Quantum Q systems: from cluster algebras to quantum current algebras
Quantum Q systems: from cluster algebras to quantum current algebras
复制标题
量子 Q 系统：从簇代数到量子电流代数
DOI：
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发表时间：
2016
影响因子：
1.2
通讯作者：
R. Kedem
中科院分区：
文献类型：
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作者：
P. Di Francesco;R. Kedem
This paper gives a new algebraic interpretation for the algebra generated by the quantum cluster variables of the Ardocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$A_r$$end{document} 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)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$U_{sqrt{q}}({mathfrak {n}}[u,u^{-1}])subset U_{sqrt{q}}(widehat{{mathfrak {sl}}}_2)$$end{document}, 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.
