Affine vertex operator superalgebra L_osp(1|2)(l,0) at admissible level

Let L_osp(1|2)(l,0) be the simple affine vertex operator superalgebra with admissible level l. We prove that the category of weak L_osp(1|2)(l,0)-modules on which the positive part of osp(1|2) acts locally nilpotent is semisimple. Then we prove that Q-graded vertex operator superalgebras (L_osp(1|2)(l,0),ω_ξ) with new Virasoro elements ω_ξ are rational and the irreducible modules are exactly the admissible modules for osp(1|2), where 0<ξ<1 is a rational number. Furthermore, we determine the Zhu's algebras A(L_osp(1|2)(l,0)) and their bimodules A(L(l,j)) for (L_osp(1|2)(l,0),ω_ξ), where j is the admissible weight. As an application, we calculate the fusion rules among the irreducible ordinary modules of (L_osp(1|2)(l,0),ω_ξ).