nLab Dynkin diagram

under construction

Idea

A Dynkin diagram is a labeled graph that possesses a one-to-one correspondence with a finite indecomposable reduced root system, thus with a simple complex finite dimensional Lie algebra or Cartan matrix?.

The construction of a Dynkin diagram from the Cartan Matrix?, $A = (a_{i,j})_{i,j=1}^n$ is obtained from the following procedure:

1. Number of vertices = Number of simple roots = size of $A$ = n

2. If $a_{i,j} = a_{i,j} = -1$ then $i,j$ are connected by a nonlabeled edge.

3. If $a_{i,j} = -1$ and $a_{j,i} = -l$ then $i,j$ are connected by $l$ arrows labeled by <.

Properties

Classification of simple Lie groups

classification of simple Lie groups:

graphics grabbed from Schwichtenberg

ADE-Classification

Those Dynkin diagrams in the ADE classification are the following

ADE classification and McKay correspondence

Dynkin diagram/ Dynkin quiver	dihedron, Platonic solid	finite subgroups of SO(3)	finite subgroups of SU(2)	simple Lie group
$A_{n \geq 1}$		cyclic group $\mathbb{Z}_{n+1}$	cyclic group $\mathbb{Z}_{n+1}$	special unitary group $SU(n+1)$
A1		cyclic group of order 2 $\mathbb{Z}_2$	cyclic group of order 2 $\mathbb{Z}_2$	SU(2)
A2		cyclic group of order 3 $\mathbb{Z}_3$	cyclic group of order 3 $\mathbb{Z}_3$	SU(3)
A3 = D3		cyclic group of order 4 $\mathbb{Z}_4$	cyclic group of order 4 $2 D_2 \simeq \mathbb{Z}_4$	SU(4) $\simeq$ Spin(6)
D4	dihedron on bigon	Klein four-group $D_4 \simeq \mathbb{Z}_2 \times \mathbb{Z}_2$	quaternion group $2 D_4 \simeq$ Q8	SO(8), Spin(8)
D5	dihedron on triangle	dihedral group of order 6 $D_6$	binary dihedral group of order 12 $2 D_6$	SO(10), Spin(10)
D6	dihedron on square	dihedral group of order 8 $D_8$	binary dihedral group of order 16 $2 D_{8}$	SO(12), Spin(12)
$D_{n \geq 4}$	dihedron, hosohedron	dihedral group $D_{2(n-2)}$	binary dihedral group $2 D_{2(n-2)}$	special orthogonal group, spin group $SO(2n)$, $Spin(2n)$
$E_6$	tetrahedron	tetrahedral group $T$	binary tetrahedral group $2T$	E6
$E_7$	cube, octahedron	octahedral group $O$	binary octahedral group $2O$	E7
$E_8$	dodecahedron, icosahedron	icosahedral group $I$	binary icosahedral group $2I$	E8

Dynkin Index

Let $\mathcal{g}$ be a finite simple complex Lie algebra with a Killing form $k$ on $\mathcal{g}$ given by the trace in the adjoint representation, $k(x,y) = Tr R_{ad}(x)R_{ad}(y)$ for $x,y \in \mathcal{g}$.

For any irreducible finite representation $R_{\lambda}$ of $\mathcal{g}$, $TrR_\lambda(x)R_\lambda(y) = I_\lambda k(x,y)$ Where $I_\lambda$ is the Dynkin Index of $R_\lambda$.

The Dynkin index can also be defined in terms of the eigenvalue $C_\lambda$ of the quadratic Casimir operator: $I_\lambda = \frac{dim(R_\lambda)}{dim(\mathcal{g}}C_\lambda$.

Remark

In mathematical physics,in the context of embeddings of gauge fields, the Dynkin index is used in the calculation of topological charges (instanton number). To demonstrate this, let $A_\mu$ be a gauge potential on the Euclidean space $E^4$ given by

$$A_\mu(x) = A_{\mu}^{\alpha}X_\alpha$$

where $X_\alpha$ are the generators of a compact gauge group $G$. If the gauge field $A_\mu$ is an embedding of $\tilde{G}$ in $G$ then the Dynkin index of the embedding is denoted as $j_{\tilde{G} \over G}$ . The topological charge is

$$q = j_{\tilde{G} \over G} q_1$$

where $q_1$ is the charge of $A_\mu$ treated as a $\tilde{G}$ gauge field. The proof was carried out for $\tilde{G} \simeq SU(2)$ by Bitar and Sorba see Myers, de Roo & Sorba 79, Sec. 2, which can be extended to arbitrary simple compact groups.

Related concepts

• Dynkin quiver

• ADE classification

References

General

Named after Eugene B. Dynkin.

See also:

• Wikipedia, Dynkin diagram

Dynkin index

• Bianchi M. et al. (2004) Dynkin Index. In: Duplij S., Siegel W., Bagger J. (eds) Concise Encyclopedia of Supersymmetry. Springer, Dordrecht. publisher

• C. Meyers, Mees de Roo, P. Sorba, Group-theoretical aspects of instantons. Nuov Cim A 52, 519–530 (1979) (doi:10.1007/BF02770858)