Symmetry, integrability and deformations of long-range interacting Hamiltonians

The notion of coalgebra symmetry in Hamiltonian systems is analysed. It is shown how the complete integrability of some long-range interacting Hamilto nians can be extracted from their associated coalgebra structure with no us e of a quantum R-matrix. Within this framework, integrable deformations can be considered as direct consequences of the introduction of coalgebra defo rmations (quantum algebras). As an example, the Gaudin magnet is derived fr om a sl(2) coalgebra, and a completely integrable deformation of this Hamil tonian is obtained through a twisted gl(2) quantum algebra.