From subfactors to categories and topology II: The quantum double of tensor categories and subfactors

Abstract

For every tensor category C there is a braided tensor category Z( C) , the ‘center’ of C . It is well known to be related to Drinfel'd's notion of the quantum double of a finite dimensional Hopf algebra H by an equivalence Z(H -mod) ≃ ⊗ br D(H) -mod of braided tensor categories. In the Hopf algebra situation, whenever D( H)-mod is semisimple (which is the case iff D( H) is semisimple iff H is semisimple and cosemisimple iff S 2=id and char F∤ dim H ) it is modular in the sense of Turaev, i.e. its S-matrix is invertible. (This was proven by Etingof and Gelaki in characteristic zero. We give a fairly general proof in the appendix.) The present paper is concerned with a generalization of this and other results to the quantum double (center) of more general tensor categories. 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 (i.e. F= C and there is a positive ∗-operation on the morphisms) or semisimple and spherical over an algebraically closed field F . In the latter case we assume dim C≡∑ id(X i) 2≠0 , where the summation runs over the isomorphism classes of simple objects. We prove that Z( C) (i) is a semisimple spherical (or ∗-) category and (ii) is weakly monoidally Morita equivalent (in the sense of Müger (J. Pure Appl. Algebrea 180 (2003) 81–157)) to C⊗ F C op . This implies dim Z( C)=( dim C) 2 . (iii) We analyze the simple objects of Z( C) in terms of certain finite dimensional algebras, of which Ocneanu's tube algebra is the smallest. We prove the conjecture of Gelfand and Kazhdan according to which the number of simple objects of Z( C) coincides with the dimension of the state space H S 1×S 1 of the torus in the triangulation TQFT built from C . (iv) We prove that Z( C) is modular and we compute Δ ±( Z( C))≡∑ iθ(X i) ±1d(X i) 2= dim C . (v) Finally, if C is already modular then Z( C) ≃ ⊗ br C⊗ F C ̃ ≃ ⊗ C⊗ F C op , where C ̃ is the tensor category C with the braiding c ̃ X,Y=c Y,X −1 .