a t-dependent vector field taking values in a finite-dimensional real Lie algebra of Hamiltonian vector fields with respect to a Poisson structure. In this new paper (arXiv:1410.7336), after reviewing the classification of finite-dimensional real Lie algebras of Hamiltonian vector fields on the plane, we present new Lie-Hamilton systems with physical, biological and mathematical applications. New results cover Cayley-Klein Riccati equations, the hereafter called planar diffusion Riccati systems and complex Bernoulli equations, all of them with t-dependent real coefficients. Furthermore, we study the existence of local diffeomorphisms among new and already known Lie-Hamilton systems on the plane. In particular, we show that the Cayley-Klein Riccati equations describe as particular cases well-known coupled Riccati equations, second-order Kummer-Schwarz equations, Milne-Pinney equations, the harmonic oscillator with t-dependent frequency and other systems of physical and mathematical relevance.

In this new paper (arXiv:1408.3689) it is shown that the canonical classical r-matrix arising from the Drinfel’d double structure underlying the two-fold centrally extended (2+1) Galilean and Newton-Hooke Lie algebras (with either zero or non-zero cosmological constant Λ, respectively) originates as a well-defined non-relativistic contraction of a specific class of canonical r-matrices associated with the Drinfel’d double structure of the (2+1) (anti)-de Sitter Lie algebra. The full quantum group structure associated with such (2+1) Galilean and Newton-Hooke Drinfel’d doubles is presented, and the corresponding noncommutative spacetimes are shown to contain a commuting ‘absolute time’ coordinate together with two noncommutative space coordinates, whose commutator is a function of the cosmological constant Λ and of the (central) ‘quantum time’ coordinate. Thus, the Chern-Simons approach to Galilean (2+1) gravity can be consistently understood as the appropriate non-relativistic limit of the Lorentzian theory, and their associated quantum group symmetries (which do not fall into the family of so-called kappa-deformations) can also be derived from the (anti)-de Sitter quantum doubles through a well-defined quantum group contraction procedure.

In this paper (arXiv:1407.1401) we quantize an N-dimensional classical Hamiltonian system that can be regarded as a deformation of the Coulomb problem. Moreover, the kinetic energy term in the Hamiltonian is just the one corresponding to an N-dimensional Taub-NUT space, a fact that makes this system relevant from a geometric viewpoint. Since the classical Hamiltonian is known to be maximally superintegrable, we propose a quantization prescription that preserves such superintegrability in the quantum mechanical setting. We show that, to this end, one must choose as the kinetic part of the Hamiltonian the conformal Laplacian of the underlying Riemannian manifold, which combines  the usual Laplace-Beltrami operator on the Taub-NUT manifold and a multiple of its scalar curvature. As a consequence, we obtain a novel exactly solvable deformation of the quantum Coulomb problem, whose spectrum is computed in closed form, and showing that the well-known maximal degeneracy of the flat system is preserved in the deformed case. Several interesting algebraic and physical features of this new exactly solvable quantum system are analysed.

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.