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

Ya V. Lisitsyn, A. V. Shapovalov

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    1 Citation (Scopus)


    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 languageEnglish
    Pages (from-to)1172-1186
    Number of pages15
    JournalTheoretical and Mathematical Physics
    Issue number3
    Publication statusPublished - Sep 2000

    ASJC Scopus subject areas

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

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