Affine vertex operator superalgebra Lsl(2|1)̂(k,0) at boundary admissible level

Let L_sl(2|1)(k,0) be the simple affine vertex operator superalgebra associated to the affine Lie superalgebra sl(2|1) with admissible level k. We conjecture that L_sl(2|1)(k,0) is rational in the category O at boundary admissible level k and there are finitely many irreducible weak L_sl(2|1)(k,0)-modules in the category O, where the irreducible modules are exactly the admissible modules of level k for sl(2|1). In this paper, we first prove this conjecture at boundary admissible level -1/2. Then we give an example to show that outside of the boudary levels, L_sl(2|1)(k,0) is not rational in the category O. Furthermore, we consider the Q-graded vertex operator superalgebras (L_sl(2|1)(k,0),ω_ξ) associated to a family of new Virasoro elements ω_ξ, where $0<ξ<1$ is a rational number. We determine the Zhu's algebra Aω_ξ(L_sl(2|1)(-1/2,0)) of (L_sl(2|1)(-1/2,0),ω_ξ) and prove that (L_sl(2|1)(-1/2,0),ω_ξ) is rational and C₂-cofinite. Finally, we consider the case of non-boundary admissible level 1/2 to support our conjecture, that is, we show that there are infinitely many irreducible weak L_sl(2|1)(1/2,0)-modules in the category O and (L_sl(2|1)(1/2,0),ω_ξ) is not rational.