We consider an infinite quiver Q(g)\documentclass[12pt]{minimal}
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\begin{document}$$Q({\mathfrak {g}})$$\end{document} and a family of periodic quivers Qm(g)\documentclass[12pt]{minimal}
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\begin{document}$$Q_m({\mathfrak {g}})$$\end{document} for a finite-dimensional simple Lie algebra g\documentclass[12pt]{minimal}
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\begin{document}$${\mathfrak {g}}$$\end{document} and m∈Z>1\documentclass[12pt]{minimal}
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\begin{document}$$m \in {\mathbb Z}_{>1}$$\end{document}. The quiver Q(g)\documentclass[12pt]{minimal}
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\begin{document}$$Q({\mathfrak {g}})$$\end{document} is essentially same as what introduced in Hernandez and Leclerc (J Eur Math Soc 18:1113–1159, 2016) for the quantum affine algebra g^\documentclass[12pt]{minimal}
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\begin{document}$${\hat{{\mathfrak {g}}}}$$\end{document}. We construct the Weyl group W(g)\documentclass[12pt]{minimal}
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\begin{document}$$W({\mathfrak {g}})$$\end{document} as a subgroup of the cluster modular group for Qm(g)\documentclass[12pt]{minimal}
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\begin{document}$$Q_m({\mathfrak {g}})$$\end{document}, 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 Uq(g^)\documentclass[12pt]{minimal}
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\begin{document}$$U_q({\hat{{\mathfrak {g}}}})$$\end{document} (Frenkel and Reshetikhin in Contemp Math 248:163–205, 1999), and to the lattice g\documentclass[12pt]{minimal}
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\begin{document}$${\mathfrak {g}}$$\end{document}-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(g)\documentclass[12pt]{minimal}
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\begin{document}$$Q({\mathfrak {g}})$$\end{document} correspond to the τ\documentclass[12pt]{minimal}
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\begin{document}$$\tau $$\end{document}-function for the lattice g\documentclass[12pt]{minimal}
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\begin{document}$${\mathfrak {g}}$$\end{document}-Toda field equation.
