ADE classification

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the ADE classification (originally A-D-E classifications) is a situation where certain kinds of objects are in correspondence with <a href="/facts/Simply_laced_Dynkin_diagram/oBhUwRIR">simply laced Dynkin diagrams</a>. 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 <a href="/facts/Simply_laced_Dynkin_diagram/oBhUwRIR">simply laced Dynkin diagrams</a> comprises

A
          
            n
          
        
        ,
        
        
          D
          
            n
          
        
        ,
        
        
          E
          
            6
          
        
        ,
        
        
          E
          
            7
          
        
        ,
        
        
          E
          
            8
          
        
        .
      
    
    {\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 <a href="/facts/Root_system/TFaEdwMU">root system</a> forming angles of 
 
 
 
 π
 
 /
 
 2
 =
 
 90
 
 ∘
 
 
 
 
 {\displaystyle \pi /2=90^{\circ }}
 
 (no edge between the vertices) or 
 
 
 
 2
 π
 
 /
 
 3
 =
 
 120
 
 ∘
 
 
 
 
 {\displaystyle 2\pi /3=120^{\circ }}
 
 (single edge between the vertices). These are two of the four families of Dynkin diagrams (omitting 
 
 
 
 
 B
 
 n
 
 
 
 
 {\displaystyle B_{n}}
 
 and 
 
 
 
 
 C
 
 n
 
 
 
 
 {\displaystyle C_{n}}
 
), and three of the five exceptional Dynkin diagrams (omitting 
 
 
 
 
 F
 
 4
 
 
 
 
 {\displaystyle F_{4}}
 
 and 
 
 
 
 
 G
 
 2
 
 
 
 
 {\displaystyle G_{2}}
 
).
This list is non-redundant if one takes 
 
 
 
 n
 ≥
 4
 
 
 {\displaystyle n\geq 4}
 
 for 
 
 
 
 
 D
 
 n
 
 
 .
 
 
 {\displaystyle D_{n}.}
 
 If one extends the families to include redundant terms, one obtains the <a href="/facts/Exceptional_isomorphism/BTsHIJ70">exceptional isomorphisms</a>

D
          
            3
          
        
        ≅
        
          A
          
            3
          
        
        ,
        
          E
          
            4
          
        
        ≅
        
          A
          
            4
          
        
        ,
        
          E
          
            5
          
        
        ≅
        
          D
          
            5
          
        
        ,
      
    
    {\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 <a href="/facts/Finite_Coxeter_group/bgzHFm1Q">finite Coxeter groups</a>, by the same diagrams: in this case the Dynkin diagrams exactly coincide with the Coxeter diagrams, as there are no multiple edges.

ADE classification open-in-new

ADE classification