TY - JOUR

T1 - Quantum Q systems

T2 - from cluster algebras to quantum current algebras

AU - Di Francesco, Philippe

AU - Kedem, Rinat

N1 - Funding Information:
We thank O. Babelon, F. Bergeron, J.-E. Bourgine, I. Cherednik, A. Negut, V. Pasquier, and O. Schiffmann for discussions at various stages of this work. R.K.’s research is supported by NSF Grant DMS-1404988. P.D.F. is supported by the NSF Grant DMS-1301636 and the Morris and Gertrude Fine endowment. R.K. would like to thank the Institut de Physique Théorique (IPhT) of Saclay, France, for hospitality during various stages of this work. The authors also acknowledge hospitality and support from Galileo Galilei Institute, Florence, Italy, as part of the scientific program on “Statistical Mechanics, Integrability and Combinatorics”, from the Centre de Recherche Mathématique de l’Université de Montreal during the thematic semester: “AdS/CFT, Holography, Integrability”, as well as of the Kavli Institute for Theoretical Physics, Santa Barbara, California, during the program “New approaches to non-equilibrium and random systems”, supported by the NSF Grant PHY11-25915.
Publisher Copyright:
© 2016, Springer Science+Business Media Dordrecht.

PY - 2017/2/1

Y1 - 2017/2/1

N2 - 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.

AB - 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.

KW - Discrete integrable systems

KW - Drinfeld algebra

KW - Q-systems

KW - Quantum determinants

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U2 - 10.1007/s11005-016-0902-2

DO - 10.1007/s11005-016-0902-2

M3 - Article

AN - SCOPUS:84995395293

SN - 0377-9017

VL - 107

SP - 301

EP - 341

JO - Letters in Mathematical Physics

JF - Letters in Mathematical Physics

IS - 2

ER -