Lever rule

In mathematics, the Dynkin index

$x_{\lambda }$

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

${\rm {tr}}(t_{a}t_{b})=2x_{\lambda }g_{ab}$

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

${\rm {tr}}(t_{a}t_{b})=2g_{ab}$

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

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

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 $|\lambda |$. 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