Dynkin index

In mathematics, the Dynkin index

$x_{\lambda}$

of a representation with highest weight $|\lambda|$ of a compact simple Lie algebra $\mathfrak g$ that has a highest weight $\lambda$ is defined by

${\rm tr}(t_at_b)= 2x_\lambda g_{ab}$

evaluated in the representation $|\lambda|$. Here $t_a$ are the matrices representing the generators, and $g_{ab}$ is given by

${\rm tr}(t_at_b)= 2g_{ab}$

evaluated in the defining representation.

By taking traces, we find that

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

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 quadratic Casimir in the representation $|\lambda|$. The index $x_{\lambda}$ is always a positive integer.

In the particular case where $\lambda$ is the highest root, meaning that $|\lambda|$ is the adjoint representation, $x_{\lambda}$ 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