Cluster realization of Weyl groups and q-characters of quantum affine algebras

Abstract

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