  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.
