Hirota equation and the quantum plane

TL;DR

This paper explores the geometric integrability of Hirota's discrete KP equation using projective geometry over division rings, linking it to quantum algebra structures like the quantum plane and Hopf algebras.

Contribution

It introduces a geometric framework for Hirota's equation via Desargues maps and connects this to quantum algebra through the functional pentagon equation and Weyl relations.

Findings

Introduction of maps satisfying the functional pentagon equation

Establishment of Weyl commutation relations from ultra-locality

Definition of a coproduct in the quantum plane bi-algebra

Abstract

We discuss geometric integrability of Hirota's discrete KP 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 A 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 bi-algebra, which can be extended to the corresponding Hopf algebra.