Date: Wed, 15 Mar 1995 05:02:08 -0400 (AST)
Subject: reversing the Grothendeick ring construction? 

Date: Tue, 14 Mar 95 12:21:01 CST
From: David Yetter <dyetter@math.ksu.edu>

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