On the Generalized Cluster Algebras of Geometric Type

Symmetry Integrability and Geometry-Methods and Applications

article

Liqian Bai¹  Xueqing Chen²  Ming Ding³  Fan Xu⁴ 

[1] School of Mathematics and Statistics, Northwestern Polytechnical University;Department of Mathematics, University of Wisconsin-Whitewater;School of Mathematics and Information Science, Guangzhou University;Department of Mathematical Sciences, Tsinghua University

关键词: cluster algebra;    generalized cluster algebra;    lower bound;    upper bound;    standard monomial 2020 Mathematics Subject Classification 13F60;    05E16;   

DOI  :  10.3842/SIGMA.2020.092

来源: National Academy of Science of Ukraine

【 摘 要 】

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.

【 授权许可】

Unknown