Citation: Maurer, Fabian, and Ulrich Thiel. "Computing the center of a fusion category." arXiv preprint arXiv:2406.13438 (2024).
Web: https://arxiv.org/abs/2406.13438
Tags: Mathematical, Monoidal-categories, Non-finite/semisimple, Modular-tensor-categories
In this paper, the authors present an algorithm for computing the Drinfeld center of a fusion category. The main takeaway from the paper is that you should NOT do a brute-force search through the objects in your category looking for half-braidings. Naively this is possible, because the categorical constraints on the half-braidings can be reduced to algebraic expressions. However, the authors give a great example over Vec_{S3} of how messy these algebraic expressions can be. Certainly not doable by hand, and hardly doable (they say) by computer.
This claim that brute force searches are bad should be taken with a grain of salt. A big selling point of this paper is that they work over non algebraically closed fields, which can cause all sorts of problems and subtleties. In particular, they look at the Ising category. It's F-symbols are all defined over Q(sqrt(2)), and so it gives a fusion category over Q(sqrt(2)). However, the braiding and twist factors are only defined over Q adjoint a 16th root of unity! This means that, as a category over Q(sqrt(2)), Ising is not a spherical fusion category and it is not a braided fusion category. They show that the Drinfeld center of Ising as a Q(sqrt(2)) fusion category only has five simple objects, and the global quantum dimension of this doubled Ising is much bigger than expected.
Perhaps the reason brute force doesn't work is because they are considering these nasty situations, and in the algebraically closed case brute force would be more reasonable.
The way the authors claim we should compute Drinfeld centers is via the induction function, defined as the adjoint of the forgetful functor. This functor is very nice, and takes every object in a category to some object in its Drinfeld center. The relevant theorem here is that every simple object in the Drinfeld center is the subobject of the induction functor acting on some simple object in the original category. This means that the algorithm works by first computing the induction functor on all simple objects (which is doable, since the induction functor has an explicit description), then breaking the images down into direct sums, and the recording the results.
A key reference for this work is an earlier paper by Brugieres and Virelizier, which treats the Drinfeld center in high-detail:
> Bruguieres, Alain, and Alexis Virelizier. "On the center of fusion categories." Pacific Journal of Mathematics 264.1 (2013): 1-30.