Tensor categories of weight modules of sl2$\widehat{\mathfrak {sl}}_2$ at admissible level

Abstract The category of weight modules of the simple affine vertex algebra of at an admissible level is neither finite nor semisimple and modules are usually not lower‐bounded and have infinite‐dimensional conformal weight subspaces. However, this vertex algebra enjoys a duality with , the simple principal ‐algebra of at level (the super conformal algebra) where the levels are related via . Every weight module of is lower‐bounded and has finite‐dimensional conformal weight spaces. The main technical result is that every weight module of is ‐cofinite. The existence of a vertex tensor category follows and the theory of vertex superalgebra extensions implies the existence of vertex tensor category structure on for any admissible level . As applications, the fusion rules of ordinary modules with any weight module are computed, and it is shown that is a ribbon category if and only if is, in particular, it follows that for admissible levels and and , the category is a ribbon category.