Let Lₒ₋ (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 Lₒ₋ (₂|₁) (k, 0) is rational in the category O at boundary admissible level k and there are finitely many irreducible weak Lₒ₋ (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 -12. Then we give an example to show that outside of the boudary levels, Lₒ₋ (₂|₁) (k, 0) is not rational in the category O. Furthermore, we consider the Q-graded vertex operator superalgebras (Lₒ₋ (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 A⏨_⏝ (Lₒ₋ (₂|₁) (-12, 0) ) of (Lₒ₋ (₂|₁) (-12, 0), ω_ξ) and prove that (Lₒ₋ (₂|₁) (-12, 0), ω_ξ) is rational and C₂-cofinite. Finally, we consider the case of non-boundary admissible level 12 to support our conjecture, that is, we show that there are infinitely many irreducible weak Lₒ₋ (₂|₁) (12, 0) -modules in the category O and (Lₒ₋ (₂|₁) (12, 0), ω_ξ) is not rational.
Li et al. (Wed,) studied this question.