In mathematics , the Dynkin index of an irreducible representation R is defined as
![{\ displaystyle \ mathrm {track} (T ^ {a} T ^ {b}) = \ delta ^ {ab} T_ {R}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/938295772682331f9d238fbd9074e4e1461378c0)
where are the generators of the representation. The term takes its name in honor of the Russian mathematician Eugene Dynkin .
![{\ displaystyle T ^ {a}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb8b02d8588d09a61be646fe251831701bbd4986)
For a representation of the Lie algebra with the highest weight , the Dynkin index is defined as
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77)
![\ lambda](https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a)
![{\ displaystyle \ chi _ {\ lambda}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dbd0e416e02a215304e4b13837c55e1a06acbfd9)
![{\ displaystyle \ chi _ {\ lambda} = {\ frac {\ dim (| \ lambda |)} {2 \ dim (g)}} (\ lambda, \ lambda +2 \ rho)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f627a15770e96e82bffd4cdbc5f1d2a4f2e154a8)
wherein the Weyl vector
![{\ displaystyle \ rho = {\ frac {1} {2}} \ sum _ {\ alpha \ in \ Delta ^ {+}} \ alpha}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d8200d78709bcf4f9dda4f00c6071992c2984a1)
is equal to half the sum of all positive roots of . If, as a special case, the greatest root, that is, is the adjoint representation , then the Dynkin index is equal to the dual Coxeter number .
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77)
![\ lambda](https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a)
![{\ displaystyle | \ lambda |}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e92d2419f2d2ad612edb802a7727b4f4eb07f3c8)
![{\ displaystyle \ chi _ {\ lambda}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dbd0e416e02a215304e4b13837c55e1a06acbfd9)
literature
- Philippe Di Francesco, Pierre Mathieu, David Sénéchal: Conformal Field Theory. Springer-Verlag, New York 1997, ISBN 0-387-94785-X .