Note on Character Varieties and Cluster Algebras
We use Bonahon-Wong's trace map to study character varieties of the once-punctured torus and of the 4-punctured sphere. We clarify a relationship with the cluster algebra associated with ideal triangulations of surfaces, and we show that the Goldman Poisson algebra of loops on surfaces is recov...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2019 |
| Автор: | |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут математики НАН України
2019
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/210072 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Note on Character Varieties and Cluster Algebras / K. Hikami // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 50 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-210072 |
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Hikami, K. 2025-12-02T09:39:01Z 2019 Note on Character Varieties and Cluster Algebras / K. Hikami // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 50 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 13F60; 30F60; 33E17; 57Q15 arXiv: 1711.03379 https://nasplib.isofts.kiev.ua/handle/123456789/210072 https://doi.org/10.3842/SIGMA.2019.003 We use Bonahon-Wong's trace map to study character varieties of the once-punctured torus and of the 4-punctured sphere. We clarify a relationship with the cluster algebra associated with ideal triangulations of surfaces, and we show that the Goldman Poisson algebra of loops on surfaces is recovered from the Poisson structure of the cluster algebra. It is also shown that cluster mutations give the automorphism of the character varieties. Motivated by a work of Chekhov-Mazzocco-Rubtsov, we revisit confluences of punctures on the sphere from a cluster algebraic viewpoint, and we obtain associated affine cubic surfaces constructed by van der Put-Saito based on the Riemann-Hilbert correspondence. Further studies are quantizations of character varieties by use of quantum cluster algebra. The author would like to thank Thang Le for the communications during the Workshop "Low-Dimensional Topology and Number Theory" at Mathematisches Forschungsinstitut Oberwolfach in August 2017. He thanks the organizers for the invitation. Thanks are also due to the speakers of "Geometry of Moduli Spaces and Integrable Systems" at Gakushuin University, Tokyo, in September 2017. This work is supported in part by JSPS KAKENHI Grant Numbers JP16H03927, JP17K05239, JP17K18781, and JP16H02143. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Note on Character Varieties and Cluster Algebras Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Note on Character Varieties and Cluster Algebras |
| spellingShingle |
Note on Character Varieties and Cluster Algebras Hikami, K. |
| title_short |
Note on Character Varieties and Cluster Algebras |
| title_full |
Note on Character Varieties and Cluster Algebras |
| title_fullStr |
Note on Character Varieties and Cluster Algebras |
| title_full_unstemmed |
Note on Character Varieties and Cluster Algebras |
| title_sort |
note on character varieties and cluster algebras |
| author |
Hikami, K. |
| author_facet |
Hikami, K. |
| publishDate |
2019 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
We use Bonahon-Wong's trace map to study character varieties of the once-punctured torus and of the 4-punctured sphere. We clarify a relationship with the cluster algebra associated with ideal triangulations of surfaces, and we show that the Goldman Poisson algebra of loops on surfaces is recovered from the Poisson structure of the cluster algebra. It is also shown that cluster mutations give the automorphism of the character varieties. Motivated by a work of Chekhov-Mazzocco-Rubtsov, we revisit confluences of punctures on the sphere from a cluster algebraic viewpoint, and we obtain associated affine cubic surfaces constructed by van der Put-Saito based on the Riemann-Hilbert correspondence. Further studies are quantizations of character varieties by use of quantum cluster algebra.
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| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/210072 |
| citation_txt |
Note on Character Varieties and Cluster Algebras / K. Hikami // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 50 назв. — англ. |
| work_keys_str_mv |
AT hikamik noteoncharactervarietiesandclusteralgebras |
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2025-12-07T21:24:16Z |
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2025-12-07T21:24:16Z |
| _version_ |
1850886226274418688 |