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

:$chi\_\{lambda\}=frac\{dim(|lambda|)\}\{2dim(g)\}(lambda,\; lambda\; +2\; ho)$

where the Weyl vector

:$ho=frac\{1\}\{2\}sum\_\{alphain\; Delta^+\}\; alpha$

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