Weight representations of affine Kac-Moody algebras and small quantum groups

We study the weight modules over affine Kac-Moody algebras from the view point of vertex algebras, and determine the abelian category of weight modules for the simple affine vertex algebra $L_k\left({\mathrm{sl}}_2\right)$ at any non-integral admissible level k. In particular, we show that the principal block of the category of weight modules over admissible $L_k\left({\mathrm{sl}}_2\right)$ is equivalent to that of the corresponding (unrolled) small quantum group.