Advertisement

Algebras and Representation Theory

, Volume 21, Issue 6, pp 1203–1217 | Cite as

A Quantum Analog of Generalized Cluster Algebras

  • Liqian Bai
  • Xueqing Chen
  • Ming Ding
  • Fan Xu
Article
  • 65 Downloads

Abstract

We define a quantum analog of a class of generalized cluster algebras which can be viewed as a generalization of quantum cluster algebras defined in Berenstein and Zelevinsky (Adv. Math. 195(2), 405–455 2005). In the case of rank two, we extend some structural results from the classical theory of generalized cluster algebras obtained in Chekhov and Shapiro (Int. Math. Res. Notices 10, 2746–2772 2014) and Rupel (2013) to the quantum case.

Keywords

Generalized cluster algebra Generalized quantum cluster algebra Laurent phenomenon Standard monomial 

Mathematics Subject Classification (2010)

Primary 16G20 17B67 Secondary 17B35 18E30 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Berenstein, A., Fomin, S., Zelevinsky, A.: Cluster algebras III: upper bounds and double Bruhat cells. Duke Math. J. 126(1), 1–52 (2005)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Berenstein, A., Zelevinsky, A.: Quantum cluster algebras. Adv. Math. 195 (2), 405–455 (2005)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Chekhov, L., Shapiro, M.: Teichmüller spaces of Riemann surfaces with orbifold points of arbitrary order and cluster variables. Int. Math. Res. Notices 10, 2746–2772 (2014)CrossRefGoogle Scholar
  4. 4.
    Fomin, S., Zelevinsky, A.: Cluster algebras. I. Foundations, J. Amer. Math. Soc. 15(2), 497–529 (2002)CrossRefGoogle Scholar
  5. 5.
    Fomin, S., Zelevinsky, A.: Cluster algebras. II. Finite type classification. Invent. Math. 154(1), 63–121 (2003)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Fock, V.V., Goncharov, A.B.: Cluster ensembles, quantization and the dilogarithm. Annales Sci. Éc. Norm. Supér. (4) 42(6), 865–930 (2009)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Fock, V.V., Goncharov, A.B.: The quantum dilogarithm and representations of quantum cluster varieties. Invent. Math. 175(2), 223–286 (2009)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Gekhtman, M., Shapiro, M., Vainshtein, A.: Generalized cluster structure on the Drinfeld double of G L n. C. R. Math. Acad. Sci. Paris 354(4), 345–349 (2016)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Nakanishi, T.: Quantum generalized cluster algebras and quantum dilogarithm functions of higher degrees. Theor. Math Phys. 185(3), 1759–1768 (2015)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Nakanishi, T.: Structure of seeds in generalized cluster algebras. Pacific J. Math. 277(1), 201–217 (2015)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Rupel, D.: Greedy bases in rank 2 generalized cluster algebras. arXiv:1309.2567 [math.RA] (2013)
  12. 12.
    Usnich, A.: Non-commutative Laurent phenomenon for two variables. arXiv:1006.1211 [math.AG] (2010)

Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Department of Applied MathematicsNorthwestern Polytechnical UniversityXi’anPeople’s Republic of China
  2. 2.Department of MathematicsUniversity of Wisconsin-WhitewaterWhitewaterUSA
  3. 3.School of Mathematical Sciences and LPMCNankai UniversityTianjinPeople’s Republic of China
  4. 4.Department of Mathematical SciencesTsinghua UniversityBeijingPeople’s Republic of China

Personalised recommendations