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

Ya V. Lisitsyn, A. V. Shapovalov

    Research output: Contribution to journalArticle

    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 languageEnglish
    Pages (from-to)1172-1186
    Number of pages15
    JournalTheoretical and Mathematical Physics
    Volume124
    Issue number3
    DOIs
    Publication statusPublished - Sep 2000

    Fingerprint

    Q-deformation
    Hopf Algebra
    algebra
    Quantum Integrable Systems
    Integrable Hamiltonian System
    Integrals of Motion
    Simple Lie Algebra
    Linear differential equation
    Quantum Systems
    Hamiltonian Systems
    Linear Combination
    Phase Space
    Lie Algebra
    Exact Solution
    Analogue
    Algebra
    differential equations
    analogs

    ASJC Scopus subject areas

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

    Cite this

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

    In: Theoretical and Mathematical Physics, Vol. 124, No. 3, 09.2000, p. 1172-1186.

    Research output: Contribution to journalArticle

    @article{a5847c9ef5e44ada842fcac82cc0c6db,
    title = "Integrable N-dimensional systems on the hopf algebra and q-deformations",
    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.",
    author = "Lisitsyn, {Ya V.} and Shapovalov, {A. V.}",
    year = "2000",
    month = "9",
    doi = "10.1007/BF02550996",
    language = "English",
    volume = "124",
    pages = "1172--1186",
    journal = "Theoretical and Mathematical Physics(Russian Federation)",
    issn = "0040-5779",
    publisher = "Springer New York",
    number = "3",

    }

    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

    AN - SCOPUS:0034348048

    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 -