ADE classification information

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

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 $\pi /2=90^{\circ }$ (no edge between the vertices) or $2\pi /3=120^{\circ }$ (single edge between the vertices). These are two of the four families of Dynkin diagrams (omitting $B_{n}$ and $C_{n}$ ), and three of the five exceptional Dynkin diagrams (omitting $F_{4}$ and $G_{2}$ ).

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

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.

ADE classification information

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