The pentagon relation and incidence geometry

We define a map \documentclass[12pt]{minimal}\begin{document}$S:{\mathbb {D}}^2\times {\mathbb {D}}^2 \dashrightarrow {\mathbb {D}}^2\times {\mathbb {D}}^2$\end{document} S : D 2 × D 2 ⤏ D 2 × D 2 , where \documentclass[12pt]{minimal}\begin{document}${\mathbb {D}}$\end{document} D is an arbitrary division ring (skew field), associated with the Veblen configuration, and we show that such a map provides solutions to the functional dynamical pentagon equation. We explain that fact in elementary geometric terms using the symmetry of the Veblen and Desargues configurations. We introduce also another map of a geometric origin with the pentagon property. We show equivalence of these maps with recently introduced Desargues maps which provide geometric interpretation to a non-commutative version of Hirota's discrete Kadomtsev–Petviashvili equation. Finally, we demonstrate that in an appropriate gauge the (commutative version of the) maps preserves a natural Poisson structure—the quasiclassical limit of the Weyl commutation relations. The corresponding quantum reduction is then studied. In particular, we discuss uniqueness of the Weyl relations for the ultra-local reduction of the map. We give then the corresponding solution of the quantum pentagon equation in terms of the non-compact quantum dilogarithm function.