Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 092, 14 pages      arXiv:2003.14104      https://doi.org/10.3842/SIGMA.2020.092
Contribution to the Special Issue on Cluster Algebras

On the Generalized Cluster Algebras of Geometric Type

Liqian Bai a, Xueqing Chen b, Ming Ding c and Fan Xu d
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

Received April 01, 2020, in final form September 14, 2020; Published online September 28, 2020

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.

Key words: cluster algebra; generalized cluster algebra; lower bound; upper bound; standard monomial.

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