Modular categories are not determined by their modular data

A not on modular hyperconvex space

Finitely semisimple spherical categories and modular categories are self - dual

We show that every essentially small finitely semisimple k-linear additive spherical category in which k = End(1) is a field, is equivalent to its dual over the long canonical forgetful functor. This includes the special case of modular categories. In order to prove this result, we show that the universal coend of the spherical category with respect to the long forgetful functor is self-dual as...

On Classification of Modular Categories by Rank

The feasibility of a classification-by-rank program for modular categories follows from the Rank-Finiteness Theorem. We develop arithmetic, representation theoretic and algebraic methods for classifying modular categories by rank. As an application, we determine all possible fusion rules for all rank=5 modular categories and describe the corresponding monoidal equivalence classes.

a not on modular hyperconvex space

Verlinde conjecture and modular tensor categories

Let V be a simple vertex operator algebra satisfying the following conditions: (i) V(n) = 0 for n < 0, V(0) = C1 and the contragredient module V ′ is isomorphic to V as a V -module. (ii) Every N-gradable weak V -module is completely reducible. (iii) V is C2-cofinite. We announce a proof of the Verlinde conjecture for V , that is, of the statement that the matrices formed by the fusion rules amo...