Dynkin index

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In mathematics, the Dynkin index

of a representation with highest weight of a compact simple Lie algebra that has a highest weight is defined by

evaluated in the representation . Here are the matrices representing the generators, and is given by

evaluated in the defining representation.

By taking traces, we find that

where the Weyl vector

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

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

References[edit]

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