1999/07/31 by Adam Doliwa, Paolo Maria Santini · 3 citations
Computer Science · Materials Science · Physics and Astronomy · #Advanced Algebra and Logic #Algebraic number #Conjecture #Euclidean geometry #Hyperplane #Inscribed figure #Lattice (music) #Quadrilateral #Quantum Mechanics and Non-Hermitian Physics #Quasicrystal Structures and Properties #Subspace topology #nlin.SI #solv-int
paper · pdf · doi:10.1016/s0393-0440(00)00011-5
48 pages, 6 figures; 1 section added, to appear in J. Geom. & Phys
arxiv created 2000/02/16 · openalex publication_date 2000/11/01 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We present a detailed study of the geometric and algebraic properties of the multidimensional quadrilateral lattice (a lattice whose elementary quadrilaterals are planar; the discrete analogue of a conjugate net) and of its basic reductions. To make this study, we introduce the notions of forward and backward data, which allow us to give a geometric meaning to the tau-function of the lattice, defined as the potential connecting these data. Together with the known circular lattice (a lattice whose elementary quadrilaterals can be inscribed in circles; the discrete analogue of an orthogonal conjugate net) we introduce and study two other basic reductions of the quadrilateral lattice: the symmetric lattice, for which the forward and backward data coincide, and the D-invariant lattice, characterized by the invariance of a certain natural frame along the main diagonal. We finally discuss the Egorov lattice, which is, at the same time, symmetric, circular and D-invariant. The integrability properties of all these lattices are established using geometric, algebraic and analytic means; in particular we present a D-bar formalism to construct large classes of such lattices. We also discuss quadrilateral hyperplane lattices and the interplay between quadrilateral point and hyperplane lattices in all the above reductions.