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

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

2001-11-19

arxiv.org/abs/math/0111205v1

J. Pure Appl. Alg. 180, 159-219 (2003)

Mathematics

Category Theory

latex2e, ca. 62 pp. Requires diagrams.tex V3.88

Scientific paper

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}.

Affiliated with

Also associated with

No associations

Rating

From Subfactors to Categories and Topology II. The quantum double of tensor categories and subfactors does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with From Subfactors to Categories and Topology II. The quantum double of tensor categories and subfactors, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and From Subfactors to Categories and Topology II. The quantum double of tensor categories and subfactors will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-394773