Symplectic Maps from Cluster Algebras
We consider nonlinear recurrences generated from the iteration of maps that arise from cluster algebras. More precisely, starting from a skew-symmetric integer matrix, or its corresponding quiver, one can define a set of mutation operations, as well as a set of associated cluster mutations that are...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2011 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2011
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/147396 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Symplectic Maps from Cluster Algebras / A.P. Fordy, A. Hone // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 17 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862626160670670848 |
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| author | Fordy, A.P. Hone, A. |
| author_facet | Fordy, A.P. Hone, A. |
| citation_txt | Symplectic Maps from Cluster Algebras / A.P. Fordy, A. Hone // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 17 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We consider nonlinear recurrences generated from the iteration of maps that arise from cluster algebras. More precisely, starting from a skew-symmetric integer matrix, or its corresponding quiver, one can define a set of mutation operations, as well as a set of associated cluster mutations that are applied to a set of affine coordinates (the cluster variables). Fordy and Marsh recently provided a complete classification of all such quivers that have a certain periodicity property under sequences of mutations. This periodicity implies that a suitable sequence of cluster mutations is precisely equivalent to iteration of a nonlinear recurrence relation. Here we explain briefly how to introduce a symplectic structure in this setting, which is preserved by a corresponding birational map (possibly on a space of lower dimension). We give examples of both integrable and non-integrable maps that arise from this construction. We use algebraic entropy as an approach to classifying integrable cases. The degrees of the iterates satisfy a tropical version of the map.
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| first_indexed | 2025-12-07T13:35:47Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-147396 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T13:35:47Z |
| publishDate | 2011 |
| publisher | Інститут математики НАН України |
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| spelling | Fordy, A.P. Hone, A. 2019-02-14T17:28:43Z 2019-02-14T17:28:43Z 2011 Symplectic Maps from Cluster Algebras / A.P. Fordy, A. Hone // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 17 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 37K10; 17B63; 53D17; 14T05 http://dx.doi.org/10.3842/SIGMA.2011.091 https://nasplib.isofts.kiev.ua/handle/123456789/147396 We consider nonlinear recurrences generated from the iteration of maps that arise from cluster algebras. More precisely, starting from a skew-symmetric integer matrix, or its corresponding quiver, one can define a set of mutation operations, as well as a set of associated cluster mutations that are applied to a set of affine coordinates (the cluster variables). Fordy and Marsh recently provided a complete classification of all such quivers that have a certain periodicity property under sequences of mutations. This periodicity implies that a suitable sequence of cluster mutations is precisely equivalent to iteration of a nonlinear recurrence relation. Here we explain briefly how to introduce a symplectic structure in this setting, which is preserved by a corresponding birational map (possibly on a space of lower dimension). We give examples of both integrable and non-integrable maps that arise from this construction. We use algebraic entropy as an approach to classifying integrable cases. The degrees of the iterates satisfy a tropical version of the map. This paper is a contribution to the Proceedings of the Conference “Symmetries and Integrability of Difference Equations (SIDE-9)” (June 14–18, 2010, Varna, Bulgaria). The full collection is available at http://www.emis.de/journals/SIGMA/SIDE-9.html.
 The authors would like to thank the Isaac Newton Institute, Cambridge for hospitality during the Programme on Discrete Integrable Systems, where this collaboration began. They are also grateful to the organisers of SIDE 9 in Varna for inviting us both to speak there. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Symplectic Maps from Cluster Algebras Article published earlier |
| spellingShingle | Symplectic Maps from Cluster Algebras Fordy, A.P. Hone, A. |
| title | Symplectic Maps from Cluster Algebras |
| title_full | Symplectic Maps from Cluster Algebras |
| title_fullStr | Symplectic Maps from Cluster Algebras |
| title_full_unstemmed | Symplectic Maps from Cluster Algebras |
| title_short | Symplectic Maps from Cluster Algebras |
| title_sort | symplectic maps from cluster algebras |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/147396 |
| work_keys_str_mv | AT fordyap symplecticmapsfromclusteralgebras AT honea symplecticmapsfromclusteralgebras |