On braided tensor categories

Abstract

We investigate invertible elements and gradings in braided tensor categories. This leads us to the definition of theta-, product-, subgrading and orbitcategories in order to construct new families of BTC's from given ones. We use the representation theory of Hecke algebras in order to relate the fusionring of a BTC generated by an object $X$ with a two component decomposition of its tensorsquare to the fusionring of quantum groups of type $A$ at roots of unity. We find the condition of `local isomorphie' on a special fusionring morphism implying that a BTC is obtained from the above constructions applied to the semisimplified representation category of a quantum group. This family of BTC's contains new series of twisted categories that do not stem from known Hopf algebras. Using the language of incidence graphs and the balancing structure on a BTC we also find strong constraints on the fusionring morphism. For Temperley Lieb type categories these are sufficient to show local isomorphie. Thus we obtain a classification for the subclass of Temperley Lieb type categories.