Affine Lie Algebras

  • Philippe Di Francesco
  • Pierre Mathieu
  • David Sénéchal
Part of the Graduate Texts in Contemporary Physics book series (GTCP)


This chapter is a basic introduction to affine Lie algebras, preparing the stage for their application to conformal field theory. In Sect. 14.1.1, after having introduced the affine Lie algebras per se, we show how the fundamental concepts of roots, weights, Cartan matrices, and Weyl groups are extended to the affine case. Section 14.2 introduces the outer automorphism group of affine Lie algebras, which is generated by the new symmetry transformations of the extended Dynkin diagram. The following section describes highest-weight representations, focusing on those whose highest weight is dominant. Characters for these representations are introduced in Sect. 14.4. Their modular properties are presented in the following sections, where various properties of their modular S matrices are also reported. The affine extension of finite Lie algebra embeddings is presented in Sect. 14.7. Four appendices complete the chapter. The first one contains the proof of a technical identity related to outer automorphism groups. The second appendix displays an explicit basis (in terms of semi-infinite paths) for the states in integrable representations of affine su(N). In the third one, the modular transformation properties of the affine characters are derived. The final appendix lists all the symbols pertaining to affine Lie algebras.


Weyl Group Simple Root Dynkin Diagram Outer Automorphism Dynkin Label 
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Copyright information

© Springer-Verlag New York, Inc. 1997

Authors and Affiliations

  • Philippe Di Francesco
    • 1
  • Pierre Mathieu
    • 2
  • David Sénéchal
    • 3
  1. 1.Commissariat l’Énergie Atomique Centre d’Études de SaclayService de Physique ThéoriqueGif-sur-YvetteFrance
  2. 2.Département de PhysiqueUniversité LavalQuébecCanada
  3. 3.Département de PhysiqueUniversité de SherbrookeSherbrookeCanada

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