Inspired by the quantum McKay correspondence, we consider the classical ADE Lie theory as a quantum theory over \(\mathfrak {sl}_2\). We introduce anti-symmetric characters for representations of quantum groups and investigate the Fourier duality to study the spectral theory. In the ADE Lie theory, there is a correspondence between the eigenvalues of the Coxeter element and the eigenvalues of the adjacency matrix. We formalize related notions and prove such a correspondence for representations of Verlinde algebras of quantum groups: this includes generalized Dynkin diagrams over any simple Lie algebra \(\mathfrak {g}\) at any level \(k\). This answers a recent comment of Terry Gannon on an old question posed by Victor Kac in 1994.

受量子McKay对应关系的启发，我们将经典的ADE Lie理论视为\（\ mathfrak {sl} _2 \）上的量子理论。我们引入了反对称特征来表示量子群，并研究了傅里叶对偶性以研究光谱理论。在ADE Lie理论中，Coxeter元素的特征值与邻接矩阵的特征值之间存在对应关系。我们将相关概念形式化，并证明量子群的Verlinde代数的表示形式具有这种对应关系：这包括任何简单的Lie代数\（\ mathfrak {g} \）在任何级别\（k \）上的广义Dynkin图。。这回答了特里·加农（Terry Gannon）最近对维克多·卡克（Victor Kac）在1994年提出的一个老问题的评论。