Lever rule

In mathematics, the Dynkin index

$x_{\lambda }$

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

$\mathrm{t}\mathrm{r}(t_a{t}_b)=2x_{\lambda }{g}_{ab}$

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

$\mathrm{t}\mathrm{r}(t_a{t}_b)=2g_{ab}$

evaluated in the defining representation.

By taking traces, we find that

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

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

In the particular case where $\lambda$ is the highest root, meaning that $\left|\lambda \right|$ 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