On the Generalized Cluster Algebras of Geometric Type
We develop and prove the analogs of some results shown in [Berenstein A., Fomin S., Zelevinsky A., Duke Math. J. 126 (2005), 1-52] concerning lower and upper bounds of cluster algebras to the generalized cluster algebras of geometric type. We show that lower bounds coincide with upper bounds under t...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2020 |
| Main Authors: | , , , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2020
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/211028 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | On the Generalized Cluster Algebras of Geometric Type. Liqian Bai, Xueqing Chen, Ming Ding and Fan Xu. SIGMA 16 (2020), 092, 14 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862724009861316608 |
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| author | Bai, Liqian Chen, Xueqing Ding, Ming Xu, Fan |
| author_facet | Bai, Liqian Chen, Xueqing Ding, Ming Xu, Fan |
| citation_txt | On the Generalized Cluster Algebras of Geometric Type. Liqian Bai, Xueqing Chen, Ming Ding and Fan Xu. SIGMA 16 (2020), 092, 14 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We develop and prove the analogs of some results shown in [Berenstein A., Fomin S., Zelevinsky A., Duke Math. J. 126 (2005), 1-52] concerning lower and upper bounds of cluster algebras to the generalized cluster algebras of geometric type. We show that lower bounds coincide with upper bounds under the conditions of acyclicity and coprimality. Consequently, we obtain the standard monomial bases of these generalized cluster algebras. Moreover, in the appendix, we prove that an acyclic generalized cluster algebra is equal to the corresponding generalized upper cluster algebra without the assumption of the existence of coprimality.
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| first_indexed | 2026-03-21T06:43:18Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-211028 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-21T06:43:18Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Bai, Liqian Chen, Xueqing Ding, Ming Xu, Fan 2025-12-22T09:32:30Z 2020 On the Generalized Cluster Algebras of Geometric Type. Liqian Bai, Xueqing Chen, Ming Ding and Fan Xu. SIGMA 16 (2020), 092, 14 pages 1815-0659 2020 Mathematics Subject Classification: 13F60; 05E16 arXiv:2003.14104 https://nasplib.isofts.kiev.ua/handle/123456789/211028 https://doi.org/10.3842/SIGMA.2020.092 We develop and prove the analogs of some results shown in [Berenstein A., Fomin S., Zelevinsky A., Duke Math. J. 126 (2005), 1-52] concerning lower and upper bounds of cluster algebras to the generalized cluster algebras of geometric type. We show that lower bounds coincide with upper bounds under the conditions of acyclicity and coprimality. Consequently, we obtain the standard monomial bases of these generalized cluster algebras. Moreover, in the appendix, we prove that an acyclic generalized cluster algebra is equal to the corresponding generalized upper cluster algebra without the assumption of the existence of coprimality. The authors are greatly indebted to referees for their valuable comments and recommendations, which definitely help to improve the readability and quality of the paper. Liqian Bai was supported by the NSF of China (No. 11801445), the Natural Science Foundation of Shaanxi Province (No. 2020JQ-116), and the Fundamental Research Funds for the Central Universities (No. 310201911cx027), Ming Ding was supported by the NSF of China (No. 11771217), and Fan Xu was supported by the NSF of China (No. 11471177). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications On the Generalized Cluster Algebras of Geometric Type Article published earlier |
| spellingShingle | On the Generalized Cluster Algebras of Geometric Type Bai, Liqian Chen, Xueqing Ding, Ming Xu, Fan |
| title | On the Generalized Cluster Algebras of Geometric Type |
| title_full | On the Generalized Cluster Algebras of Geometric Type |
| title_fullStr | On the Generalized Cluster Algebras of Geometric Type |
| title_full_unstemmed | On the Generalized Cluster Algebras of Geometric Type |
| title_short | On the Generalized Cluster Algebras of Geometric Type |
| title_sort | on the generalized cluster algebras of geometric type |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211028 |
| work_keys_str_mv | AT bailiqian onthegeneralizedclusteralgebrasofgeometrictype AT chenxueqing onthegeneralizedclusteralgebrasofgeometrictype AT dingming onthegeneralizedclusteralgebrasofgeometrictype AT xufan onthegeneralizedclusteralgebrasofgeometrictype |