# Hopf algebras vs Fusion categories for topological order

Disclaimer: Before I begin with the question I want to warn that some people would argue that it is a math question and not a physics question. However, it finds it origins in the study of topological order using tensor networks and hence I would argue it counts as mathematical physics.

Now, on to the real question. In the last decade it has become clear that the tensor network framework is very powerful for describing topologically ordered materials and anyons. One of the central notions in this story is that of a Matrix Product Operator (MPO) and the related MPO-symmetries. The main property of these objects is the so-called pulling-through condition (see figure below for the case of hexagonal lattices).

Legend: The black dots denote the individual operators with the red line indicating the MPO. The crossing in the middle denotes a PEPS tensor on a given lattice site.

Here one has a string-like operator that one can pull through the lattice and that acts as some kind of defect operator (or anyon). As is clear from the equation in the picture, one needs an algebraic operation to go from one tensor/operator to a product of tensors/operators, i.e. one needs a coalgebra structure. In the literature it has been argued (shown) that the resulting structure is that of a Hopf algebra.

Now on the other hand it is also well-known that anyonic theories are described by fusion categories [1, 2]. Mathematicians would tell us that by Tannaka duality these fusion categories can be obtained as representation categories of (weak) Hopf algebras.

We are almost there now. The main question is "how I can relate these two stories". The MPO's in the fusion category framework are generally labeled by simple objects. However, simple objects in those categories are representations of Hopf algebras and not elements of Hopf algebras as in the first(ish) paragraph. So I don't see how I can obtain the Hopf algebra description?

1. Originally these arguments came from topological and conformal field theories, but in recent work this has also been obtained in the tensor network framework.

2. In fact modular tensor categories, but that is not important here (at least as far as I can see).