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 DingEmail author
  • Fan Xu


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.


Generalized cluster algebra Generalized quantum cluster algebra Laurent phenomenon Standard monomial 

Mathematics Subject Classification (2010)

Primary 16G20 17B67 Secondary 17B35 18E30 


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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

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