Computing the center of a fusion category

Abstract

We present an algorithm for explicitly computing the categorical (Drinfeld) center of a pivotal fusion category. Our approach is based on decomposing the images of simple objects under the induction functor from the category to its center. We have implemented this algorithm in a general-purpose software framework TensorCategories.jl for tensor categories that we develop within the open-source computer algebra system OSCAR. We compute explicit models for the centers in form of the tuples $(X,{\hat{I}}^3)$ where $X$ is an object and ${\hat{I}}^3$ is a half-braiding. From these models we can compute the $F$-symbols and $R$-symbols. Using the data from the AnyonWiki, we were able to compute the center together with its $F$-symbols and $R$-symbols for all the 279 multiplicity-free fusion categories up to rank 5, and furthermore some chosen examples of rank 6, including the Haagerup subfactor (presented in a separate paper).

Expanded version: Added computation of the center of all multiplicity-free fusion categories up to rank 5 together with their F-symbols and R-symbols