Classical Hamiltonian systems with sl(2) coalgebra symmetry and their integrable deformations

Abstract

Several families of classical integrable systems with two degrees of freedom are derived from phase-space realizations of sl(2) Poisson coalgebras. As a remarkable fact, the existence of the N-dimensional integrable generalization of all these systems is always ensured (by construction) due to their underlying dynamical coalgebra symmetry. By following the same approach, different integrable deformations for such systems are obtained from the q-deformed analogues of sl(2). The well-known Jordan-Schwinger realization is also proven to be related to a (non-coassociative) coalgebra structure on sl(2) and the 2 N dimensional integrable Hamiltonian generated by such Jordan-Schwinger representation is obtained. Finally, the relation between complete integrability and the properties of the initial phase-space realization is elucidated through two more examples based on the Heisenberg-Weyl and so(3,2) Poisson coalgebras.