  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 $L_k(\mathfrak{sl}_2)$ at any non-integral
admissible level $k$. In particular, we show that the principal block of the
category of weight modules over admissible $L_k(\mathfrak{sl}_2)$ is equivalent
to that of the corresponding (unrolled) small quantum group.
