Noncommutative spacetimes are widely believed to model some properties of the quantum structure of spacetime at the Planck regime. In this contribution (arXiv:1702.04704) the construction of (anti-)de Sitter noncommutative spacetimes obtained through quantum groups is reviewed. In this approach the quantum deformation parameter z is related to a Planck scale, and the cosmological constant Λ plays the role of a second deformation parameter of geometric nature, whose limit Λ→0 provides the corresponding noncommutative Minkowski spacetimes.

In this paper (arXiv:1701.05783) the Eisenhart geometric formalism, which transforms an Euclidean natural Hamiltonian into a geodesic Hamiltonian with one additional degree of freedom, is applied to the four families of quadratically superintegrable systems with multiple separability in the Euclidean plane. Firstly, the separability and superintegrability of such four geodesic Hamiltonians in a three-dimensional curved space are studied and then these four systems are modified with the addition of a potential. Secondly, we study the superintegrability of certain four position-dependent mass Hamiltonians, that enjoys the same separability as the original system. All the Hamiltonians here studied describe superintegrable systems on non-Euclidean three-dimensional manifolds with a broken spherically symmetry.

In this paper (arXiv:1701.04902) the correspondence between Poisson homogeneous spaces over a Poisson-Lie group G and Lagrangian Lie subalgebras of the classical double D(𝔤) is revisited and explored in detail for the case in which 𝔤=D(𝔞) is a classical double itself. We apply these results to give an explicit description of all 2d Poisson homogeneous spaces over the group SL(2,R)≅SO(2,1), namely 2d anti de Sitter space, 2d hyperbolic space and the lightcone in 3d Minkowski space. We show how each of these spaces is obtained as a quotient with respect to a Poisson-subgroup for one of the three inequivalent Lie bialgebra structures on sl(2,R) and as a coisotropic one for the others. We then classify the Poisson homogeneous structures for 3d anti de Sitter space AdS3 and show that the ones that are quotients by a Poisson subgroup are determined by a three-parameter family of classical r-matrices for so(2,2), while the coisotropic cases are much more numerous. In particular, we present the two Poisson homogeneous structures on AdS3 that arise from two Drinfel’d double structures on SO(2,2). The first one realises AdS3 as a quotient of SO(2,2) by the Poisson-subgroup SL(2,R), while the second one, the non-commutative spacetime of the twisted κ-AdS deformation, realises AdS3 as a coisotropic Poisson homogeneous space.

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 Lie algebra, a Vessiot-Guldberg Lie algebra, of Hamiltonian vector fields relative to a Poisson structure. Their general solution can be written as an autonomous function, the superposition rule, of a generic finite family of particular solutions and a set of constants. In this paper (arXiv:1612.08901) we pioneer the study of Lie-Hamilton systems on Riemannian spaces (sphere, Euclidean, and hyperbolic plane), pseudo-Riemannian spaces (anti-de Sitter, de Sitter, and Minkowski spacetimes) as well as on semi-Riemannian spaces (Newtonian spacetimes). Their corresponding constants of motion and superposition rules are obtained explicitly in a geometric way. This work extends the (graded) contraction of Lie algebras to a contraction procedure for Lie algebras of vector fields, Hamiltonian functions, and related symplectic structures, invariants, and superposition rules.

In this paper (arXiv:1612.03169) the quantum duality principle is used to obtain explicitly the Poisson analogue of the kappa-(A)dS quantum algebra in (3+1) dimensions as the corresponding Poisson-Lie structure on the dual solvable Lie group. The construction is fully performed in a kinematical basis and deformed Casimir functions are also explicitly obtained. The cosmological constant Λ is included as a Poisson-Lie group contraction parameter, and the limit Λ→0 leads to the well-known kappa-Poincaré algebra in the bicrossproduct basis. A twisted version with Drinfel’d double structure of this kappa-(A)dS deformation is sketched.

