A systematic construction of completely integrable Hamiltonians from coalgebras

Abstract

A universal algorithm to construct N -particle (classical and quantum) completely integrable Hamiltonian systems from representations of coalgebras with Casimir element is presented. In particular, this construction shows that quantum deformations can be interpreted as generating structures for integrable deformations of Hamiltonian systems with coalgebra symmetry. In order to illustrate this general method, the so(2, 1) algebra and the oscillator algebra h 4 are used to derive new classical integrable systems including a generalization of Gaudin-Calogero systems and oscillator chains. Quantum deformations are then used to obtain some explicit integrable deformations of the previous long-range interacting systems and a (non-coboundary) deformation of the (1 + 1) Poincaré algebra is shown to provide a new Ruijsenaars-Schneider-like Hamiltonian.

Quantum Deformations and Superintegrable Motions on Spaces with Variable Curvature

“…In fact, the coalgebra approach [34] provides an N -particle symplectic realization of sl(2, R) through the N -sites coproduct of (2.4) living on sl(2, R) ⊗ · · · N ) ⊗ sl(2, R) [22]:…”

Integrable geodesic motion on 3D curved spaces from non-standard quantum deformations

“…In order to perform the generalization of this construction to 3D spaces, a three particle symplectic realization of the deformed Poisson algebra (1) has to be obtained from the 3-sites coproduct map [3], which is defined as:…”

“…By subsituting these expresions in (3) we get the three-particle Casimir which Poisson-commutes, by construction [3], with the three-particle generators (12) and also with the two-particle Casimir C (2) z (6). Hence the generic Hamiltonian…”

“…which is just the metric of the 3D Riemannian and relativistic spacetimes written in geodesic polar coordinates [10] multiplied by a global factor 1/cosh(λ 1 ρ) ≡ e −zJ (3) − that encodes the information concerning the variable curvature of the space. In the new coordinates the sectional and scalar curvatures read…”

“…In turn, the spaces with variable curvature arise from the integrable geodesic motion and they can be considered as deformations of the abovementioned spaces for which the curvature is a function of both the deformation parameter z and some intrinsic coordinates of the space. Moreover, some known and new (super)integrable potentials (such as oscillator and Kepler-type ones) defined on these curved spaces have also been obtained [2] by making use of the dynamical coalgebra symmetry [3].…”