Classical Dynamical Systems from q-algebras:"cluster" variables and explicit solutions

A general procedure to get the explicit solution of the equations of motion for N-body classical Hamiltonian systems equipped with coalgebra symmetry is introduced by defining a set of appropriate collective variables which are based on the iterations of the coproduct map on the generators of the algebra. In this way several examples of N-body dynamical systems obtained from q-Poisson algebras are explicitly solved: the q-deformed version of the sl(2) Calogero-Gaudin system (q-CG), a q-Poincare' Gaudin system and a system of Ruijsenaars type arising from the same (non co-boundary) q-deformation of the (1+1) Poincare' algebra. Also, a unified interpretation of all these systems as different Poisson-Lie dynamics on the same three dimensional solvable Lie group is given.Comment: 19 Latex pages, No figure

(1+1) Schrodinger Lie bialgebras and their Poisson-Lie groups

All Lie bialgebra structures for the (1+1)-dimensional centrally extended Schrodinger algebra are explicitly derived and proved to be of the coboundary type. Therefore, since all of them come from a classical r-matrix, the complete family of Schrodinger Poisson-Lie groups can be deduced by means of the Sklyanin bracket. All possible embeddings of the harmonic oscillator, extended Galilei and gl(2) Lie bialgebras within the Schrodinger classification are studied. As an application, new quantum (Hopf algebra) deformations of the Schrodinger algebra, including their corresponding quantum universal R-matrices, are constructed.Comment: 25 pages, LaTeX. Possible applications in relation with integrable systems are pointed; new references adde

New superintegrable models with position-dependent mass from Bertrand's Theorem on curved spaces

A generalized version of Bertrand's theorem on spherically symmetric curved spaces is presented. This result is based on the classification of (3+1)-dimensional (Lorentzian) Bertrand spacetimes, that gives rise to two families of Hamiltonian systems defined on certain 3-dimensional (Riemannian) spaces. These two systems are shown to be either the Kepler or the oscillator potentials on the corresponding Bertrand spaces, and both of them are maximally superintegrable. Afterwards, the relationship between such Bertrand Hamiltonians and position-dependent mass systems is explicitly established. These results are illustrated through the example of a superintegrable (nonlinear) oscillator on a Bertrand-Darboux space, whose quantization and physical features are also briefly addressed.Comment: 13 pages; based in the contribution to the 28th International Colloquium on Group Theoretical Methods in Physics, Northumbria University (U.K.), 26-30th July 201

Superintegrability on sl(2)-coalgebra spaces

We review a recently introduced set of N-dimensional quasi-maximally superintegrable Hamiltonian systems describing geodesic motions, that can be used to generate "dynamically" a large family of curved spaces. From an algebraic viewpoint, such spaces are obtained through kinetic energy Hamiltonians defined on either the sl(2) Poisson coalgebra or a quantum deformation of it. Certain potentials on these spaces and endowed with the same underlying coalgebra symmetry have been also introduced in such a way that the superintegrability properties of the full system are preserved. Several new N=2 examples of this construction are explicitly given, and specific Hamiltonians leading to spaces of non-constant curvature are emphasized.Comment: 12 pages. Based on the contribution presented at the "XII International Conference on Symmetry Methods in Physics", Yerevan (Armenia), July 2006. To appear in Physics of Atomic Nucle

Induced Representations of Quantum Kinematical Algebras and Quantum Mechanics

Unitary representations of kinematical symmetry groups of quantum systems are fundamental in quantum theory. We propose in this paper its generalization to quantum kinematical groups. Using the method, proposed by us in a recent paper (olmo01), to induce representations of quantum bicrossproduct algebras we construct the representations of the family of standard quantum inhomogeneous algebras $U_\lambda(iso_{\omega}(2))$. This family contains the quantum Euclidean, Galilei and Poincar\'e algebras, all of them in (1+1) dimensions. As byproducts we obtain the actions of these quantum algebras on regular co-spaces that are an algebraic generalization of the homogeneous spaces and $q$--Casimir equations which play the role of $q$--Schr\"odinger equations.Comment: LaTeX 2e, 20 page

