Title: Affine Lie algebras and vertex operator algebras
Abstract:In this chapter we shall discuss standard (or integrable highest weight) representations of affine Kac-Moody algebras from the point of view of vertex operator algebra theory. This will also provide t...In this chapter we shall discuss standard (or integrable highest weight) representations of affine Kac-Moody algebras from the point of view of vertex operator algebra theory. This will also provide the setting for the next chapter. We restrict our attention to a simple Lie algebra G of type A, D or E. Let G be the corresponding affine Lie algebra and let l be a positive integer. We show that a certain distinguished one of the level l standard G-modules L(l,0) has the structure of a vertex operator algebra and that every level l standard G-module, at least every such module which can be obtained from the tensor product of l basic modules (i.e., level 1 standard modules for affine Lie algebras of type Â, D or Ê), is an irreducible module for this vertex operator algebra. Conversely, every irreducible module for the vertex operator algebra L(l, 0) is also a standard (G-module of level l. The Virasoro algebra for the vertex operator algebra L(l,0) comes from a canonical vertex operator associated with the (suitably normalized) Casimir operator for G. Using an abelian intertwining algebra (see Chapter 12) associated with the direct sum of l copies of the weight lattice, we also construct intertwining operators for irreducible L(l, 0)-modules. There is some overlap between the results in this chapter and those in [FZ], which uses a different approach, involving Verma modules.Read More
Publication Year: 1993
Publication Date: 1993-01-01
Language: en
Type: book-chapter
Indexed In: ['crossref']
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