Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 3 (2007), 026, 20 pages      math-ph/0611040      https://doi.org/10.3842/SIGMA.2007.026
Contribution to the Proceedings of the O'Raifeartaigh Symposium

Quantum Deformations and Superintegrable Motions on Spaces with Variable Curvature

Orlando Ragnisco a, Ángel Ballesteros b, Francisco J. Herranz b and Fabio Musso a
a) Dipartimento di Fisica, Università di Roma Tre and Instituto Nazionale di Fisica Nucleare sezione di Roma Tre, Via Vasca Navale 84, I-00146 Roma, Italy
b) Departamento de Física, Universidad de Burgos, E-09001 Burgos, Spain

Received November 12, 2006, in final form January 22, 2007; Published online February 14, 2007

Abstract
An infinite family of quasi-maximally superintegrable Hamiltonians with a common set of (2N-3) integrals of the motion is introduced. The integrability properties of all these Hamiltonians are shown to be a consequence of a hidden non-standard quantum sl(2,R) Poisson coalgebra symmetry. As a concrete application, one of this Hamiltonians is shown to generate the geodesic motion on certain manifolds with a non-constant curvature that turns out to be a function of the deformation parameter z. Moreover, another Hamiltonian in this family is shown to generate geodesic motions on Riemannian and relativistic spaces all of whose sectional curvatures are constant and equal to the deformation parameter z. This approach can be generalized to arbitrary dimension by making use of coalgebra symmetry.

Key words: integrable systems; quantum groups; curvature; contraction; harmonic oscillator; Kepler-Coulomb; hyperbolic; de Sitter.

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