We give a complete classification of Airy structures for finite-dimensional simple Lie algebras over \({\mathbb {C}}\), and to some extent also over \({\mathbb {R}}\), up to isomorphisms and gauge transformations. The result is that the only algebras of this type which admit any Airy structures are \(\mathfrak {sl}_2\), \(\mathfrak {sp}_4\) and \(\mathfrak {sp}_{10}\). Among these, each admits exactly two non-equivalent Airy structures. Our methods apply directly also to semisimple Lie algebras. In this case it turns out that the number of non-equivalent Airy structures is countably infinite. We have derived a number of interesting properties of these Airy structures and constructed many examples. Techniques used to derive our results may be described, broadly speaking, as an application of representation theory in semiclassical analysis.

我们给出了有限维简单李代数在\({\mathbb {C}}\) 上的艾里结构的完整分类，并且在某种程度上也在\({\mathbb {R}}\) 上，直到同构和量规变换。结果是这种类型的唯一允许任何艾里结构的代数是\(\mathfrak {sl}_2\) , \(\mathfrak {sp}_4\)和\(\mathfrak {sp}_{10}\ ). 其中，每个都承认两个不等价的 Airy 结构。我们的方法也直接适用于半简单李代数。在这种情况下，结果证明非等效艾里结构的数量是可数无限的。我们已经推导出了这些艾里结构的许多有趣的特性，并构建了许多示例。从广义上讲，用于推导出我们的结果的技术可以描述为表示理论在半经典分析中的应用。