Dynkin index

From Wikipedia, the free encyclopedia

In mathematics, the Dynkin index

χ λ

of a representation $| λ |$ of the Lie algebra g that has a highest weight $λ$ is defined as follows

$\chi_{\lambda}=\frac{\dim(|\lambda|)}{2\dim(g)}(\lambda, \lambda +2\rho)$

where the Weyl vector

$\rho=\frac{1}{2}\sum_{\alpha\in \Delta^+} \alpha$

is equal to half of the sum of all the positive roots of g. In the particular case where $λ$ is the highest root, meaning that $| λ |$ is the adjoint representation, $χ λ$ is equal to the dual Coxeter number.

[edit] References

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

Categories: Representation theory of Lie algebras

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