Dynkin Index

In mathematics, the Dynkin index $I()$ of a finite-dimensional highest-weight representation of a compact simple Lie algebra $\mathfrak g$ with highest weight $\lambda$ is defined by

$\text_= 2I(\lambda) \text_,$

where $V_0$ is the 'defining representation', which is most often taken to be the fundamental representation if the Lie algebra under consideration is a matrix Lie algebra.

The notation $\text_V$ is the trace form on the representation $\rho: \mathfrak \rightarrow \text(V)$. By Schur's lemma, since the trace forms are all invariant forms, they are related by constants, so the index is well-defined.

Since the trace forms are bilinear forms, we can take traces to obtain

:$I(\lambda)=\frac(\lambda, \lambda +2\rho)$

where the Weyl vector

:$\rho=\frac\sum_ \alpha$

is equal to half of the sum of all the positive roots of $\mathfrak g$. The expression $(\lambda, \lambda +2\rho)$ is the value of the quadratic Casimir in the representation $V_\lambda$. The index $I(\lambda)$ is always a positive integer.

In the particular case where $\lambda$ is the highest root, so that $V_\lambda$ is the adjoint representation, the Dynkin index $I(\lambda)$ is equal to the dual Coxeter number.

See also

* Killing form

References

* Philippe Di Francesco, Pierre Mathieu, David Sénéchal, ''Conformal Field Theory'', 1997 Springer-Verlag New York, {{isbn, 0-387-94785-X

Representation theory of Lie algebras