$q$-Painlevé equations on cluster Poisson varieties via toric geometry
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2020
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| _version_ | 1866916117888368640 |
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| author | Mizuno, Yuma |
| author_facet | Mizuno, Yuma |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2008_11219 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | $q$-Painlevé equations on cluster Poisson varieties via toric geometry Mizuno, Yuma Mathematical Physics Algebraic Geometry Exactly Solvable and Integrable Systems 13F60, 34M55, 39A13, 14M25 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. |
| title | $q$-Painlevé equations on cluster Poisson varieties via toric geometry |
| topic | Mathematical Physics Algebraic Geometry Exactly Solvable and Integrable Systems 13F60, 34M55, 39A13, 14M25 |
| url | https://arxiv.org/abs/2008.11219 |