dorsal/arxiv
View SchemaThe symmetric, D-invariant and Egorov reductions of the quadrilateral lattice
| Authors | Adam Doliwa, Paolo Maria Santini |
|---|---|
| Categories | |
| ArXiv ID | solv-int/9907012 |
| URL | https://arxiv.org/abs/solv-int/9907012 |
| DOI | 10.1016/S0393-0440(00)00011-5 |
Abstract
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.
{
"annotation_id": "d63e9a57-85af-4f93-a13b-561193bbc112",
"date_created": "2026-03-02T18:02:51.577000Z",
"date_modified": "2026-03-02T18:02:51.577000Z",
"file_hash": "389249ccb99906fc2039e8ea13df9a3160f3ef75f7b221c120e6b6b2f388cda5",
"private": false,
"record": {
"abstract": "We present a detailed study of the geometric and algebraic properties of the\nmultidimensional quadrilateral lattice (a lattice whose elementary\nquadrilaterals are planar; the discrete analogue of a conjugate net) and of its\nbasic reductions. To make this study, we introduce the notions of forward and\nbackward data, which allow us to give a geometric meaning to the tau-function\nof the lattice, defined as the potential connecting these data. Together with\nthe known circular lattice (a lattice whose elementary quadrilaterals can be\ninscribed in circles; the discrete analogue of an orthogonal conjugate net) we\nintroduce and study two other basic reductions of the quadrilateral lattice:\nthe symmetric lattice, for which the forward and backward data coincide, and\nthe D-invariant lattice, characterized by the invariance of a certain natural\nframe along the main diagonal. We finally discuss the Egorov lattice, which is,\nat the same time, symmetric, circular and D-invariant. The integrability\nproperties of all these lattices are established using geometric, algebraic and\nanalytic means; in particular we present a D-bar formalism to construct large\nclasses of such lattices. We also discuss quadrilateral hyperplane lattices and\nthe interplay between quadrilateral point and hyperplane lattices in all the\nabove reductions.",
"arxiv_id": "solv-int/9907012",
"authors": [
"Adam Doliwa",
"Paolo Maria Santini"
],
"categories": [
"solv-int",
"nlin.SI"
],
"doi": "10.1016/S0393-0440(00)00011-5",
"title": "The symmetric, D-invariant and Egorov reductions of the quadrilateral lattice",
"url": "https://arxiv.org/abs/solv-int/9907012"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "ded47da0-b331-4e1d-9539-d6884f58a655",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}