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Angel Ballesteros

Publications and source records attributed to Angel Ballesteros.

At least 73 records · Page 4Linked to original sources
arXiv2013

On quantum deformations of (anti-)de Sitter algebras in (2+1) dimensions

Quantum deformations of (anti-)de Sitter algebras in (2+1) dimensions are revisited, and several features of these quantum structures are reviewed. 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 and groups is simultaneously given by considering the cosmological constant as an explicit Lie algebra parameter in the (A)dS algebras. By making use of this classification, a three-parameter generalization of the κ-deformation for the (2+1) (A)dS algebras and quantum spacetimes is given. Finally, the same problem is studied in (3+1) dimensions, where a two-parameter generalization of the κ-(A)dS deformation that preserves the space isotropy is found.

hep-th↗
arXiv2013

Quantum algebras as quantizations of dual Poisson-Lie groups

A systematic computational approach for the explicit construction of any quantum Hopf algebra (U_z(g),Δ_z) starting from the Lie bialgebra (g,δ) that gives the first-order deformation of the coproduct map Δ_z is presented. The procedure is based on the fact that any quantum algebra can be viewed as the quantization of the unique Poisson-Lie structure (G^\ast,Λ_g) on the dual group G^\ast, which is obtained by exponentiating the Lie algebra g^\ast defined by the dual map δ^\ast. From this perspective, the coproduct for U_z(g) is just the pullback of the group law for G^\ast, and the Poisson analogues of the quantum commutation rules for U_z(g) are given by the unique Poisson-Lie structure Λ_g on G^\ast whose linearization is the Poisson analogue of the initial Lie algebra g. This approach is shown to be very useful in order to construct quantum deformations explicitly since, once a Lie bialgebra (g,δ) is given, the full dual Poisson-Lie group (G^\ast,Λ) can be obtained either by applying standard Poisson-Lie group techniques or by implementing the algorithm here presented with the aid of symbolic manipulation programs. As a consequence, the quantization of (G^\ast,Λ) will give rise to the full U_z(g) quantum algebra, provided that ordering problems are appropriately fixed. The applicability of this approach is explicitly demonstrated by constructing several instances of quantum deformations of physically relevant Lie algebras as sl(2,R), the (2+1) Anti de Sitter algebra so(2,2) and the Poincaré algebra in (3+1) dimensions.

math-ph↗
arXiv2013

Superintegrable quantum oscillator and Kepler-Coulomb systems on curved spaces

An overview of maximally superintegrable classical Hamitonians on spherically symmetric spaces is presented. 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.

quant-ph↗
arXiv2013

Drinfel'd doubles for (2+1)-gravity

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.

math-ph↗
arXiv2013

The anisotropic oscillator on the two-dimensional sphere and the hyperbolic plane

An integrable generalization on the two-dimensional sphere S^2 and the hyperbolic plane H^2 of the Euclidean anisotropic oscillator Hamiltonian with "centrifugal" terms given by $H=1/2(p_1^2+p_2^2)+ δq_1^2+(δ+ Ω)q_2^2 +\frac{λ_1}{q_1^2}+\frac{λ_2}{q_2^2}$ is presented. The resulting generalized Hamiltonian H_κ depends explicitly on the constant Gaussian curvature κ of the underlying space, in such a way that all the results here presented hold simultaneously for S^2 (κ>0), H^2 (κ<0) and E^2 (κ=0). Moreover, H_κ is explicitly shown to be integrable for any values of the parameters δ, Ω, λ_1 and λ_2. Therefore, H_κ can also be interpreted as an anisotropic generalization of the curved Higgs oscillator, that is recovered as the isotropic limit Ω=0 of H_κ. Furthermore, numerical integration of some of the trajectories for H_κ are worked out and the dynamical features arising from the introduction of a curved background are highlighted. The superintegrability issue for H_κ is discussed by focusing on the value Ω=3δ, which is one of the cases for which the Euclidean Hamiltonian H is known to be superintegrable (the 1:2 oscillator). We show numerically that for Ω=3δ the curved Hamiltonian H_κ presents nonperiodic bounded trajectories, which seems to indicate that H_κ provides a non-superintegrable generalization of H. We compare this result with a previously known superintegrable curved analogue H'_κ of the 1:2 Euclidean oscillator showing that the Ω=3δ specialization of H_κ does not coincide with H'_κ. Finally, the geometrical interpretation of the curved "centrifugal" terms appearing in H_κ is also discussed in detail.

