In mathematics, the ADE classification (originally A-D-E classifications) is a situation where certain kinds of objects are in correspondence with simply laced Dynkin diagrams. The question of giving a common origin to these classifications, rather than a posteriori verification of a parallelism, was posed in (Arnold 1976). The complete list of simply laced Dynkin diagrams comprises

Here "simply laced" means that there are no multiple edges, which corresponds to all simple roots in the root system forming angles of ${\displaystyle \pi /2=90^{\circ }}$ (no edge between the vertices) or ${\displaystyle 2\pi /3=120^{\circ }}$ (single edge between the vertices). These are two of the four families of Dynkin diagrams (omitting ${\displaystyle B_{n}}$ and ${\displaystyle C_{n}}$), and three of the five exceptional Dynkin diagrams (omitting ${\displaystyle F_{4}}$ and ${\displaystyle G_{2}}$).

This list is non-redundant if one takes ${\displaystyle n\geq 4}$ for ${\displaystyle D_{n}.}$ If one extends the families to include redundant terms, one obtains the exceptional isomorphisms

and corresponding isomorphisms of classified objects.

The A, D, E nomenclature also yields the simply laced finite Coxeter groups, by the same diagrams: in this case the Dynkin diagrams exactly coincide with the Coxeter diagrams, as there are no multiple edges.

In terms of complex semisimple Lie algebras:

In terms of compact Lie algebras and corresponding simply laced Lie groups:

The same classification applies to discrete subgroups of ${\displaystyle SU(2)}$, the binary polyhedral groups; properly, binary polyhedral groups correspond to the simply laced affine Dynkin diagrams ${\displaystyle {\tilde {A}}_{n},{\tilde {D}}_{n},{\tilde {E}}_{k},}$ and the representations of these groups can be understood in terms of these diagrams. This connection is known as the McKay correspondence after John McKay. The connection to Platonic solids is described in (Dickson 1959). The correspondence uses the construction of McKay graph.