FROM SUBFACTORS TO CATEGORIES AND TOPOLOGY II. THE QUANTUM DOUBLE OF TENSOR CATEGORIES AND

SUBFACTORS

MICHAEL MÜGER

Abstract. 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 = ∑_{i} d(X_{i})^{2} is non-zero, where the summation runs over the isomorphism classes of simple objects. We prove: (i) Z(C) is a semisimple spherical (or ∗-) category. (ii) Z(C) is weakly monoidally Morita equivalent to C × C^{op}. This implies dim Z(C) = (dim C)^{2}. (iii) Z(C) is modular and the Gauss sums are given by Δ_{+}/ − (Z(C)) = ∑_{i} θ(X_{i})^{±1}d(X_{i})^{2} = dim C. (iv) If C is already modular then Z(C) ≃ CXC̃, where C̃ is the tensor category C with the braiding c̃_{X,Y} = c_{Y,X}^{−1}.