Representations of Affine Kac-Moody Algebras

  • Jouko Mickelsson
Part of the Plenum Monographs in Nonlinear Physics book series (PMNP)


In the first chapter we explained how simple finite-dimensional Lie algebras can be completely characterized in terms of their Cartan matrices or Dynkin diagrams. The same holds for an arbitrary semisim-ple finite-dimensional Lie algebra. A semisimple Lie algebra is a direct sum of simple ideals which are pairwise orthogonal with respect to the Killing form. It follows that the Cartan matrix of a semisimple Lie algebra decomposes to a block diagonal form, each block representing a simple ideal. Similarly, the Dynkin diagram is a disconnected union of Dynkin diagrams of simple Lie algebras. Next we shall study certain infinite-dimensional Lie algebras which have many similarities with the simple finite-dimensional Lie algebras. In particular, they can be described in terms of generalized Cartan matrices. These algebras were independently introduced in Kac [1968] and Moody [1968].


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Copyright information

© Springer Science+Business Media New York 1989

Authors and Affiliations

  • Jouko Mickelsson
    • 1
  1. 1.University of JyväskyläJyväskyläFinland

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