ADE Classification

In mathematics, the ADE classification (originally A-D-E classifications) is the complete list of simply laced Dynkin diagrams or other mathematical objects satisfying analogous axioms; "simply laced" means that there are no multiple edges, which corresponds to all simple roots in the root system forming angles of (no edge between the vertices) or (single edge between the vertices). The list comprises

These comprise two of the four families of Dynkin diagrams (omitting and ), and three of the five exceptional Dynkin diagrams (omitting and ).

This list is non-redundant if one takes for If one extends the families to include redundant terms, one obtains the exceptional isomorphisms

and corresponding isomorphisms of classified objects.

The question of giving a common origin to these classifications, rather than an a posteriori verification of a parallelism, was posed in (Arnold 1976).

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.