JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2022, Vol. 57 ›› Issue (10): 34-38.doi: 10.6040/j.issn.1671-9352.0.2021.697

Previous Articles    

Linear independence of cluster monomials of cluster algebras of type G2 via categorification

GAO Xin-zhao, XIE Yun-li*   

  1. School of Mathematics, Southwest Jiaotong University, Chengdu 611756, Sichuan, China
  • Published:2022-10-06

PDF (PC)

164

Abstract: Using the method of categorization, Fomin-Zelevinskys conjecture about the linear independence of cluster monomials in all cluster algebras holds for cluster algebras of type G2 is proved, by establishing the Caldero-Chapoton formula from the indecomposable rigid objects in the category of finitely generated modules of the cluster tilted algebra to the cluster variables of the corresponding cluster algebra.

Key words: cluster algebra, cluster monomial, cluster category, cluster tilted algebra, Caldero-Chapoton formula

CLC Number: 

  • O154.1
[1] FOMIN S, ZELEVINSKY A. Cluster algebras I: foundations[J]. Journal of the American Mathematical Society, 2002, 15(2):497-529.
[2] FOMIN S, ZELEVINSKY A. Cluster algebras IV: coefficients[J]. Compositio Mathematica, 2007, 143(1):112-164.
[3] GROSS M, HACKING P, KEEL S, et al. Canonical bases for cluster algebras[J]. Journal of the American Mathematical Society, 2018, 31(2):497-608.
[4] CALDERO P, KELLER B. From triangulated categories to cluster algebras[J]. Inventiones Mathematicae, 2008, 172:169-211.
[5] DEMONET L. Categorification of skew-symmetrizable cluster algebras[J]. Algebras and Representation Theory, 2010, 14(6):1087-1162.
[6] GEIB C, LECLERC B, SCHRÖER J. Generic bases for cluster algebras and the Chamber Ansatz[J]. Journal of the American Mathematical Society, 2012, 25(1):121-176.
[7] IRELLI C G, DANIEL L-F A. Quivers with potentials associated to triangulated surfaces, part III: tagged triangulations and cluster monomials[J]. Compositio Mathematica, 2012, 148(6):1833-1866.
[8] FOMIN S, SHAPIRO M, THURSTON D. Cluster algebras and triangulated surfaces, part I: cluster complexes[J]. Acta Mathematica, 2008, 201(1):83-146.
[9] MUSIKER G, SCHIFFLER R, WILLIAMS L. Bases for cluster algebras from surfaces[J]. Compositio Mathematica, 2012, 149(2):217-263.
[10] DERKSEN H, WEYMAN J, ZELEVINSKY A. Quivers with potentials and their representations II: applications to cluster algebras[J]. Journal of the American Mathematical Society, 2010, 23(3):749-790.
[11] PLAMONDON P-G. Cluster algebras via cluster categories with infinite-dimensional morphism spaces[J]. Compositio Mathematica, 2012, 147(6):1921-1954.
[12] IRELLI G C, KELLER B, DANIEL L-F A, et al. Linear independence of cluster monomials for skew-symmetric cluster algebras[EB/OL].(2012-03-06)[2020-04-01]. arXiv:1203.1307[math.RT].
[13] HAPPEL D. Auslander-Reiten triangles in derived categories of finite-dimensional algebras[J]. Proceedings of the American Mathematical Society, 1991, 12(3):641-648.
[14] LADKANI S. 2-Calabi-Yau-tilted algebras that are not Jacobian[EB/OL].(2014-03-26)[2020-04-01]. arXiv:1403.6814 [math.RT].
[15] GEIB C, LECLERC B, SCHRÖER J. Quivers with relations for symmetrizable Cartan matrices I: foundations[J]. Inventiones Mathematicae, 2016, 209(1):61-158.
[16] IYAMA O, YOSHINO Y. Mutation in triangulated categories and rigid Cohen-Macaulay modules[J]. Inventiones Mathematicae, 2008, 172(1):117-168.
[17] ADACHI T, IYAMA O, REITEN I. τ-tilting theory[J]. Compositio Mathematica, 2014, 150(3):415-452.
[18] CHANG Wen, ZHANG Jie, ZHU Bin. On support -tilting modules over endomorphism algebras of rigid objects[J]. Acta Mathematica Sinica, 2015, 31(9):1508-1516.
[19] LIU Pin, XIE Yunli. On the relation between maximal rigid objects and -tilting modules[J]. Colloquium Mathematicum, 2016, 142:169-178.
[20] PALU Y. Cluster characters for 2-Calabi-Yau triangulated categories[J]. Annales de lInstitut Fourier, 2008, 58(6):2221-2248.
[21] PLAMONDON P-G. Cluster characters for cluster categories with infinite-dimensional morphism spaces[J]. Advances in Mathematics, 2011, 227(1):1-39.
[22] CALDERO P, KELLER B. From triangulated categories to cluster algebras[J]. Inventiones Mathematicae, 2008, 172(1):169-211.
[23] PALU Y. Cluster algebras as Hall algebras of quiver representations[J]. Commentarii Mathematici Helvetic, 2006, 81(3):596-616.
[1] FENG Qing, HUANG Ju. Two constructions of monoidal categories [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2016, 51(4): 30-34.
[2] LIU Yang1, WANG Zheng-ping2, XU Qing-bing2. The Gabriel monad and the localization of R-module category [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2014, 49(2): 24-28.
[3] LIU Hong-jin, LIU Li-min. Complements of partial silting objects in triangulated categories [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(4): 67-71.
[4] HUANG Ju, FAN Xin-xiang. Limit categories of complete categories and η-extensions [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2020, 55(8): 43-47.
[5] YANG Ting, XIE Yun-li. Higher rigid subcategories and t-structures [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2020, 55(12): 69-75.
[6] ZHENG Min, CHEN Qing-hua. Ki-group of t-structure of triangulated category [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2020, 55(8): 48-53.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
No Suggested Reading articles found!
Copyright 2018 © Editorial office of JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)
Supported by Beijing Magtech Co.Ltd