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Quantum Q systems: from cluster algebras to quantum current algebras

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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 \(U_{\sqrt{q}}({\mathfrak {n}}[u,u^{-1}])\subset U_{\sqrt{q}}(\widehat{{\mathfrak {sl}}}_2)\), 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.

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Notes

  1. Throughout this paper, we shall refer to equation (2.1) as the “M-system” relation, in analogy with the Q-system relation (see below for a precise connection).

  2. In cluster algebras, all inverses of the generators are adjoined by definition, hence the localization.

  3. To be precise, to compare with the usual definition of [9], the generators are renormalized cluster variables.

  4. This is a delta function in the sense that \(\oint \frac{du}{2i\pi u} f(u)\delta (u/z)=f(z)\), where the contour integral picks out the constant term of the current \(f(u)\delta (u/z)\).

  5. After acceptance of the paper we became aware of an earlier work by Miki [20] who introduces difference operators similar to the \({\alpha }=1\) version of \({\mathcal M}_{{\alpha },n}^{q,t}\).

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Acknowledgements

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.

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  1. Department of Mathematics, University of Illinois, MC-382, Urbana, IL, 61821, USA

    Philippe Di Francesco & Rinat Kedem

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  1. Philippe Di Francesco
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  2. Rinat Kedem
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Correspondence to Rinat Kedem.

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Di Francesco, P., Kedem, R. Quantum Q systems: from cluster algebras to quantum current algebras. Lett Math Phys 107, 301–341 (2017). https://doi.org/10.1007/s11005-016-0902-2

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