A systematic construction of completely integrable Hamiltonians from coalgebras

A universal algorithm to construct N-particle (classical and quantum) completely integrable Hamiltonian systems from representations of coalgebras with Casimir elements 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) Poincare algebra is shown to provide a new Ruijsenaars-Schneider-like Hamiltonian.