For a symmetrizable Kac–Moody Lie algebra \({\mathfrak {g}}\), we construct a family of weighted quivers \(Q_m({\mathfrak {g}})\) (\(m \ge 2\)) whose cluster modular group \(\Gamma _{Q_m({\mathfrak {g}})}\) contains the Weyl group \(W({\mathfrak {g}})\) as a subgroup. We compute explicit formulae for the corresponding cluster \({{\mathcal {A}} }\)- and \({{\mathcal {X}} }\)-transformations. As a result, we obtain green sequences and the cluster Donaldson–Thomas transformation for \(Q_m({\mathfrak {g}})\) in a systematic way when \({\mathfrak {g}}\) is of finite type. Moreover if \({\mathfrak {g}}\) is of classical finite type with the Coxeter number h, the quiver \(Q_{kh}({\mathfrak {g}})\) (\(k \ge 1\)) 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.

对于对称Kac–Moody Lie代数\（{\ mathfrak {g}} \\），我们构造了一系列加权颤动\（Q_m（{\ mathfrak {g}}）\）（\（m \ ge 2 \）），其集群模块化组\（\ Gamma _ {Q_m（{\ mathfrak {g}}）} \）包含Weyl组\（W（{\ mathfrak {g}}）\）作为子组。我们为相应的簇\（{{\ mathcal {A}}} \） -和\（{{\ mathcal {X}}} \） -转换计算显式公式。结果，当\（{\ mathfrak {g}} \）为有限类型时，我们以系统的方式获得了绿色序列和\（Q_m（{\ mathfrak {g}}）\）的聚类唐纳森-托马斯变换。而且如果\（{\ mathfrak {g}} \）是经典有限类型，其Coxeter数为h，颤动\（Q_ {kh}（{\ mathfrak {g}}）\）（\（k \ ge 1 \））是等同于颤动的颤动，颤动是编码一次穿孔的磁盘的较高Teichmüller空间的簇结构的颤动，在边界上有2 k个标记点，直至冻结的顶点。这种对应关系诱导了Weyl基团的直接产物对一般标记表面的较高Teichmüller空间的作用。我们最终证明，从几何学角度来看，该动作与在Goncharov和Shen中构建的动作相吻合（Adv Math 327：225–348，2018年）。