Vertex operator algebras associated to type G affine Lie algebras

In this paper, we study representations of the vertex operator algebra L(k,0) at one-third admissible levels for the affine algebra of type . We first determine singular vectors and then obtain a description of the associative algebra A(L(k,0)) using the singular vectors. We then prove that there are only finitely many irreducible A(L(k,0))-modules from the category O. Applying the A(V)-theory, we prove that there are only finitely many irreducible weak L(k,0)-modules from the category O and that such an L(k,0)-module is completely reducible. Our result supports the conjecture made by Adamović and Milas (1995) [2].