nlin.SI↗
arXiv2012

Non-coboundary Poisson-Lie structures on the book group

All possible Poisson-Lie (PL) structures on the 3D real Lie group generated by a dilation and two commuting translations are obtained. Its classification is fully performed by relating these PL groups with the corresponding Lie bialgebra structures on the corresponding "book" Lie algebra. By construction, all these Poisson structures are quadratic Poisson-Hopf algebras for which the group multiplication is a Poisson map. In contrast to the case of simple Lie groups, it turns out that most of the PL structures on the book group are non-coboundary ones. Moreover, from the viewpoint of Poisson dynamics, the most interesting PL book structures are just some of these non-coboundaries, which are explicitly analysed. In particular, we show that the two different q-deformed Poisson versions of the sl(2,R) algebra appear as two distinguished cases in this classification, as well as the quadratic Poisson structure that underlies the integrability of a large class of 3D Lotka-Volterra equations. Finally, the quantization problem for these PL groups is sketched.

math-ph↗
arXiv2012

Classification of real three-dimensional Poisson-Lie groups

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.

math-ph↗
arXiv2011

Integrable deformations of Lotka-Volterra systems

The Hamiltonian structure of a class of three-dimensional (3D) Lotka-Volterra (LV) equations is revisited from a novel point of view by showing that the quadratic Poisson structure underlying its integrability structure is just a real three-dimensional Poisson-Lie group. As a consequence, the Poisson coalgebra map that is given by the group multiplication provides the keystone for the explicit construction of a new family of 3N-dimensional integrable systems that, under certain constraints, contain N sets of deformed versions of the 3D LV equations. Moreover, by considering the most generic Poisson-Lie structure on this group, a new two-parametric integrable perturbation of the 3D LV system through polynomial and rational perturbation terms is explicitly found.

nlin.SI↗
arXiv2011

Superintegrable Oscillator and Kepler Systems on Spaces of Nonconstant Curvature via the Stäckel Transform

The Stäckel transform is applied to the geodesic motion on Euclidean space, through the harmonic oscillator and Kepler-Coloumb potentials, in order to obtain maximally superintegrable classical systems on N-dimensional Riemannian spaces of nonconstant curvature. By one hand, the harmonic oscillator potential leads to two families of superintegrable systems which are interpreted as an intrinsic Kepler-Coloumb system on a hyperbolic curved space and as the so-called Darboux III oscillator. On the other, the Kepler-Coloumb potential gives rise to an oscillator system on a spherical curved space as well as to the Taub-NUT oscillator. Their integrals of motion are explicitly given. The role of the (flat/curved) Fradkin tensor and Laplace-Runge-Lenz N-vector for all of these Hamiltonians is highlighted throughout the paper. The corresponding quantum maximally superintegrable systems are also presented.

math-ph↗
arXiv2011

Quantum mechanics on spaces of nonconstant curvature: the oscillator problem and superintegrability

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 Schroedinger 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.

