Module category and $C_2$-cofiniteness of affine vertex operator superalgebras

In this paper, we investigate the Lie algebra structures of weight one subspaces of $C_2$-cofinite vertex operator superalgebras. We also show that for any positive integer $k$, vertex operator superalgebras $L_{sl(1|n+1)}(k,0)$ and $L_{osp(2|2n)}(k,0)$ have infinitely many irreducible admissible modules. As a consequence, we give a proof of the fact that $L_{\mathfrak g}(k,0)$ is $C_2$-cofinite if and only if $\mathfrak g$ is either a simple Lie algebra, or $\mathfrak g=osp(1|2n)$, and $k$ is a nonnegative integer. As an application, we show that $L_{G(3)}(1,0)$ is a vertex operator superalgebra such that the category of $L_{G(3)}(1,0)$-modules is semisimple but $L_{G(3)}(1,0)$ is not $C_2$-cofinite.