Quantum Q systems: from cluster algebras to quantum current algebras

Research output: Contribution to journalArticle

Abstract

This paper gives a new algebraic interpretation for the algebra generated by the quantum cluster variables of the A r 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
Volume107
Issue number2
DOIs
StatePublished - Feb 1 2017

Fingerprint

Current Algebra
Cluster Algebra
current algebra
Quantum Algebra
algebra
Quantum Affine Algebra
Algebra
quotients
Conserved Quantity
determinants
set theory
Determinant
Quotient
polynomials
Polarization
generators
Generator
Polynomial
Subset
polarization

Keywords

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

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

Quantum Q systems : from cluster algebras to quantum current algebras. / Di Francesco, Philippe; Kedem, Rinat.

In: Letters in Mathematical Physics, Vol. 107, No. 2, 01.02.2017, p. 301-341.

Research output: Contribution to journalArticle

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