# nLab Dynkin diagram

Contents

under construction

### Context

#### Graph theory

graph theory

graph

category of simple graphs

### Extra structure

#### Representation theory

representation theory

geometric representation theory

# Contents

## 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

graphics grabbed from Schwichtenberg

Those Dynkin diagrams in the ADE classification are the following

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
$SU(n+1)$
A1cyclic group of order 2
$\mathbb{Z}_2$
cyclic group of order 2
$\mathbb{Z}_2$
SU(2)
A2cyclic 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)
D4dihedron 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)
D5dihedron on
triangle
dihedral group of order 6
$D_6$
binary dihedral group of order 12
$2 D_6$
SO(10), Spin(10)
D6dihedron 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
$SO(2n)$, $Spin(2n)$
$E_6$tetrahedrontetrahedral 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.