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.

In this paper (arXiv:1302.0684) the quantum deformations of (anti-)de Sitter  (A)dS algebras in (2+1) dimensions are revisited. In particular, the classification problem of (2+1) (A)dS Lie bialgebras is presented and the associated noncommutative quantum (A)dS spaces are also analysed. Moreover, the flat limit (or vanishing cosmological constant) of all these structures leading to (2+1) quantum Poincaré algebras is studied. Some results on the analogous (3+1) problem are sketched.

This new paper (arXiv:1212.3809) deals with a systematic computational approach for the explicit construction of any quantum Hopf algebra starting from the Lie bialgebra that gives the first-order deformation of the coproduct map. The procedure is based on the fact that any quantum algebra can be viewed as the quantization of a unique Poisson-Lie structure on the dual group.

In this paper (arXiv:1207.0071) a new integrable generalization on the two-dimensional sphere and the hyperbolic plane of the Euclidean anisotropic oscillator Hamiltonian with “centrifugal” terms is presented. The dynamical features arising from the introduction of a curved background are highlighted, and the superintegrability properties of the Hamiltonian are studied.

In the new paper “Comment on “new integrable family in n-dimensional homogeneous Lotka-Volterra systems with Abelian Lie Algebra” [J. Phys. Soc. Jpn. 72 (2003) 973]” we present the new (generalized) Poisson structure and the Hamiltonian functions for the so-called generalized ladder system (GLS), obtaining also its first integrals as functionally independent Casimir functions for the associated Poisson algebra and thus proving the complete Liouville integrability of the GLS.

In this paper (arXiv:1202.2077) all real three dimensional Poisson-Lie groups are explicitly constructed and fully classified under group automorphisms by making use of their one-to-one correspondence with the complete classification of real three-dimensional Lie bialgebras given in [X. Gomez, J. Math. Phys. vol. 41, p. 4939 (2000)]. Many of these 3D Poisson-Lie groups are non-coboundary structures, whose Poisson brackets are given here for the first time. Casimir functions for all three-dimensional PL groups are given, and some features of several PL structures are commented.