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Symmetry, Integrability and Geometry: Methods and Applications, 2020, Volume 16, 092, 14 pp.
DOI: https://doi.org/10.3842/SIGMA.2020.092 (Mi sigma1629) 		 
This article is cited in 3 scientific papers (total in 3 papers)

On the Generalized Cluster Algebras of Geometric Type

Liqian Baia, Xueqing Chenb, Ming Dingc, Fan Xud

a School of Mathematics and Statistics, Northwestern Polytechnical University, Xi'an, Shaanxi 710072, P. R. China
b Department of Mathematics, University of Wisconsin-Whitewater, 800 West Main Street, Whitewater, WI 53190, USA
c School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, P. R. China
d Department of Mathematical Sciences, Tsinghua University, Beijing 100084, P. R. China
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DOI: https://doi.org/10.3842/SIGMA.2020.092
Abstract: 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.
Keywords: cluster algebra, generalized cluster algebra, lower bound, upper bound, standard monomial.
Funding agency Grant number
National Natural Science Foundation of China 11801445
11771217
11471177
Shaanxi Province Natural Science Foundation of China 2020JQ-116
Fundamental Research Funds for the Central Universities of China 310201911cx027
Liqian Bai was supported by 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 NSF of China (No. 11771217) and Fan Xu was supported by NSF of China (No. 11471177).
Received: April 1, 2020; in final form September 14, 2020; Published online September 28, 2020
Bibliographic databases:
Document Type: Article
MSC: 13F60, 05E16
Language: English
Citation: Liqian Bai, Xueqing Chen, Ming Ding, Fan Xu, “On the Generalized Cluster Algebras of Geometric Type”, SIGMA, 16 (2020), 092, 14 pp.
Citation in format AMSBIB
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\paper On the Generalized Cluster Algebras of Geometric Type
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