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arXiv · 2008.11219

$q$-Painlev\'e equations on cluster Poisson varieties via toric geometry

Yuma Mizuno
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Abstract

We provide a relation between the geometric framework for $q$-Painlev\'{e} equations and cluster Poisson varieties by using toric models of rational surfaces associated with $q$-Painlev\'{e} equations. We introduce the notion of seeds of $q$-Painlev\'{e} 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 $q$-Painlev\'{e} equations given by Sakai. We realize $q$-Painlev\'{e} systems as automorphisms on cluster Poisson varieties associated with seeds of $q$-Painlev\'{e} type.

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BibTeXRIS

Yuma Mizuno. 2020-08-25. $q$-Painlev\'e equations on cluster Poisson varieties via toric geometry. https://doi.org/10.1007/s00029-023-00906-2

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