Dynkin index

In mathematics, the Dynkin index $I(\lambda )$ of a finite-dimensional highest-weight representation of a compact simple Lie algebra $\mathfrak{𝔤}$ with highest weight $\lambda$ is defined by $${\text{Tr}}_{V_{\lambda }}=2I(\lambda ){\text{Tr}}_{V_0},$$

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 ${\text{Tr}}_V$ is the trace form on the representation $\rho :\mathfrak{𝔤}\to \text{End}(V)$. 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

$I(\lambda )=\frac{\dim V_{\lambda }}{2\dim \mathfrak{𝔤}}(\lambda ,\lambda +2\rho )$

where the Weyl vector

$\rho =\frac{1}{2}\sum _{\alpha \in {\mathrm{\Delta }}^{+}}\alpha$

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