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| In [[mathematics]], the '''Dynkin index'''
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| :<math>x_{\lambda}</math>
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| of a representation with highest weight <math>|\lambda|</math> of a compact simple [[Lie algebra]] ''g'' that has a [[highest weight]] <math>\lambda</math> is defined by
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| :<math> {\rm tr}(t_at_b)= 2x_\lambda g_{ab}</math> | |
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| evaluated in the representation <math>|\lambda|</math>. Here <math>t_a</math> are the matrices representing the generators, and
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| <math>g_{ab}</math> is
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| :<math> {\rm tr}(t_at_b)= 2g_{ab}</math>
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| evaluated in the defining representation.
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| By taking traces, we find that
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| :<math>x_{\lambda}=\frac{\dim(|\lambda|)}{2\dim(g)}(\lambda, \lambda +2\rho)</math>
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| where the [[Weyl vector]]
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| :<math>\rho=\frac{1}{2}\sum_{\alpha\in \Delta^+} \alpha</math> | |
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| is equal to half of the sum of all the [[positive root]]s of ''g''. The expression <math>(\lambda, \lambda +2\rho)</math> is the value quadratic Casimir in the representation <math>|\lambda|</math>. The index <math>x_{\lambda}</math> is always a positive integer.
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| In the particular case where <math>\lambda</math> is the [[highest root]], meaning that <math>|\lambda|</math> is the [[Adjoint representation of a Lie group|adjoint representation]], <math>x_{\lambda}</math> is equal to the [[dual Coxeter number]].
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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
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| [[Category:Representation theory of Lie algebras]]
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