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In mathematics , the Dynkin index
x
λ
{\displaystyle x_{\lambda }}
of a representation with highest weight
|
λ
|
{\displaystyle |\lambda |}
Lie algebra
g
{\displaystyle {\mathfrak {g}}}
highest weight
λ
{\displaystyle \lambda }
t
r
(
t
a
t
b
)
=
2
x
λ
g
a
b
{\displaystyle {\rm {tr}}(t_{a}t_{b})=2x_{\lambda }g_{ab}}
evaluated in the representation
|
λ
|
{\displaystyle |\lambda |}
t
a
{\displaystyle t_{a}}
g
a
b
{\displaystyle g_{ab}}
t
r
(
t
a
t
b
)
=
2
g
a
b
{\displaystyle {\rm {tr}}(t_{a}t_{b})=2g_{ab}}
evaluated in the defining representation.
By taking traces, we find that
x
λ
=
dim
|
λ
|
2
dim
g
(
λ
,
λ
+
2
ρ
)
{\displaystyle x_{\lambda }={\frac {\dim |\lambda |}{2\dim {\mathfrak {g}}}}(\lambda ,\lambda +2\rho )}
where the Weyl vector
ρ
=
1
2
∑
α
∈
Δ
+
α
{\displaystyle \rho ={\frac {1}{2}}\sum _{\alpha \in \Delta ^{+}}\alpha }
is equal to half of the sum of all the positive roots of
g
{\displaystyle {\mathfrak {g}}}
(
λ
,
λ
+
2
ρ
)
{\displaystyle (\lambda ,\lambda +2\rho )}
|
λ
|
{\displaystyle |\lambda |}
x
λ
{\displaystyle x_{\lambda }}
In the particular case where
λ
{\displaystyle \lambda }
highest root , meaning that
|
λ
|
{\displaystyle |\lambda |}
adjoint representation ,
x
λ
{\displaystyle x_{\lambda }}
dual Coxeter number .
References Philippe Di Francesco, Pierre Mathieu, David Sénéchal, Conformal Field Theory , 1997 Springer-Verlag New York, ISBN 0-387-94785-X 
