Dynkin index

In mathematics, the Dynkin index $I({\lambda })$ of finite-dimensional highest-weight representations of a compact simple Lie algebra ${\mathfrak {g}}$ 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 $\rho :{\mathfrak {g}}\rightarrow {\text{End}}(V)$. 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

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

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 $V_{\lambda }$.

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