On a non-commutative sixth $q$-Painlevé system: from discrete system to surface theory
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915928307924992 |
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| author | Bobrova, Irina |
| author_facet | Bobrova, Irina |
| contents | In this paper, we describe the non-commutative formal geometry underlying a certain class of discrete integrable systems. Our main example is a non-commutative analog, labeled $q$-P$(A_3)$, of the sixth $q$-Painlevé equation. The system $q$-P$(A_3)$ is constructed by postulating an extended birational representation of the extended affine Weyl group $\widetilde{W}$ of type $D_5^{(1)}$ and by selecting the same translation element in $\widetilde{W}$ as in the commutative case. Starting from this non-commutative discrete system, we develop a non-commutative version of Sakai$'$s surface theory, which allows us to derive the same birational representation that we initially postulated. Moreover, we recover the well-known cascade of multiplicative discrete Painlevé equations rooted in $q$-P$(A_3)$ and establish a connection between $q$-P$(A_3)$ and the non-commutative $d$-Painlevé systems introduced in I. Bobrova. Affine Weyl groups and non-Abelian discrete systems: an application to the $d$-Painlevé equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_22466 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a non-commutative sixth $q$-Painlevé system: from discrete system to surface theory Bobrova, Irina Exactly Solvable and Integrable Systems Mathematical Physics Rings and Algebras In this paper, we describe the non-commutative formal geometry underlying a certain class of discrete integrable systems. Our main example is a non-commutative analog, labeled $q$-P$(A_3)$, of the sixth $q$-Painlevé equation. The system $q$-P$(A_3)$ is constructed by postulating an extended birational representation of the extended affine Weyl group $\widetilde{W}$ of type $D_5^{(1)}$ and by selecting the same translation element in $\widetilde{W}$ as in the commutative case. Starting from this non-commutative discrete system, we develop a non-commutative version of Sakai$'$s surface theory, which allows us to derive the same birational representation that we initially postulated. Moreover, we recover the well-known cascade of multiplicative discrete Painlevé equations rooted in $q$-P$(A_3)$ and establish a connection between $q$-P$(A_3)$ and the non-commutative $d$-Painlevé systems introduced in I. Bobrova. Affine Weyl groups and non-Abelian discrete systems: an application to the $d$-Painlevé equations. |
| title | On a non-commutative sixth $q$-Painlevé system: from discrete system to surface theory |
| topic | Exactly Solvable and Integrable Systems Mathematical Physics Rings and Algebras |
| url | https://arxiv.org/abs/2507.22466 |