Dynkin index

In mathematics, the Dynkin index

${\displaystyle x_{\lambda }}$

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

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

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

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

evaluated in the defining representation.

By taking traces, we find that

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

where the Weyl vector

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

is equal to half of the sum of all the positive roots of ${\displaystyle {\mathfrak {g}}}$. The expression ${\displaystyle (\lambda ,\lambda +2\rho )}$ is the value of the quadratic Casimir in the representation ${\displaystyle |\lambda |}$. The index ${\displaystyle x_{\lambda }}$ is always a positive integer.

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