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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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