Non-Tannakian Categories in Quantum Field Theory

Abstract

This article is an excellent survey of results on braided tensorcategories. After reviewing definitions, the author discussesTannaka type duality results for semisimple, Tannakian and weaklyTannakian categories. The core of the paper, however, deals withnon Tannakian categories: specifically, a quotient procedure isintroduced which provides a semisimple quotient of any rigidtensor category, providing in particular a consistent truncationof the category of representations of any Hopf algebra (notnecessarily semisimple). Examples of non Tannakian categoriesarising from {\rm SU}(2) WZW models and by quotienting{\rm Rep} (U\sb q({\rm sl}\sb 2)) at a root of unity arediscussed, and shown to be isomorphic for q =\exp(iπ/(k+2)). A uniqueness result for reconstruction ofU\sb q({\rm sl}\sb 2) from the truncated category is proved.The introduction and conclusion contain discussions of the roleof non Tannakian tensor categories in quantum fieldtheory.\par {For the entire collection see MR 93h:81002.}