Hirota equation and the quantum plane

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.Comment: 19 pages, 9 figures; added references with relevant comments (v2