Commuting integrable maps from a deformed D$_4$ cluster algebra
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915737321340928 |
|---|---|
| author | Hone, A. N. W. Kim, W. Mase, T. |
| author_facet | Hone, A. N. W. Kim, W. Mase, T. |
| contents | In this paper we revisit an integrable map of the plane which we obtained recently as a two-parameter family of deformed mutations in the cluster algebra of type D$_4$. The rational first integral for this map defines an invariant foliation of the plane by level curves, and we explain how this corresponds to a rational elliptic surface of rank 2. This leads us to construct another (independent) integrable map, commuting with the first, such that both maps lift to compositions of mutations in an enlarged cluster algebra, whose underlying quiver is equivalent to the one found by Okubo for the $q$-Painlevé VI equation. The degree growth of the two commuting maps is calculated in two different ways: firstly, from the tropical (max-plus) equations for the d-vectors of the cluster variables; and secondly, by constructing the minimal space of initial conditions for the two maps, via blowing up $\mathbb{P}^1 \times \mathbb{P}^1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_05270 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Commuting integrable maps from a deformed D$_4$ cluster algebra Hone, A. N. W. Kim, W. Mase, T. Exactly Solvable and Integrable Systems Mathematical Physics In this paper we revisit an integrable map of the plane which we obtained recently as a two-parameter family of deformed mutations in the cluster algebra of type D$_4$. The rational first integral for this map defines an invariant foliation of the plane by level curves, and we explain how this corresponds to a rational elliptic surface of rank 2. This leads us to construct another (independent) integrable map, commuting with the first, such that both maps lift to compositions of mutations in an enlarged cluster algebra, whose underlying quiver is equivalent to the one found by Okubo for the $q$-Painlevé VI equation. The degree growth of the two commuting maps is calculated in two different ways: firstly, from the tropical (max-plus) equations for the d-vectors of the cluster variables; and secondly, by constructing the minimal space of initial conditions for the two maps, via blowing up $\mathbb{P}^1 \times \mathbb{P}^1$. |
| title | Commuting integrable maps from a deformed D$_4$ cluster algebra |
| topic | Exactly Solvable and Integrable Systems Mathematical Physics |
| url | https://arxiv.org/abs/2508.05270 |