We consider an infinite quiver $$Q({\mathfrak {g}})$$ Q ( g ) and a family of periodic quivers $$Q_m({\mathfrak {g}})$$ Q m ( g ) for a finite-dimensional simple Lie algebra $${\mathfrak {g}}$$ g and $$m \in {\mathbb Z}_{>1}$$ m ∈ Z > 1 . The quiver $$Q({\mathfrak {g}})$$ Q ( g ) is essentially same as what introduced in Hernandez and Leclerc (J Eur Math Soc 18:1113–1159, 2016) for the quantum affine algebra $${\hat{{\mathfrak {g}}}}$$ g ^ . We construct the Weyl group $$W({\mathfrak {g}})$$ W ( g ) as a subgroup of the cluster modular group for $$Q_m({\mathfrak {g}})$$ Q m ( g ) , in a similar way as (Inoue et al. in Cluster realizations of Weyl groups and higher Teichmüller theory. arXiv:1902.02716 ), and study its applications to the q -characters of quantum non-twisted affine algebras $$U_q({\hat{{\mathfrak {g}}}})$$ U q ( g ^ ) (Frenkel and Reshetikhin in Contemp Math 248:163–205, 1999), and to the lattice $${\mathfrak {g}}$$ g -Toda field theory (Inoue and Hikami in Nucl Phys B 581:761–775, 2000). In particular, when q is a root of unity, we prove that the q -character is invariant under the Weyl group action. We also show that the A -variables for $$Q({\mathfrak {g}})$$ Q ( g ) correspond to the $$\tau $$ τ -function for the lattice $${\mathfrak {g}}$$ g -Toda field equation.

中文翻译：

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Weyl群的簇实现和量子仿射代数的q特征

我们考虑一个无限的箭袋 $$Q({\mathfrak {g}})$$ Q ( g ) 和一个周期性箭袋 $$Q_m({\mathfrak {g}})$$ Q m ( g )有限维简单李代数 $${\mathfrak {g}}$$ g 和 $$m \in {\mathbb Z}_{>1}$$ m ∈ Z > 1 。颤动 $$Q({\mathfrak {g}})$$ Q ( g ) 本质上与 Hernandez 和 Leclerc (J Eur Math Soc 18:1113–1159, 2016) 中介绍的量子仿射代数 $$ 相同{\hat{{\mathfrak {g}}}}$$ g ^ 。我们构造外尔群 $$W({\mathfrak {g}})$$ W ( g ) 作为 $$Q_m({\mathfrak {g}})$$ Q m ( g ) ，与（Inoue et al. in Cluster implementations of Weyl group and Higher Teichmüller 理论。arXiv:1902.02716）类似，并研究其在量子非扭曲仿射代数的 q 字符中的应用 $$U_q({\hat{{\mathfrak {g}}}})$$ U q ( g ^ ) (Frenkel and Reshetikhin in Contemp Math 248 :163–205, 1999)，以及格 $${\mathfrak {g}}$$ g -Toda 场论（Inoue 和 Hikami in Nucl Phys B 581:761–775, 2000）。特别地，当 q 是单位根时，我们证明 q 字符在 Weyl 群作用下是不变的。我们还表明 $$Q({\mathfrak {g}})$$ Q ( g ) 的 A 变量对应于晶格 $${\mathfrak {g} 的 $$\tau $$ τ 函数}$$ g -Toda 场方程。我们证明了 q 字符在 Weyl 群作用下是不变的。我们还表明 $$Q({\mathfrak {g}})$$ Q ( g ) 的 A 变量对应于晶格 $${\mathfrak {g} 的 $$\tau $$ τ 函数}$$ g -Toda 场方程。我们证明了 q 字符在 Weyl 群作用下是不变的。我们还表明 $$Q({\mathfrak {g}})$$ Q ( g ) 的 A 变量对应于晶格 $${\mathfrak {g} 的 $$\tau $$ τ 函数}$$ g -Toda 场方程。