Vertex Operator Algebras Associated to Type G Affine Lie Algebras II

Abstract

We continue the study of the vertex operator algebra L(k, 0) associated to a type affine Lie algebra at admissible one-third integer levels, , initiated in Axtell and Lee [3]. We show there are only finitely many irreducible L(k, 0)-modules from the category 𝒪. The proof relies on knowledge of an explicit formula for the singular vectors. After obtaining this formula, we are able to prove there are only finitely many irreducible A(L(k, 0))-modules from the category 𝒪. The main result then follows from the bijective correspondence in A(V)-theory.

Key Words:

• Admissible levels
• Affine Lie algebras
• Irreducible representations
• Sigular vectors
• Vertex operator algebras

2010 Mathematics Subject Classification:

• 17B67
• 17B69

ACKNOWLEDGMENTS

The author wishes to thank K.-H. Lee and A. Feingold for helpful conversations and encouragement.

This work was supported in part by NRF Grant # 2010-0010753.

Notes

Communicated by K. Misra.