nLab
Dynkin diagram

Contents

under construction

Context

Graph theory

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 nA = (a_{i,j})_{i,j=1}^n is obtained from the following procedure:

  1. Number of vertices = Number of simple roots = size of AA = n

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

  3. If a i,j=1a_{i,j} = -1 and a j,i=la_{j,i} = -l then i,ji,j are connected by ll 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 n1A_{n \geq 1}cyclic group
n+1\mathbb{Z}_{n+1}
cyclic group
n+1\mathbb{Z}_{n+1}
special unitary group
SU(n+1)SU(n+1)
A1cyclic group of order 2
2\mathbb{Z}_2
cyclic group of order 2
2\mathbb{Z}_2
SU(2)
A2cyclic group of order 3
3\mathbb{Z}_3
cyclic group of order 3
3\mathbb{Z}_3
SU(3)
A3
=
D3
cyclic group of order 4
4\mathbb{Z}_4
cyclic group of order 4
2D 2 42 D_2 \simeq \mathbb{Z}_4
SU(4)
\simeq
Spin(6)
D4dihedron on
bigon
Klein four-group
D 4 2× 2D_4 \simeq \mathbb{Z}_2 \times \mathbb{Z}_2
quaternion group
2D 42 D_4 \simeq Q8
SO(8), Spin(8)
D5dihedron on
triangle
dihedral group of order 6
D 6D_6
binary dihedral group of order 12
2D 62 D_6
SO(10), Spin(10)
D6dihedron on
square
dihedral group of order 8
D 8D_8
binary dihedral group of order 16
2D 82 D_{8}
SO(12), Spin(12)
D n4D_{n \geq 4}dihedron,
hosohedron
dihedral group
D 2(n2)D_{2(n-2)}
binary dihedral group
2D 2(n2)2 D_{2(n-2)}
special orthogonal group, spin group
SO(2n)SO(2n), Spin(2n)Spin(2n)
E 6E_6tetrahedrontetrahedral group
TT
binary tetrahedral group
2T2T
E6
E 7E_7cube,
octahedron
octahedral group
OO
binary octahedral group
2O2O
E7
E 8E_8dodecahedron,
icosahedron
icosahedral group
II
binary icosahedral group
2I2I
E8

Dynkin Index

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

For any irreducible finite representation R λR_{\lambda} of \mathcal{g}, TrR λ(x)R λ(y)=I λk(x,y)TrR_\lambda(x)R_\lambda(y) = I_\lambda k(x,y) Where I λI_\lambda is the Dynkin Index of R λR_\lambda.

The Dynkin index can also be defined in terms of the eigenvalue C λC_\lambda of the quadratic Casimir operator: I λ=dim(R λ)dim(C λ 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 μA_\mu be a gauge potential on the Euclidean space E 4E^4 given by

A μ(x)=A μ αX α A_\mu(x) = A_{\mu}^{\alpha}X_\alpha

where X αX_\alpha are the generators of a compact gauge group GG. If the gauge field A μA_\mu is an embedding of G˜\tilde{G} in GG then the Dynkin index of the embedding is denoted as j G˜Gj_{\tilde{G} \over G} . The topological charge is

q=j G˜Gq 1 q = j_{\tilde{G} \over G} q_1

where q 1q_1 is the charge of A μA_\mu treated as a G˜\tilde{G} gauge field. The proof was carried out for G˜SU(2)\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.

References

General

See also:

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

Last revised on August 3, 2021 at 03:53:36. See the history of this page for a list of all contributions to it.