On braided tensor categories

TL;DR

This paper explores the structure of braided tensor categories, introduces new constructions like theta- and orbitcategories, and classifies Temperley-Lieb type categories using fusion ring analysis and quantum group representations.

Contribution

It defines new categories from existing ones and establishes conditions for their construction, providing a classification for Temperley-Lieb type categories.

Findings

New families of braided tensor categories constructed from existing ones.

Conditions for local isomorphie linking categories to quantum groups.

Complete classification of Temperley-Lieb type 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.