Let $L(-{ 1/2}(l+1),0)$ be the simple vertex operator algebra associated to an affine Lie algebra of type $A_{ l}^{ (1)}$ with the lowest admissible half-integer level $-{ 1/2}(l+1)$, for even l. We study the category of weak modules for that vertex operator algebra which are in category $\cal{ O}$ as modules for the associated affine Lie algebra. We classify irreducible objects in that category and prove semisimplicity of that category.
Categories: math.QA math.RT