For a symmetrizable Kac–Moody Lie algebra g, we construct a family of weighted quivers Qm(g) (m≥ 2) whose cluster modular group ΓQm(g) contains the Weyl group W(g) as a subgroup. We compute explicit formulae for the corresponding cluster A- and X-transformations. As a result, we obtain green sequences and the cluster Donaldson–Thomas transformation for Qm(g) in a systematic way when g is of finite type. Moreover if g is of classical finite type with the Coxeter number h, the quiver Qkh(g) (k≥ 1) is mutation-equivalent to a quiver encoding the cluster structure of the higher Teichmüller space of a once-punctured disk with 2k marked points on the boundary, up to frozen vertices. This correspondence induces the action of direct products of Weyl groups on the higher Teichmüller space of a general marked surface. We finally prove that this action coincides with the one constructed in Goncharov and Shen (Adv Math 327:225–348, 2018) from the geometrical viewpoint.
ASJC Scopus subject areas
- Physics and Astronomy(all)