title: Dynkin index
text: # Dynkin index
  
  In mathematics, the Dynkin index $I({\lambda })$ of a finite-dimensional highest-weight representation of a compact simple Lie algebra ${\mathfrak {g}}$ with highest weight $\lambda$ is defined by
  
  ${\text{Tr}}_{V_{\lambda }}=2I(\lambda ){\text{Tr}}_{V_{0}},$
  
  where $V_{0}$ 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 ${\text{Tr}}_{V}$ is the trace form on the representation $\rho :{\mathfrak {g}}\rightarrow {\text{End}}(V)$. 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
  
      $I(\lambda )={\frac {\dim V_{\lambda }}{2\dim {\mathfrak {g}}}}(\lambda ,\lambda +2\rho )$
  
  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 $V_{\lambda }$. The index $I(\lambda )$ is always a positive integer.
  
  In the particular case where $\lambda$ is the highest root, so that $V_{\lambda }$ is the adjoint representation, the Dynkin index $I(\lambda )$ is equal to the dual Coxeter number.
  
  