q-Painlevé equations on cluster Poisson varieties via toric geometry

Selecta Mathematica Pub Date : 2024-01-27 DOI:10.1007/s00029-023-00906-2

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Abstract

We provide a relation between the geometric framework for q-Painlevé equations and cluster Poisson varieties by using toric models of rational surfaces associated with q-Painlevé equations. We introduce the notion of seeds of q-Painlevé type by the negative semi-definiteness of symmetric bilinear forms associated with seeds, and classify the mutation equivalence classes of these seeds. This classification coincides with the classification of q-Painlevé equations given by Sakai. We realize q-Painlevé systems as automorphisms on cluster Poisson varieties associated with seeds of q-Painlevé type.

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通过环几何研究群泊松变体上的 q-Painlevé 方程

摘要我们利用与 q-Painlevé 方程相关的有理曲面的环模型，提供了 q-Painlevé 方程的几何框架与簇泊松变种之间的关系。我们通过与种子相关的对称双线性形式的负半定义性引入了 q-Painlevé 型种子的概念，并对这些种子的突变等价类进行了分类。这一分类与酒井（Sakai）给出的 q-Painlevé 方程分类不谋而合。我们把 q-Painlevé 系统看作是与 q-Painlevé 类型种子相关的簇泊松变体上的自动形态。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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