# 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

${\rm {tr}}(t_{a}t_{b})=2x_{\lambda }g_{ab}$ evaluated in the representation $|\lambda |$ . Here $t_{a}$ are the matrices representing the generators, and $g_{ab}$ is given by

${\rm {tr}}(t_{a}t_{b})=2g_{ab}$ evaluated in the defining representation.

By taking traces, we find that

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