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

Let L(n-l+1/2,0) be the vertex operator algebra associated to an affine Lie algebra of type B_l^(1) at level n-l+1/2, for a positive integer n. We classify irreducible L(n-l+1/2,0)-modules and show that every L(n-l+1/2,0)-module is completely reducible. In the special case n=1, we study a category of weak L(-l+3/2,0)-modules which are in the category $\mathcal{O}$ as modules for the associated affine Lie algebra. We classify irreducible objects in that category and prove semisimplicity of that category.