Module category and C2-cofiniteness of affine vertex operator superalgebras

In this paper, we investigate the Lie algebra structures of weight one subspaces C2-cofinite vertex operator superalgebras. We also show that for any positive integer k, superalgebras Lsl(1|n+1)(k,0) and Losp(2|2n)(k,0) have inequivalent infinitely many irreducible admissible modules. As a consequence, give proof fact Lg(k,0) is if only g either simple algebra, or g=osp(1|2n), k nonnegative integer. an application, LG(3)(1,0) superalgebra such category ordinary LG(3)(1,0)-modules semisimple but not C2-cofinite.