Affine vertex operator superalgebra L₎ₒ₏ (₁|₂) (l, 0) at admissible level

Let L₎ₒ₏ (₁|₂) (l, 0) be the simple affine vertex operator superalgebra with admissible level l. We prove that the category of weak L₎ₒ₏ (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₎ₒ₏ (₁|₂) (l, 0), _) with new Virasoro elements _ are rational and the irreducible modules are exactly the admissible modules for osp (1|2), where 0<<1 is a rational number. Furthermore, we determine the Zhu's algebras A (L₎ₒ₏ (₁|₂) (l, 0) ) and their bimodules A (L (l, j) ) for (L₎ₒ₏ (₁|₂) (l, 0), _), where j is the admissible weight. As an application, we calculate the fusion rules among the irreducible ordinary modules of (L₎ₒ₏ (₁|₂) (l, 0), _).