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

We construct the class of integrable classical and quantum systems on the Hopf algebras describing $n$ interacting particles. We obtain the general structure of an integrable Hamiltonian system for the Hopf algebra $A(g)$ of a simple Lie algebra $g$ and prove that the integrals of motion depend only on linear combinations of $k$ coordinates of the phase space, $2\cdot\mathrm{ind}g\leq k\leq\mathbf g\cdot\mathrm{ind}g$, where $\mathrm{ind} g$ and $\mathbf g$ are the respective index and Coxeter number of the Lie algebra $g$. The standard procedure 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 Hopf algebra $A\bigl(sl(2)\bigr)$ is constructed using the method of noncommutative integration of linear differential equations.