五边形关系与入射几何
The pentagon relation and incidence geometry
摘要：
我们定义一个映射S：D^2 x D^2 -> D^2 x D^2，其中D是一个任意的除环（斜场），与凡勃伦组态有关，我们证明了这样的映射提供了泛函动力学五边形方程的解。我们解释了这一事实，在初等几何条件下使用的对称性的凡勃伦和Desargues配置。本文还介绍了另一个具有五边形性质的几何原点映射。我们显示等价的这些地图最近推出的Desargues地图提供几何解释的非交换版本的广田离散Kadomtsev Petviashvili方程。最后，我们证明了在一个适当的规范（交换版本）的映射保持一个自然的泊松结构-准经典极限的Weyl交换关系。然后研究相应的量子约化。特别地，我们讨论了Weyl关系的超局部约化的映射的唯一性。然后，我们给出了量子五边形方程在非紧量子双对数函数下的相应解。
英文摘要：
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.
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