From Subfactors to Categories and Topology II. The quantum double of tensor categories and subfactors

TL;DR

This paper generalizes the concept of the quantum double for tensor categories, establishing their properties, structure, and relation to topological quantum field theories, extending prior results from Hopf algebras to broader categorical contexts.

Contribution

It proves that the center of certain tensor categories is semisimple, spherical, and modular, and relates it to known algebraic and topological structures, extending previous work on quantum doubles.

Findings

Z(C) is a semisimple spherical (or *-) category.

Z(C) is Morita equivalent to C x C^op, with dim Z(C) = (dim C)^2.

Number of simple objects in Z(C) matches the torus state space dimension.

Abstract

We are concerned with the center (=quantum double) of tensor categories and prove generalizations of several results proven previously for quantum doubles of Hopf algebras. 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. there is a positive *-operation on the morphisms) 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: (i) Z(C) is a semisimple spherical (or *-) category. (ii) Z(C) is weakly monoidally Morita equivalent (in the sense of math.CT/0111204) to C X 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\times S^1} of the torus in the triangulation TQFT built from C. (iv) We prove that Z(C) is modular and we compute the Gauss sums Delta_+/-(Z(C))=sum_i theta(X_i)^{+/- 1}d(X_i)^2=dim C. (v) Finally, if C is already modular then Z(C)\simeq C X C~, where C~ is the tensor category C with the braiding c~_{X,Y}=c_{Y,X}^{-1}.