Dynkin index

In mathematics, the Dynkin index

$\chi_{\lambda}$

of a representation $|\lambda|$ of the Lie algebra g that has a highest weight $\lambda$ is defined as follows

$\chi_{\lambda}=\frac{\dim(|\lambda|)}{2\dim(g)}(\lambda, \lambda %2B2\rho)$

$\rho=\frac{1}{2}\sum_{\alpha\in \Delta^%2B} \alpha$

is equal to half of the sum of all the positive roots of g. In the particular case where $\lambda$ is the highest root, meaning that $|\lambda|$ is the adjoint representation, $\chi_{\lambda}$ is equal to the dual Coxeter number.

References

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