A universal algorithm to construct N -particle (classical and quantum) completely integrable Hamiltonian systems from representations of coalgebras with Casimir element is presented. In particular, this construction shows that quantum deformations can be interpreted as generating structures for integrable deformations of Hamiltonian systems with coalgebra symmetry. In order to illustrate this general method, the so(2, 1) algebra and the oscillator algebra h 4 are used to derive new classical integrable systems including a generalization of Gaudin-Calogero systems and oscillator chains. Quantum deformations are then used to obtain some explicit integrable deformations of the previous long-range interacting systems and a (non-coboundary) deformation of the (1 + 1) Poincaré algebra is shown to provide a new Ruijsenaars-Schneider-like Hamiltonian.

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 start with the local classification of finite-dimensional real Lie algebras of vector fields on the plane obtained in [A. González-López, N. Kamran and P.J. Olver, Proc. London Math. Soc. 64, 339 (1992)] and we interpret their results as a local classification of Lie systems. Moreover, by determining which of these real Lie algebras consist of Hamiltonian vector fields with respect to a Poisson structure, we provide the complete local classification of Lie-Hamilton systems on the plane. We present and study through our results new Lie-Hamilton systems of interest which are used to investigate relevant non-autonomous differential equations, e.g. we get explicit local diffeomorphisms between such systems. 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.

An infinite family of classical superintegrable Hamiltonians defined on the Ndimensional spherical, Euclidean and hyperbolic spaces are shown to have a common set of (2N − 3) functionally independent constants of the motion. Among them, two different subsets of N integrals in involution (including the Hamiltonian) can always be explicitly identified. As particular cases, we recover in a straightforward way most of the superintegrability properties of the Smorodinsky-Winternitz and generalized Kepler-Coulomb systems on spaces of constant curvature and we introduce as well new classes of (quasi-maximally) superintegrable potentials on these spaces. Results here presented are a consequence of the sl(2, R) Poisson coalgebra symmetry of all the Hamiltonians, together with an appropriate use of the phase spaces associated to Poincaré and Beltrami coordinates.

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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. It 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.

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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.This implies that the space of Ad-invariant symmetric bilinear forms of this Lie algebra is two-dimensional. If one identifies the duals of J a and P a with, respectively, P a and J a , the pairings corresponding to C 1 and C 2 are given, in this order, by J a , P b s = 0, J a , J b s = g ab , P a , P b s = χ g ab . J a , P b t = g ab , J a , J b t = 0, P a , P b t = 0, (2.5)

The full spectrum and eigenfunctions of the quantum version of a nonlinear oscillator defined on an N -dimensional space with nonconstant curvature are rigorously found. Since the underlying curved space generates a position-dependent kinetic energy, three different quantization prescriptions are worked out by imposing that the maximal superintegrability of the system has to be preserved after quantization. The relationships among these three Schrödinger problems are described in detail through appropriate similarity transformations. These three approaches are used to illustrate different features of the quantization problem on N -dimensional curved spaces or, alternatively, of position-dependent mass quantum Hamiltonians. This quantum oscillator is, to the best of our knowledge, the first example of a maximally superintegrable quantum system on an N -dimensional space with nonconstant curvature.