2012/08/31 by Adam Doliwa · 1 citation
Mathematics · Physics and Astronomy · #Action (physics) #Advanced Topics in Algebra #Affine transformation #Algebra over a field #Algebraic geometry #Algebraic structures and combinatorial models #Geometry #Homogeneous space #Mathematics #Nonlinear Waves and Solitons #Pentagon #Physics #Point (geometry) #Projective geometry #Projective plane #Pure mathematics #Type (biology) #nlin.SI
paper · pdf · doi:10.1090/conm/593/11871
published as Algebraic and Geometric Aspects of Integrable Systems and Random Matrices, Contemporary Mathematics, vol. 593, Amer. Math. Soc., Providence, RI, 2013, pp. 205-230 · 19 pages, 9 figures; added references with relevant comments (v2)
arxiv created 2012/11/21 · openalex publication_date 2013/01/01 · arxiv updated 2013/08/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We discuss geometric integrability of Hirota’s discrete Kadomtsev–Petviashvilii equation in the framework of projective geometry over division rings using the recently introduced notion of Desargues maps. We also present the Darboux-type transformations, and we review symmetries of the Desargues maps from the point of view of root lattices of type <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the action of the corresponding affine Weyl group. Such a point of view facilities to study the relation of Desargues maps and the discrete conjugate nets. Recent investigation of geometric integrability of Desargues maps allowed to introduce two maps satisfying functional pentagon equation. Moreover, the ultra-locality requirement imposed on the maps leads to Weyl commutation relations. We show that the pentagonal property of the maps allows to define a coproduct in the quantum plane bialgebra, which can be extended to the corresponding Hopf algebra.