The pentagon relation and incidence geometry

Adam Doliwa, Sergey Sergeev

    Research output: Contribution to journalArticle

    1 Citation (Scopus)

    Abstract

    We define a map D2 × D2 → D2 × D2, where D is an arbitrary division ring (skew field), associated with the Veblen configuration, and we show that such a map provides solutions to the functional dynamical pentagon equation. We explain that fact in elementary geometric terms using the symmetry of the Veblen and Desargues configurations. We introduce also another map of a geometric origin with the pentagon property. We show equivalence of these maps with recently introduced Desargues maps which provide geometric interpretation to a non-commutative version of Hirota's discrete Kadomtsev-Petviashvili equation. Finally, we demonstrate that in an appropriate gauge the (commutative version of the) maps preserves a natural Poisson structure-the quasiclassical limit of the Weyl commutation relations. The corresponding quantum reduction is then studied. In particular, we discuss uniqueness of the Weyl relations for the ultra-local reduction of the map. We give then the corresponding solution of the quantum pentagon equation in terms of the non-compact quantum dilogarithm function.

    Original languageEnglish
    Article number1.4882285
    Pages (from-to)1-20
    Number of pages20
    JournalJournal of Mathematical Physics
    Volume55
    Issue number6
    DOIs
    Publication statusPublished - 2014

    Fingerprint

    Incidence Geometry
    Pentagon
    incidence
    geometry
    Division ring or skew field
    Dilogarithm
    Kadomtsev-Petviashvili Equation
    Configuration
    Poisson Structure
    commutation
    Discrete Equations
    uniqueness
    configurations
    division
    equivalence
    Gauge
    Uniqueness
    Equivalence
    Symmetry
    rings

    Cite this

    Doliwa, Adam ; Sergeev, Sergey. / The pentagon relation and incidence geometry. In: Journal of Mathematical Physics. 2014 ; Vol. 55, No. 6. pp. 1-20.
    @article{f28fe01c834345318468db5b78809cf5,
    title = "The pentagon relation and incidence geometry",
    abstract = "We define a map D2 × D2 → D2 × D2, where D is an arbitrary division ring (skew field), associated with the Veblen configuration, and we show that such a map provides solutions to the functional dynamical pentagon equation. We explain that fact in elementary geometric terms using the symmetry of the Veblen and Desargues configurations. We introduce also another map of a geometric origin with the pentagon property. We show equivalence of these maps with recently introduced Desargues maps which provide geometric interpretation to a non-commutative version of Hirota's discrete Kadomtsev-Petviashvili equation. Finally, we demonstrate that in an appropriate gauge the (commutative version of the) maps preserves a natural Poisson structure-the quasiclassical limit of the Weyl commutation relations. The corresponding quantum reduction is then studied. In particular, we discuss uniqueness of the Weyl relations for the ultra-local reduction of the map. We give then the corresponding solution of the quantum pentagon equation in terms of the non-compact quantum dilogarithm function.",
    keywords = "Poisson's equation, Laplace-equations, Functional equations, noncummutative field theory, rings",
    author = "Adam Doliwa and Sergey Sergeev",
    year = "2014",
    doi = "10.1063/1.4882285",
    language = "English",
    volume = "55",
    pages = "1--20",
    journal = "Journal of Mathematics",
    issn = "0022-2488",
    publisher = "American Institute of Physics",
    number = "6",

    }

    Doliwa, A & Sergeev, S 2014, 'The pentagon relation and incidence geometry', Journal of Mathematical Physics, vol. 55, no. 6, 1.4882285, pp. 1-20. https://doi.org/10.1063/1.4882285

    The pentagon relation and incidence geometry. / Doliwa, Adam; Sergeev, Sergey.

    In: Journal of Mathematical Physics, Vol. 55, No. 6, 1.4882285, 2014, p. 1-20.

    Research output: Contribution to journalArticle

    TY - JOUR

    T1 - The pentagon relation and incidence geometry

    AU - Doliwa, Adam

    AU - Sergeev, Sergey

    PY - 2014

    Y1 - 2014

    N2 - We define a map D2 × D2 → D2 × D2, where D is an arbitrary division ring (skew field), associated with the Veblen configuration, and we show that such a map provides solutions to the functional dynamical pentagon equation. We explain that fact in elementary geometric terms using the symmetry of the Veblen and Desargues configurations. We introduce also another map of a geometric origin with the pentagon property. We show equivalence of these maps with recently introduced Desargues maps which provide geometric interpretation to a non-commutative version of Hirota's discrete Kadomtsev-Petviashvili equation. Finally, we demonstrate that in an appropriate gauge the (commutative version of the) maps preserves a natural Poisson structure-the quasiclassical limit of the Weyl commutation relations. The corresponding quantum reduction is then studied. In particular, we discuss uniqueness of the Weyl relations for the ultra-local reduction of the map. We give then the corresponding solution of the quantum pentagon equation in terms of the non-compact quantum dilogarithm function.

    AB - We define a map D2 × D2 → D2 × D2, where D is an arbitrary division ring (skew field), associated with the Veblen configuration, and we show that such a map provides solutions to the functional dynamical pentagon equation. We explain that fact in elementary geometric terms using the symmetry of the Veblen and Desargues configurations. We introduce also another map of a geometric origin with the pentagon property. We show equivalence of these maps with recently introduced Desargues maps which provide geometric interpretation to a non-commutative version of Hirota's discrete Kadomtsev-Petviashvili equation. Finally, we demonstrate that in an appropriate gauge the (commutative version of the) maps preserves a natural Poisson structure-the quasiclassical limit of the Weyl commutation relations. The corresponding quantum reduction is then studied. In particular, we discuss uniqueness of the Weyl relations for the ultra-local reduction of the map. We give then the corresponding solution of the quantum pentagon equation in terms of the non-compact quantum dilogarithm function.

    KW - Poisson's equation

    KW - Laplace-equations

    KW - Functional equations

    KW - noncummutative field theory

    KW - rings

    UR - http://www.scopus.com/inward/record.url?scp=84929000003&partnerID=8YFLogxK

    UR - http://www.mendeley.com/research/pentagon-relation-incidence-geometry

    U2 - 10.1063/1.4882285

    DO - 10.1063/1.4882285

    M3 - Article

    VL - 55

    SP - 1

    EP - 20

    JO - Journal of Mathematics

    JF - Journal of Mathematics

    SN - 0022-2488

    IS - 6

    M1 - 1.4882285

    ER -