quant-ph↗
arXiv2011

A new exactly solvable quantum model in N dimensions

An N-dimensional position-dependent mass Hamiltonian (depending on a parameter λ) formed by a curved kinetic term and an intrinsic oscillator potential is considered. It is shown that such a Hamiltonian is exactly solvable for any real positive value of the parameter λ. Algebraically, this Hamiltonian can be thought of as a new maximally superintegrable λ-deformation of the N-dimensional isotropic oscillator and, from a geometric viewpoint, this system is just the intrinsic oscillator potential on an N-dimensional hyperbolic space with nonconstant curvature. The spectrum of this model is shown to be hydrogenlike, and their eigenvalues and eigenfunctions are explicitly obtained by deforming appropriately the symmetry properties of the N-dimensional harmonic oscillator. A further generalization of this construction giving rise to new exactly solvable models is envisaged.

quant-ph↗
arXiv2010

Integrable Henon-Heiles Hamiltonians: a Poisson algebra approach

The three integrable two-dimensional Henon-Heiles systems and their integrable perturbations are revisited. A family of new integrable perturbations is found, and N-dimensional completely integrable generalizations of all these systems are constructed by making use of sl(2,R)+h(3) as their underlying Poisson symmetry algebra. In general, the procedure here introduced can be applied in order to obtain N-dimensional integrable generalizations of any 2D integrable potential of the form V(q_1^2, q_2), and the formalism gives the explicit form of all the integrals of the motion. Further applications of this algebraic approach in different contexts are suggested.

math-ph↗
arXiv2010

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.

math-ph↗
arXiv2010

On two superintegrable nonlinear oscillators in N dimensions

We consider the classical superintegrable Hamiltonian system given by $H=T+U={p^2}/{2(1+λq^2)}+{ω^2 q^2}/{2(1+λq^2)}$, 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 λ. 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 λ 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.

math-ph↗
arXiv2010

Integrable perturbations of the N-dimensional isotropic oscillator

Two new families of completely integrable perturbations of the N-dimensional isotropic harmonic oscillator Hamiltonian are presented. Such perturbations depend on arbitrary functions and N free parameters and their integrals of motion are explicitly constructed by making use of an underlying h_6-coalgebra symmetry. Several known integrable Hamiltonians in low dimensions are obtained as particular specializations of the general results here presented. An alternative route for the integrability of all these systems is provided by a suitable canonical transformation which, in turn, opens the possibility of adding (N-1) `Rosochatius' terms that preserve the complete integrability of all these models.

nlin.SI↗
arXiv2010

Three-dimensional gravity and Drinfel'd doubles: spacetimes and symmetries from quantum deformations

We show how the constant curvature spacetimes of 3d gravity and the associated symmetry algebras can be derived from a single quantum deformation of the 3d Lorentz algebra sl(2,R). We investigate the classical Drinfel'd double of a "hybrid" deformation of sl(2,R) that depends on two parameters (η,z). With an appropriate choice of basis and real structure, this Drinfel'd double agrees with the 3d anti-de Sitter algebra. The deformation parameter ηis related to the cosmological constant, while z is identified with the inverse of the speed of light and defines the signature of the metric. We generalise this result to de Sitter space, the three-sphere and 3d hyperbolic space through analytic continuation in ηand z; we also investigate the limits of vanishing ηand z, which yield the flat spacetimes (Minkowski and Euclidean spaces) and Newtonian models, respectively.

gr-qc↗
arXiv2009

Hamiltonian systems admitting a Runge-Lenz vector and an optimal extension of Bertrand's theorem to curved manifolds

Bertrand's theorem asserts that any spherically symmetric natural Hamiltonian system in Euclidean 3-space which possesses stable circular orbits and whose bounded trajectories are all periodic is either a harmonic oscillator or a Kepler system. In this paper we extend this classical result to curved spaces by proving that any Hamiltonian on a spherically symmetric Riemannian 3-manifold which satisfies the same conditions as in Bertrand's theorem is superintegrable and given by an intrinsic oscillator or Kepler system. As a byproduct we obtain a wide panoply of new superintegrable Hamiltonian systems. The demonstration relies on Perlick's classification of Bertrand spacetimes and on the construction of a suitable, globally defined generalization of the Runge-Lenz vector.

math-ph↗