Integrable N-dimensional systems on the Hopf algebra and q-deformations

We construct the class of integrable classical and quantum systems on the H opf algebras describing a interacting particles. We obtain the general stru cture of an integrable Hamiltonian system for the Hopf algebra A(g) of a si mple Lie algebra g and prove that the integrals of motion depend only on li near combinations of k coordinates of the phase space 2 . ind g less than o r equal to k less than or equal to g . ind g, where ind g and g are the res pective index and Coxeter number of the Lie algebra g. The standard procedu re of q-deformation results in the quantum integrable system. We apply this general scheme to the algebras sl(2), sl(3), and o(3,1). An exact solution for the quantum analogue of the N-dimensional Hamiltonian system on the Ho pf algebra A(sl(2)) is constructed using the method of noncommutative integ ration of linear differential equations.