Abstract
We initiate a study of the dependence of the choice of ground ring on the problem on whether a cluster algebra is equal to its upper cluster algebra. A condition for when there is equality of the cluster algebra and upper cluster algebra is given by using a variation of Muller’s theory of cluster localization. An explicit example exhibiting dependence on the ground ring is provided. We also present a maximal green sequence for this example.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Belavin A A, Drinfeld V G. Solutions of the classical Yang-Baxter equation for simple Lie algebras. Funct Anal Appl, 1982, 16: 159–180
Berenstein A, Fomin S, Zelevinsky A. Cluster algebras, III: Upper bounds and double Bruhat cells. Duke Math J, 2005, 126: 1–52
Brüstle T, Dupont G, Pérotin M. On maximal green sequences. Int Math Res Not IMRN, 2014, 2014: 4547–4586
Bucher E, Machacek J, Runberg E, et al. Building maximal green sequences via component preserving mutations. ArXiv:1902.02262
Canakci I, Lee K Y, Schiffler R. On cluster algebras from unpunctured surfaces with one marked point. Proc Amer Math Soc Ser B, 2015, 2: 35–49
Eisner I. Exotic cluster structures on SL 5. J Phys A, 2014, 47: 474002
Eisner I. Exotic cluster structures on SL n with Belavin-Drinfeld data of minimal size, I: The structure. Israel J Math, 2017, 218: 391–443
Eisner I. Exotic cluster structures on SL n with Belavin-Drinfeld data of minimal size, II: Correspondence between cluster structures and Belavin-Drinfeld triples. Israel J Math, 2017, 218: 445–487
Fomin S, Williams L, Zelevinsky A. Introduction to Cluster Algebras. Chapters 1–3. ArXiv:1608.05735
Fomi S, Zelevinsky A. Cluster algebras, I: Foundations. J Amer Math Soc, 2002, 15: 497–529
Gekhtman M, Shapiro M, Vainshtein A. Cluster Algebras and Poisson Geometry. Mathematical Surveys and Monographs, vol. 167167. Providence: Amer Math Soc, 2010
Gekhtman M, Shapiro M, Vainshtein A. Cluster structures on simple complex Lie groups and Belavin-Drinfeld classi-cation. Mosc Math J, 2012, 12: 293–312
Gekhtman M, Shapiro M, Vainshtein A. Cremmer-Gervais cluster structure on SL n. Proc Natl Acad Sci USA, 2014, 111: 9688–9695
Gekhtman M, Shapiro M, Vainshtein A. Exotic Cluster Structures on SL n: The Cremmer-Gervais Case. Memoirs of the American Mathematical Society, vol. 246. Providence: Amer Math Soc, 2017
Machacek J, Ovenhouse N. Log-canonical coordinates for Poisson brackets and rational changes of coordinates. J Geom Phys, 2017, 121: 288–296
Muller G. Locally acyclic cluster algebras. Adv Math, 2013, 233: 207–247
Muller G. A = U for locally acyclic cluster algebras. SIGMA Symmetry Integrability Geom Methods Appl, 2014, 10: Article 094
Scott J S. Grassmannians and cluster algebras. Proc London Math Soc (3), 2006, 92: 345–380
The Stacks Project Authors. Stacks Project. https://doi.org/stacks.math.columbia.edu, 2018
Acknowledgements
This work was supported by National Science Foundation of USA (Grant No. DMS-1702115).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bucher, E., Machacek, J. & Shapiro, M. Upper cluster algebras and choice of ground ring. Sci. China Math. 62, 1257–1266 (2019). https://doi.org/10.1007/s11425-018-9486-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-018-9486-6