The pentagon relation and incidence geometry

Abstract

Abstract. We define a map S : D 2 × D 2 D 2 × 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.

Non-Commutative Rational Yang–Baxter Maps

“…We remark, see [9,10], that three-dimensional consistency of the equations considered here is a consequence of four-dimensional compatibility of the non-commutative Hirota's discrete KP system [8], where the counterpart of the functional Yang-Baxter equation is the functional pentagon equation [11]. Since the solutions of the pentagon equation presented in [11] allow for quantization (understood as a reduction from the noncommutative case by adding certain commutation relations preserved by the integrable evolution), we expect that also the non-commutative rational Yang-Baxter map obtained above can be quantized in such a way also. It would be instructive to understand various applications of the Hirota discrete KP systems and its reductions reviewed in [21] from that perspective.…”

Non-commutative lattice-modified Gel’fand–Dikii systems

“…In particular, as shown by [6,89], the four dimensional consistency of the geometric construction of multidimensional lattice of planar quadrilaterals [32] is related to Zamolodchikov's tetrahedron equation [94], which is a multidimensional analogue of the quantum Yang-Baxter equation [5,56,52]. Recently, the four dimensional consistency of Desargues maps has been related [34,31] to certain solutions of the functional pentagon quation [93] and of its quantum reduction, which is of fundamental importance in the modern analytic theory of quantum groups [90].…”

Simplex and Polygon Equations

“…which appears as the Biedenharn-Elliott identity for Wigner 6j-symbols and Racah coefficients in the representation theory of the rotation group [19], as an identity for fusion matrices in conformal field theory [84], as a consistency condition for the associator in quasi-Hopf algebras [27,28] (also see [3,4,10,33,36,40,41]), as an identity for the Rogers dilogarithm function [87] and matrix generalizations [53], for the quantum dilogarithm [5,17,20,37,55,100], and in various other contexts (see, e.g., [26,52,56,57,60,67,72]). In particular, it is satisfied by the Kac-Takesaki operator (T f )(g, g ) = f (gg , g ), g, g ∈ G, G a group, where it expresses the associativity of the group operation (see, e.g., [101]).…”