The pentagon relation and incidence geometry

Abstract

We define a map S: D^2 x D^2 --> D^2 x D^2, where 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.

20 pages, 5 figures; Section 3 reorganized and expanded, typos corrected (v2)

References in corpus (6)

• The quantum dilogarithm and representations quantum cluster varieties
• T-systems and Y-systems in integrable systems
• Quantum geometry of 3-dimensional lattices
• The affine Weyl group symmetry of Desargues maps and of the non-commutative Hirota-Miwa system
• Quantum Teichmüller space from quantum plane
• Quantization of three-wave equations

Cited by in corpus (13)

• Higher Segal spaces I
• Simplex and Polygon Equations
• Non-commutative rational Yang-Baxter maps
• Non-commutative bi-rational maps satisfying Zamolodchikov equation, and Desargues lattices
• Non-commutative lattice modified Gel'fand-Dikii systems
• The Coxeter relations and KP map for non-commuting symbols
• Matrix factorizations and pentagon maps
• Desargues maps and their reductions
• Non-commutative double-sided continued fractions
• Pentagon Relations in Direct Sums and Grassmann Algebras
• Hirota equation and the quantum plane
• Hopf algebra structure of generalized quasi-symmetric functions in partially commutative variables
• Non-commutative q-Painleve VI equation