A counterpart of the Verlinde algebra for the small quantum group

Abstract

Let $\bar{Pr}$ denote the ideal spanned by the characters of projective modules in the Grothendieck ring of the category of finite dimensional modules over the small quantum group $u_l$. We show that $\bar{Pr}$ admits a description completely parallel to that of the Verlinde algebra of the fusion category, restricted over $u_l$, with the character of the Steinberg module playing the role of the identity.