Hostname: page-component-76fb5796d-dfsvx Total loading time: 0 Render date: 2024-04-26T01:03:03.142Z Has data issue: false hasContentIssue false
  • English
  • Français

QUANTISATION SPACES OF CLUSTER ALGEBRAS

Part of: Arithmetic rings and other special rings

Published online by Cambridge University Press:  04 September 2017

FLORIAN GELLERT  and
PHILIPP LAMPE
FLORIAN GELLERT
Affiliation:
Faculty of Mathematics, Bielefeld University, PO Box 100 131, 33501 Bielefeld, Germany e-mails: florian.gellert@math.uni-bielefeld.de, lampe@math.uni-bielefeld.de
PHILIPP LAMPE
Affiliation:
Faculty of Mathematics, Bielefeld University, PO Box 100 131, 33501 Bielefeld, Germany e-mails: florian.gellert@math.uni-bielefeld.de, lampe@math.uni-bielefeld.de
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The article concerns the existence and uniqueness of quantisations of cluster algebras. We prove that cluster algebras with an initial exchange matrix of full rank admit a quantisation in the sense of Berenstein-Zelevinsky and give an explicit generating set to construct all quantisations.

MSC classification

Primary: 13F60: Cluster algebras
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 60 , Issue 2 , May 2018 , pp. 273 - 284
Copyright
Copyright © Glasgow Mathematical Journal Trust 2017 

References

REFERENCES

1. Berenstein, A., Fomin, S. and Zelevinsky, A., Cluster algebras. III. Upper bounds and double Bruhat cells, Duke Math. J. 126 (1) (2005), 1–52.Google Scholar
2. Berenstein, A. and Zelevinsky, A., Quantum cluster algebras, Adv. Math. 195 (2) (2005), 405–455.CrossRefGoogle Scholar
3. Burban, I., Iyama, O., Keller, B. and Reiten, I., Cluster tilting for one-dimensional hypersurface singularities. Adv. Math. 217 (6) (2008), 2443–2484.CrossRefGoogle Scholar
4. Caldero, P. and Chapoton, F., Cluster algebras as Hall algebras of quiver representations, Comment. Math. Helv. 81 (3) (2006), 595–616.Google Scholar
5. Cayley, A., Sur les déterminants gauches, J. Reine Angew. Math. 38 (1849), 93–96.Google Scholar
6. Fock, V. V. and Goncharov, A. B., Cluster ensembles, quantization and the dilogarithm, Ann. Sci. École Norm. Sup., série 4, 42 (6) (2009), 865–930.CrossRefGoogle Scholar
7. Fomin, S., Shapiro, M. and Thurston, D., Cluster algebras and triangulated surfaces, I. Cluster complexes, Acta Math. 201 (1) (2008), 83–146.CrossRefGoogle Scholar
8. Fomin, S. and Thurston, D., Cluster algebras and triangulated surfaces. Part II: Lambda lengths. Preprint: arXiv 1210.5569 (2012). To appear in Memoirs of Amer. Math. Soc.Google Scholar
9. Fomin, S. and Zelevinsky, A., Cluster algebras. I. Foundations. J. Amer. Math. Soc. 15 (2) (2002), 497–529 (electronic).Google Scholar
10. Fomin, S. and Zelevinsky, A., Cluster algebras. II. Finite type classification, Invent. Math. 154 (1) (2003), 63–121.Google Scholar
11. Fomin, S. and Zelevinsky, A., Cluster algebras. IV. Coefficients, Compos. Math. 143 (1) (2007), 112–164.Google Scholar
12. Geiß, C., Leclerc, B. and Schröer, J., Cluster structures on quantum coordinate rings, Selecta Math. (N.S.) 19 (2) (2013), 337–397.Google Scholar
13. Gekhtman, M., Shapiro, M. and Vainshtein, A., Cluster algebras and Poisson geometry, Mosc. Math. J. 3 (3) (2003), 899–934.CrossRefGoogle Scholar
14. Gellert, F., Sage functions for the quantisation of cluster algebras. http://math.uni-bielefeld.de/~fgellert/quantisation.phpGoogle Scholar
15. Goodearl, K. R. and Yakimov, M. T., Quantum cluster algebra structures on quantum nilpotent algebras. Mem. Amer. Math. Soc. 247 (1169) (2017), vii+119 pp.Google Scholar
16. Grabowski, J. E., Graded cluster algebras, J. Algebr. Combin. 42 (4) (2015), 1111–1134.CrossRefGoogle Scholar
17. Grabowski, J. E. and Launois, S., Graded quantum cluster algebras and an application to quantum grassmannians, Proc. London Math. Soc. 109 (3) (2014), 697–732.CrossRefGoogle Scholar
18. Hernandez, D. and Leclerc, B., Cluster algebras and quantum affine algebras, Duke Math. J. 154 (2) (2010), 265–341.Google Scholar
19. Knuth, D. E., Overlapping Pfaffians, Electron. J. Combin. 3 (2), Research Paper 5 (1996).Google Scholar
20. Kimura, Y. and Qin, F., Graded quiver varieties, quantum cluster algebras and dual canonical basis, Adv. Math. 262 (2014), 261–312.CrossRefGoogle Scholar
21. Lampe, P., A quantum cluster algebra of Kronecker type and the dual canonical basis, Int. Math. Res. Not. IMRN 2011 (13) (2011), 2970–3005.Google Scholar
22. Lampe, P., Quantum cluster algebras of type A and the dual canonical basis, Proc. London Math. Soc. 108 (2014), 1–43.Google Scholar
23. Leclerc, B., Dual canonical bases, quantum shuffles and q-characters, Math. Z. 246 (4) (2004), 691–732.Google Scholar
24. Lusztig, G., Introduction to quantum groups, Progress in mathematics, vol. 110 (Birkhäuser Boston Inc., Boston, MA, 1993).Google Scholar
25. Rupel, D., The Feigin tetrahedron, SIGMA 11 (024) (2015), 30 pages.Google Scholar
26. Zelevinsky, A., Quantum cluster algebras, Lecture, Infinite Analysis, vol. 11 (Winter School, Osaka University, Japan, 2011).Google Scholar