Dynkin index

In mathematics, the Dynkin index ${\displaystyle I({\lambda })}$ of a finite-dimensional highest-weight representation of a compact simple Lie algebra ${\displaystyle {\mathfrak {g}}}$ with highest weight ${\displaystyle \lambda }$ is defined by

$${\displaystyle {\text{Tr}}_{V_{\lambda }}=2I(\lambda ){\text{Tr}}_{V_{0}},}$$

where ${\displaystyle V_{0}}$ is the 'defining representation', which is most often taken to be the fundamental representation if the Lie algebra under consideration is a matrix Lie algebra.

The notation ${\displaystyle {\text{Tr}}_{V}}$ is the trace form on the representation ${\displaystyle \rho$ :{\mathfrak {g}}\rightarrow {\text{End}}(V)} . By Schur's lemma, since the trace forms are all invariant forms, they are related by constants, so the index is well-defined.

Since the trace forms are bilinear forms, we can take traces to obtain

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

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 ${\displaystyle V_{\lambda }}$. The index ${\displaystyle I(\lambda )}$ is always a positive integer.

In the particular case where ${\displaystyle \lambda }$ is the highest root, so that ${\displaystyle V_{\lambda }}$ is the adjoint representation, the Dynkin index ${\displaystyle I(\lambda )}$ is equal to the dual Coxeter number.