Does anyone know of any work on "reversing the Grothendieck ring construction" in the sense of constructing (and classifying) k-linear monoidal categories with a given Grothendieck ring? The only results I know in this direction are results of Kazhdan and Kerler which show that any C-linear tortile category with the same Grothendieck ring as Rep(U(sl_2)) must be C-linear monoidally equivalent to Rep(U_q(sl_2)) for some q, and a result (probably old, though I know no source other than my own derivation) that shows that semisimple C-linear monoidal categories with Grothendieck ring Z[G] for a finite group G are in 1-1 correspondence with pairs (\alpha, \phi) where \alpha is a U(1)-valued 3-cocycle, and \phi is a complex number (the multiple of Id_1 which give the component at 1 of the right- and left- identity transformations). (This latter shows that "twisted" Dijkgraaf-Witten theory is a TQFT of Turaev-Viro type.) Surely there is a citation in the literature for "my" result cited above, but where? Are there any other results along these lines? --David Yetter