Dynkin index

In mathematics, the Dynkin index of a finite-dimensional highest-weight representation of a compact simple Lie algebra with highest weight is defined by

where 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 is the trace form on the representation . 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

where the Weyl vector

is equal to half of the sum of all the positive roots of . The expression is the value of the quadratic Casimir in the representation . The index is always a positive integer.

In the particular case where is the highest root, so that is the adjoint representation, the Dynkin index is equal to the dual Coxeter number.

See also

References

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