ADE classification

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

${\displaystyle A_{n},\,D_{n},\,E_{6},\,E_{7},\,E_{8}.}$

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

${\displaystyle D_{3}\cong A_{3},E_{4}\cong A_{4},E_{5}\cong D_{5},}$

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.