Dynkin Diagram - Rank 2 Dynkin Diagrams

Rank 2 Dynkin Diagrams

Dynkin diagrams are equivalent to generalized Cartan matrices, as shown in this table of rank 2 Dynkin diagrams with their corresponding 2x2 Cartan matrices.

For rank 2, the Cartan matrix form is:

A multi-edged diagram corresponds to the nondiagonal Cartan matrix elements -a21, -a12, with the number of edges drawn equal to max(-a21, -a12), and an arrow pointing towards nonunity elements.

A generalized Cartan matrix is a square matrix such that:

  1. For diagonal entries, .
  2. For non-diagonal entries, .
  3. if and only if

The Cartan matrix determines whether the group is of finite type (if it is a Positive-definite matrix, i.e. all eigenvalues are positive), of affine type (if it is not positive-definite but positive-semidefinite, i.e. all eigenvalues are non-negative), or of indefinite type. The indefinite type often is further subdivided, for example a Coxeter group is Lorentzian if it has one negative eigenvalue and all other eigenvalues are positive. Moreover, multiple sources refer to hyberbolic Coxeter groups, but there are several non-equivalent definitions for this term. In the discussion below, hyperbolic Coxeter groups are a special case of Lorentzian, satisfying an extra condition. Note that for rank 2, all negative determinant Cartan matrices correspond to hyperbolic Coxeter group. But in general, most negative determinant matrices are neither hyperbolic nor Lorentzian.

Finite branches have (-a21, -a12)=(1,1), (2,1), (3,1), and affine branches (with a zero determinant) have (-a21, -a12) =(2,2) or (4,1).

Rank 2 Dynkin diagrams
Group
name		 Dynkin diagram Cartan matrix Symmetry
order		 Related
simply-laced
group3
(Standard)
multi-edged
graph
 Valued
graph1		 Coxeter
graph2		 Determinant

(4-a21*a12)
Finite (Determinant>0)
A1xA1 4 2
A2
(undirected)		 3 3
B2 2 4
C2 2 4
BC2
(undirected)		 2 4
G2 1 6
G2
(undirected)		 1 6
Affine (Determinant=0)
A1(1) 0 ∞
A2(2) 0 ∞
Hyperbolic (Determinant<0)
-1 -
-2 -
-2 -
-3 -
-4 -
-4 -
-5 -
4-ab<0 -

Note1: For hyperbolic groups, (a12*a21>4), the multiedge style is abandoned in favor of an explicit labeling (a21, a12) on the edge. These are usually not applied to finite and affine graphs.

Note2: For undirected groups, Coxeter diagrams are interchangeable. They are usually labeled by their order of symmetry, with order-3 implied with no label.

Note3: Many multi-edged groups can be obtained from a higher ranked simply-laced group by applying a suitable folding operation.

Read more about this topic:  Dynkin Diagram

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