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.