Dynkin index

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_at_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_at_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