Dynkin index

In mathematics, the Dynkin index [math]\displaystyle{ I({\lambda}) }[/math] of a finite-dimensional highest-weight representation of a compact simple Lie algebra [math]\displaystyle{ \mathfrak g }[/math] with highest weight [math]\displaystyle{ \lambda }[/math] is defined by [math]\displaystyle{ \text{Tr}_{V_\lambda}= 2I(\lambda) \text{Tr}_{V_0}, }[/math]

where [math]\displaystyle{ V_0 }[/math] 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 [math]\displaystyle{ \text{Tr}_V }[/math] is the trace form on the representation [math]\displaystyle{ \rho: \mathfrak{g} \rightarrow \text{End}(V) }[/math]. 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

[math]\displaystyle{ I(\lambda)=\frac{\dim V_\lambda}{2\dim\mathfrak g}(\lambda, \lambda +2\rho) }[/math]

where the Weyl vector

[math]\displaystyle{ \rho=\frac{1}{2}\sum_{\alpha\in \Delta^+} \alpha }[/math]

is equal to half of the sum of all the positive roots of [math]\displaystyle{ \mathfrak g }[/math]. The expression [math]\displaystyle{ (\lambda, \lambda +2\rho) }[/math] is the value of the quadratic Casimir in the representation [math]\displaystyle{ V_\lambda }[/math]. The index [math]\displaystyle{ I(\lambda) }[/math] is always a positive integer.

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

See also

• Killing form

References

• Philippe Di Francesco, Pierre Mathieu, David Sénéchal, Conformal Field Theory, 1997 Springer-Verlag New York, ISBN 0-387-94785-X