A counterpart of the Verlinde algebra for the small quantum group

Abstract

Let $\overline {\Pr}$ denote the ideal spanned by the characters of projective modules in the Grothendieck ring of the category $\overline {\mathscr{C}\sb f}$ of finite dimensional modules over the small quantum group $U\sp {\rm fin}\sb q(\mathfrak {g})$. We show that $\overline {\Pr}$ admits a description completely parallel to that of the Verlinde algebra of the fusion category (see [AP]), with the character of the Steinberg module playing the role of the identity.