  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.
