Q-systems as cluster algebras II: Cartan Matrix of Finite Type and the Polynomial Property

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We define the cluster algebra associated with the Q-system for the Kirillov-Reshetikhin characters of the quantum affine algebra Uq(g{fraktur}̂) for any simple Lie algebra, generalizing the simply-laced case treated in (Kedem in Q-systems as cluster algebras. arXiv:0712.2695 [math.RT], 2007). We describe some special properties of this cluster algebra, and explain its relation to the deformed Q-systems which appeared on our proof of the combinatorial-KR conjecture. We prove that the polynomiality of the cluster variables in terms of the "initial cluster seeds", including solutions of the Q-system, is a consequence of the Laurent phenomenon and the boundary conditions. We also define the cluster algebra associated with T-systems, or general systems which take the form of T-systems in the bipartite case. Such systems describe the recursion relations satisfied by the q-characters of Kirillov-Reshetikhin modules and also appear in the categorification picture in terms of preprojective algebras of Geiss, Leclerc and Schröer. We give a formulation of both Q-systems and generalized T-systems as cluster algebras with coefficients. This provides a proof of the polynomiality of solutions of all such "generalized T-systems" with appropriate boundary conditions.

Original languageEnglish (US)
Pages (from-to)183-216
Number of pages34
JournalLetters in Mathematical Physics
Issue number3
StatePublished - Oct 2009


  • Cluster algebras
  • Kirillov-Reshetikhin modules
  • Q-systems

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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