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...
Gespeichert in:
| Datum: | 2011 |
|---|---|
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Інститут математики НАН України
2011
|
| Schriftenreihe: | Symmetry, Integrability and Geometry: Methods and Applications |
| Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/147396 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Symplectic Maps from Cluster Algebras / A.P. Fordy, A. Hone // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 17 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-147396 |
|---|---|
| record_format |
dspace |
| fulltext |
|
| spelling |
irk-123456789-1473962019-02-15T01:24:06Z Symplectic Maps from Cluster Algebras Fordy, A.P. Hone, A. 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. 2011 Article 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 http://dspace.nbuv.gov.ua/handle/123456789/147396 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| 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. |
| format |
Article |
| author |
Fordy, A.P. Hone, A. |
| spellingShingle |
Fordy, A.P. Hone, A. Symplectic Maps from Cluster Algebras Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Fordy, A.P. Hone, A. |
| author_sort |
Fordy, A.P. |
| title |
Symplectic Maps from Cluster Algebras |
| title_short |
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_sort |
symplectic maps from cluster algebras |
| publisher |
Інститут математики НАН України |
| publishDate |
2011 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/147396 |
| 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. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
| work_keys_str_mv |
AT fordyap symplecticmapsfromclusteralgebras AT honea symplecticmapsfromclusteralgebras |
| first_indexed |
2025-07-11T02:00:48Z |
| last_indexed |
2025-07-11T02:00:48Z |
| _version_ |
1837314080124174336 |