Quantum Q systems: from cluster algebras to quantum current algebras

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This paper gives a new algebraic interpretation for the algebra generated by the quantum cluster variables of the Ar quantum Q-system (Di Francesco and Kedem in Int Math Res Not IMRN 10:2593–2642, 2014). We show that the algebra can be described as a quotient of the localization of the quantum algebra Uq(n[u,u-1])⊂Uq(sl^2), in the Drinfeld presentation. The generating current is made up of a subset of the cluster variables which satisfy the Q-system, which we call fundamental. The other cluster variables are given by a quantum determinant-type formula, and are polynomials in the fundamental generators. The conserved quantities of the discrete evolution (Di Francesco and Kedem in Adv Math 228(1):97–152, 2011) described by quantum Q-system generate the Cartan currents at level 0, in a non-standard polarization. The rest of the quantum affine algebra is also described in terms of cluster variables.

Original languageEnglish (US)
Pages (from-to)301-341
Number of pages41
JournalLetters in Mathematical Physics
Issue number2
StatePublished - Feb 1 2017


  • Discrete integrable systems
  • Drinfeld algebra
  • Q-systems
  • Quantum determinants

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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