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Dynkin index

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In mathematics, the Dynkin index of finite-dimensional highest-weight representations of a compact simple Lie algebra relates their trace forms via

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.

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[citation needed]

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 .

See also

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References

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