Non-Tannakian Categories in Quantum Field Theory

Abstract

We review the definitions of braided tensor-categories and relate them in the semisimple case to the structural data given by braid- and fusion-matrices. A number of duality-relations involving semisimple, Tannakian and weakly-Tannakian categories are summarized. We introduce a GNS-type construction to define the quotient of a rigid, abelian tensor-category onto a semisimple category. This yields a consistent calculus of truncated 6-j-symbols derived from non-semisimple Hopfalgebras. We illustrate how non-Tannakian categories are obtained in the examples of U q (sl 2), q a root of unity, and SU(2)-WZW-models. We show that the two categories are equal for \( q\; = \;\exp \left( {\frac{{i\pi }}{{k + 2}}} \right) \) and that U q (sl 2) is unique as a dual Hopf algebra if the fundamental representation is required to be two-dimensional. This leads us to formulate a duality-problem for non-Tannakian categories.