Universal R-matrix for null-plane quantized Poincar{\'e} algebra

The universal ${\cal R}$--matrix for a quantized Poincar{\'e} algebra ${\cal P}(3+1)$ introduced by Ballesteros et al is evaluated. The solution is obtained as a specific case of a formulated multidimensional generalization to the non-standard (Jordanian) quantization of $sl(2)$.Comment: 9 pages, LaTeX, no figures. The example on page 5 has been supplemented with the full descriptio

N-dimensional sl(2)-coalgebra spaces with non-constant curvature

An infinite family of ND spaces endowed with sl(2)-coalgebra symmetry is introduced. For all these spaces the geodesic flow is superintegrable, and the explicit form of their common set of integrals is obtained from the underlying sl(2)-coalgebra structure. In particular, ND spherically symmetric spaces with Euclidean signature are shown to be sl(2)-coalgebra spaces. As a byproduct of this construction we present ND generalizations of the classical Darboux surfaces, thus obtaining remarkable superintegrable ND spaces with non-constant curvature.Comment: 11 pages. Comments and new references have been added; expressions for scalar curvatures have been corrected and simplifie

On two superintegrable nonlinear oscillators in N dimensions

We consider the classical superintegrable Hamiltonian system given by $H=T+U={p^2}/{2(1+\lambda q^2)}+{{\omega}^2 q^2}/{2(1+\lambda q^2)}$, where U is known to be the "intrinsic" oscillator potential on the Darboux spaces of nonconstant curvature determined by the kinetic energy term T and parametrized by {\lambda}. We show that H is Stackel equivalent to the free Euclidean motion, a fact that directly provides a curved Fradkin tensor of constants of motion for H. Furthermore, we analyze in terms of {\lambda} the three different underlying manifolds whose geodesic motion is provided by T. As a consequence, we find that H comprises three different nonlinear physical models that, by constructing their radial effective potentials, are shown to be two different nonlinear oscillators and an infinite barrier potential. The quantization of these two oscillators and its connection with spherical confinement models is briefly discussed.Comment: 11 pages; based on the contribution to the Manolo Gadella Fest-60 years-in-pucelandia, "Recent advances in time-asymmetric quantum mechanics, quantization and related topics" hold in Valladolid (Spain), 14-16th july 201

Integrable deformations of oscillator chains from quantum algebras

A family of completely integrable nonlinear deformations of systems of N harmonic oscillators are constructed from the non-standard quantum deformation of the sl(2,R) algebra. Explicit expressions for all the associated integrals of motion are given, and the long-range nature of the interactions introduced by the deformation is shown to be linked to the underlying coalgebra structure. Separability and superintegrability properties of such systems are analysed, and their connection with classical angular momentum chains is used to construct a non-standard integrable deformation of the XXX hyperbolic Gaudin system.Comment: 15 pages, LaTe

New time-type and space-type non-standard quantum algebras and discrete symmetries

Starting from the classical r-matrix of the non-standard (or Jordanian) quantum deformation of the sl(2,R) algebra, new triangular quantum deformations for the real Lie algebras so(2,2), so(3,1) and iso(2,1) are simultaneously constructed by using a graded contraction scheme; these are realized as deformations of conformal algebras of (1+1)-dimensional spacetimes. Time-type and space-type quantum algebras are considered according to the generator that remains primitive after deformation: either the time or the space translation, respectively. Furthermore by introducing differential-difference conformal realizations, these families of quantum algebras are shown to be the symmetry algebras of either a time or a space discretization of (1+1)-dimensional (wave and Laplace) equations on uniform lattices; the relationship with the known Lie symmetry approach to these discrete equations is established by means of twist maps.Comment: 17 pages, LaTe