under construction
geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
Be?linson-Bernstein localization?
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:
Number of vertices = Number of simple roots = size of $A$ = n
If $a_{i,j} = a_{i,j} = -1$ then $i,j$ are connected by a nonlabeled edge.
If $a_{i,j} = -1$ and $a_{j,i} = -l$ then $i,j$ are connected by $l$ arrows labeled by <.
classification of simple Lie groups:
graphics grabbed from Schwichtenberg
Those Dynkin diagrams in the ADE classification are the following
ADE classification and McKay correspondence
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: $I_\lambda = \frac{dim(R_\lambda)}{dim(\mathcal{g}}C_\lambda$.
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
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
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.
See also:
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)
Last revised on June 30, 2023 at 18:41:55. See the history of this page for a list of all contributions to it.