Heisenberg XXZ model and quantum Galilei group

The 1D Heisenberg spin chain with anisotropy of the XXZ type is analyzed in terms of the symmetry given by the quantum Galilei group Gamma_q(1). We show that the magnon excitations and the s=1/2, n-magnon bound states are determined by the algebra. Thus the Gamma_q(1) symmetry provides a description that naturally induces the Bethe Ansatz. The recurrence relations determined by Gamma_q(1) permit to express the energy of the n-magnon bound states in a closed form in terms of Tchebischeff polynomials.