### Abstract

We define a map D^{2} × D^{2} → D^{2} × D^{2}, 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 language | English |
---|---|

Article number | 1.4882285 |

Pages (from-to) | 1-20 |

Number of pages | 20 |

Journal | Journal of Mathematical Physics |

Volume | 55 |

Issue number | 6 |

DOIs | |

Publication status | Published - 2014 |

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### Cite this

*Journal of Mathematical Physics*,

*55*(6), 1-20. [1.4882285]. https://doi.org/10.1063/1.4882285

}

*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.

Research output: Contribution to journal › Article

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 -