Non-simply laced McKay correspondence and triality (Q1940451)

The McKay correspondence gives a bijection between the conjugacy classes of finite subgroups of \(SU(2)\) and the simply laced complex simple Lie algebras which are classified by the Dynkin diagrams of ADE type. The paper under review gives a uniform description of this correspondence and generalize it to the case of all simple Lie algebras. Moreover, it is shown that the generalized correspondence is related to the \textit{triality} of the quaternions \(\mathbb{H}\). The main result of the paper is to establish the following natural bijection: \[ \bigg\{\text{equivalence classes of the pairs } (\widetilde{\Gamma},O_v) \bigg\}\leftrightarrow \bigg\{\text{the pairs } (\mathfrak{g},\tau) \bigg\}. \] Here \(\widetilde{\Gamma}\) is finite subgroup of \(\mathrm{SU}(2)\) and \(O_v\) is an outer automorphism of \(\widetilde{\Gamma}\) induced by \(Ad(v)\) for a \(v\in \mathrm{SU}(2)\). \(\mathfrak{g}\) is a complex simple Lie algebra and \(\tau\) is an outer automorphism of \(\mathfrak{g}\). Note that any non-simply laced simple Lie algebra can be obtained form a suitable pair \((\mathfrak{g},\tau)\) where \(\mathfrak{g}\) is simply laced simple Lie algebra by taking the fixed part of \(\mathfrak{g}\) under the action of \(\tau\).