### 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 |

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### ASJC Scopus subject areas

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

### Cite this

*Theoretical and Mathematical Physics*,

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

**Integrable N-dimensional systems on the hopf algebra and q-deformations.** / Lisitsyn, Ya V.; Shapovalov, A. V.

Research output: Contribution to journal › Article

*Theoretical and Mathematical Physics*, vol. 124, no. 3, pp. 1172-1186. https://doi.org/10.1007/BF02550996

}

TY - JOUR

T1 - Integrable N-dimensional systems on the hopf algebra and q-deformations

AU - Lisitsyn, Ya V.

AU - Shapovalov, A. V.

PY - 2000/9

Y1 - 2000/9

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0034348048&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0034348048&partnerID=8YFLogxK

U2 - 10.1007/BF02550996

DO - 10.1007/BF02550996

M3 - Article

VL - 124

SP - 1172

EP - 1186

JO - Theoretical and Mathematical Physics(Russian Federation)

JF - Theoretical and Mathematical Physics(Russian Federation)

SN - 0040-5779

IS - 3

ER -