Dynkin index

In mathematics, the Dynkin index $I(\lambda )$ of finite-dimensional highest-weight representations of a compact simple Lie algebra $𝔤$ relates their trace forms via

$$\frac{{\text{Tr}}_{V_{\lambda }}}{{\text{Tr}}_{V_{\mu }}}=\frac{I(\lambda )}{I(\mu )}.$$

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.

The notation ${\text{Tr}}_V$ is the trace form on the representation $\rho :𝔤\to \text{End}(V)$. 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 obtainTemplate:Citation needed

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

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 $𝔤$. The expression $(\lambda ,\lambda +2\rho )$ is the value of the quadratic Casimir in the representation $V_{\lambda }$.

See also

• Killing form

References

• Philippe Di Francesco, Pierre Mathieu, David Sénéchal, Conformal Field Theory, 1997 Springer-Verlag New York, Template:Isbn