### Abstract

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 · ind g ≤ k ≤ g · ind g. where ind g and 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(sl(2)) is constructed using the method of noncommutative integration of linear differential equations.

Original language | English |
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Pages (from-to) | 1172-1186 |

Number of pages | 15 |

Journal | Theoretical and Mathematical Physics |

Volume | 124 |

Issue number | 3 |

DOIs | |

Publication status | Published - Sep 2000 |

### ASJC Scopus subject areas

- Mathematical Physics
- Physics and Astronomy(all)
- Statistical and Nonlinear Physics

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## Cite this

*Theoretical and Mathematical Physics*,

*124*(3), 1172-1186. https://doi.org/10.1007/BF02550996