Given a Lie-Poisson completely integrable bi-Hamiltonian system, in the new paper arXiv:1609.07438 we present a method which allows us to construct, under certain conditions, a completely integrable bi-Hamiltonian deformation of the initial Lie-Poisson system on a non-abelian Poisson-Lie group. Moreover, we show that from the two multiplicative (Poisson-Lie) Hamiltonian structures that underly the dynamics of the deformed system and by making use of the non-abelian group law, one may obtain two completely integrable Hamiltonian systems on the direct product of the non-abelian group by itself. By construction, both systems admit reduction, via the multiplication in the non-abelian group, to the initial deformed bi-Hamiltonian system. The previous approach is applied to two relevant Lie-Poisson completely integrable bi-Hamiltonian systems: the Lorenz and Euler top systems.

In a recent paper (J.R. Morris, Quant. Stud. Math. Found. 2 (2015) 359), an inhomogeneous compactification of the extra dimension of a five-dimensional Kaluza-Klein metric has been shown to generate a position-dependent mass (PDM) in the corresponding four-dimensional system. As an application of this dimensional reduction mechanism, a specific static dilatonic scalar field has been connected with a PDM Lagrangian describing a well-known nonlinear PDM oscillator. In this paper (arXiv:1605.06829) we present more instances of this construction that lead to PDM systems with radial symmetry, and the properties of their corresponding inhomogeneous extra dimensions are compared with the ones in the nonlinear oscillator model. Finally, it is shown that the compactification introduced in this type of models can alternatively be interpreted as a novel mechanism for the dynamical generation of curvature.

We present (arXiv:1605.02384) a new exactly solvable (classical and quantum) model that can be interpreted as the generalization to the two-dimensional sphere and to the hyperbolic space of the two-dimensional anisotropic oscillator with any pair of frequencies. The new curved Hamiltonian depends on the curvature κ of the underlying space as a deformation/contraction parameter, and the Liouville integrability of the system relies on its separability in terms of geodesic parallel coordinates, which generalize the Cartesian coordinates of the plane. Moreover, the system is shown to be superintegrable for commensurate frequencies, thus mimicking the behaviour of the flat Euclidean case, which is always recovered in the κ→0 limit. The additional constant of motion in the commensurate case is, as expected, of higher-order in the momenta and can be explicitly deduced by performing the classical factorization of the Hamiltonian. The known 1:1 and 2:1 anisotropic curved oscillators are recovered as particular cases, meanwhile all the remaining ωx:ωy curved oscillators define new superintegrable systems. Furthermore, the quantum Hamiltonian is fully constructed and studied by following a quantum factorization approach. In the case of commensurate frequencies, the quantum Hamiltonian turns out to be quantum superintegrable and leads to a new exactly solvable quantum model. Its corresponding spectrum, that exhibits a maximal degeneracy, is explicitly given as an analytical deformation of the Euclidean eigenvalues in terms of both the curvature κ and the Planck constant ℏ. In fact, such spectrum is obtained as a composition of two one-dimensional (either trigonometric or hyperbolic) Pösch-Teller set of eigenvalues.

In the recent paper arXiv:1601.03357, a method to construct integrable deformations of Hamiltonian systems of ODEs endowed with Lie-Poisson symmetries is proposed by considering Poisson-Lie groups as deformations of Lie-Poisson (co)algebras. Moreover, the underlying Lie-Poisson symmetry of the initial system of ODEs is used to construct integrable coupled systems, whose integrable deformations can be obtained through the construction of the appropriate Poisson-Lie groups that deform the initial symmetry. The approach is applied in order to construct integrable deformations of both uncoupled and coupled versions of certain integrable types of Rössler and Lorenz systems. It is worth stressing that such deformations are of non-polynomial type since they are obtained through an exponentiation process that gives rise to the Poisson-Lie group from its infinitesimal Lie bialgebra structure. The full deformation procedure is essentially algorithmic and can be computerized to a large extent.