Vertex operator algebras associated to type B affine Lie algebras on admissible half-integer levels

Let be the vertex operator algebra associated to an affine Lie algebra of type at level , for a positive integer n. We classify irreducible -modules and show that every -module is completely reducible. In the special case n=1, we study a category of weak -modules which are in the category O as modules for the associated affine Lie algebra. We classify irreducible objects in that category and prove semisimplicity of that category.