In mathematics, the Dynkin index
x λ of a representation with highest weight | λ | of a compact simple Lie algebra g that has a highest weight λ is defined by
t r ( t a t b ) = 2 x λ g a b evaluated in the representation | λ | . Here t a are the matrices representing the generators, and g a b is given by
t r ( t a t b ) = 2 g a b evaluated in the defining representation.
By taking traces, we find that
x λ = dim | λ | 2 dim g ( λ , λ + 2 ρ ) where the Weyl vector
ρ = 1 2 ∑ α ∈ Δ + α is equal to half of the sum of all the positive roots of g . The expression ( λ , λ + 2 ρ ) is the value quadratic Casimir in the representation | λ | . The index x λ is always a positive integer.
In the particular case where λ is the highest root, meaning that | λ | is the adjoint representation, x λ is equal to the dual Coxeter number.
