Quantum Q systems: from cluster algebras to quantum current algebras

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• 作者：Philippe Di Francesco ; Rinat Kedem
• 关键词：Q ; systems ; Drinfeld algebra ; Discrete integrable systems ; Quantum determinants
• 刊名：Letters in Mathematical Physics
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：107
• 期：2
• 页码：301-341
• 全文大小：
• 刊物类别：Physics and Astronomy
• 刊物主题：Theoretical, Mathematical and Computational Physics; Complex Systems; Geometry; Group Theory and Generalizations;
• 出版者：Springer Netherlands
• ISSN：1573-0530
• 卷排序：107

文摘

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.