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 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

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