Poisson-Hopf algebra deformations of Lie-Hamilton systems

In this paper (arXiv:1708.08185) Hopf algebra deformations are merged with a class of Lie systems of Hamiltonian type, the so-called Lie-Hamilton systems, to devise a novel formalism: the Poisson-Hopf algebra deformations of Lie-Hamilton systems. This approach applies to any Hopf algebra deformation of any Lie-Hamilton system. Remarkably, a Hopf algebra deformation transforms a Lie-Hamilton system, whose dynamic is governed by a finite-dimensional Lie algebra of functions, into a non-Lie-Hamilton system associated with a Poisson-Hopf algebra of functions that allows for the explicit description of its t-independent constants of the motion from deformed Casimir functions. We illustrate our approach by considering the Poisson-Hopf algebra analogue of the non-standard quantum deformation of sl(2) and its applications to deform well-known Lie-Hamilton systems describing oscillator systems, Milne-Pinney equations, and several types of Riccati equations. In particular, we obtain a new position-dependent mass oscillator system with a time-dependent frequency.

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Curved momentum spaces from quantum groups with cosmological constant

In this paper (arXiv:1707.09600) we bring the concept that quantum symmetries describe theories with nontrivial momentum space properties one step further, looking at quantum symmetries of spacetime in presence of a nonvanishing cosmological constant Λ. In particular, the momentum space associated to the κ-deformation of the de Sitter algebra in (1+1) and (2+1) dimensions is explicitly constructed as a dual Poisson-Lie group manifold parametrized by Λ. Such momentum space includes both the momenta associated to spacetime translations and the `hyperbolic’ momenta associated to boost transformations, and has the geometry of (half of) a de Sitter manifold. Known results for the momentum space of the κ-Poincaré algebra are smoothly recovered in the limit Λ0, where hyperbolic momenta decouple from translational momenta. The approach here presented is general and can be applied to other quantum deformations of kinematical symmetries, including (3+1)-dimensional ones.

Non-commutative relativistic spacetimes and worldlines from 2+1 quantum (anti-)de Sitter groups

The κ-deformation of the (2+1)D anti-de Sitter, Poincaré and de Sitter groups is presented through a unified approach in which the curvature of the spacetime (or the cosmological constant) is considered as an explicit parameter. The Drinfel’d-double and the Poisson-Lie structure underlying the κ-deformation are explicitly given, and the three quantum kinematical groups are obtained as quantizations of such Poisson-Lie algebras. As a consequence, the non-commutative (2+1)D spacetimes that generalize the κ-Minkowski space to the (anti-)de Sitter ones are obtained. Moreover, noncommutative 4D spaces of (time-like) geodesics can be defined, and they can be interpreted as a novel possibility to introduce non-commutative worldlines. Furthermore, quantum (anti-)de Sitter algebras are presented both in the known basis related with 2+1 quantum gravity and in a new one which generalizes the bicrossproduct one. In this framework, the quantum deformation parameter is related with the Planck length, and the existence of a kind of “duality” between the cosmological constant and the Planck scale is also envisaged. This paper (arXiv:hep-th/0401244) is an updated review version of a 2004 manuscript with the same title and authors.

Quantum groups and noncommutative spacetimes with cosmological constant

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 Λ0 provides the corresponding noncommutative Minkowski spacetimes.

Superintegrable systems on 3-dimensional curved spaces: Eisenhart formalism and separability

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.

AdS Poisson homogeneous spaces and Drinfel’d doubles

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.

Lie Hamilton systems on curved spaces: A geometrical approach

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.