Abstract for MSRI Preprint 2002-004

From Subfactors to Categories and Topology II. The Quantum Double of Tensor Categories and Subfactors

Michael Müger

We consider $F$-linear tensor categories $C$ with simple unit and finitely many isomorphism classes of simple objects. We assume that $C$ is either a $*$-category or semisimple and spherical over an algebraically closed field $F$. In the latter case we assume that $\dim C=\sum_i d(X_i)^2$ is non-zero, where the summation runs over the isomorphism classes of simple objects. We prove:

1. $Z(C)$ is a semisimple spherical (or $*$-) category.
2. $Z(C)$ is weakly monoidally Morita equivalent to $C \times C^{op}$. This implies $\dim Z(C)=(\dim C)^2$.
3. $Z(C)$ is modular and the Gauss sums are given by $\Delta_+/-(Z(C))=\sum_i \theta(X_i)^{\pm1}d(X_i)^2=\dim C$.
4. If $C$ is already modular then $Z(C)\simeq C X \tilde C$, where $\tilde C$ is the tensor category $C$ with the braiding $\tilde c_{X,Y}=c_{Y,X}^{-1}$.