# Fusion categories in terms of graphs and relations

### Hendryk Pfeiffer

The University of British Columbia, Vancouver, Canada

## Abstract

Every fusion category $C$ that is $k$-linear over a suitable field $k$ is the category of finite-dimensional comodules of a weak Hopf algebra $H$. This weak Hopf algebra is finite-dimensional, cosemisimple and has commutative bases. It arises as the universal coend with respect to the long canonical functor $ω:C→Vect_{k}$. We show that $H$ is a quotient $H=H[G]/I$ of a weak bialgebra $H[G]$ which has a combinatorial description in terms of a finite directed graph $G$ that depends on the choice of a generator $M$ of $C$ and on the fusion coefficients of $C$. The algebra underlying $H[G]$ is the path algebra of the quiver $G×G$, and so the composability of paths in $G$ parameterizes the truncation of the tensor product of $C$. The ideal $I$ is generated by two types of relations. The first type enforces that the tensor powers of the generator $M$ have the appropriate endomorphism algebras, thus providing a Schur–Weyl dual description of $C$. If $C$ is braided, this includes relations of the form ‘$RTT=TTR$’ where $R$ contains the coefficients of the braiding on $ωM⊗ωM$, a generalization of the construction of Faddeev–Reshetikhin–Takhtajan to weak bialgebras. The second type of relations removes a suitable set of group-like elements in order to make the category of finite-dimensional comodules equivalent to $C$ over all tensor powers of the generator $M$. As examples, we treat the modular categories associated with $U_{q}(sl_{2})$.

## Cite this article

Hendryk Pfeiffer, Fusion categories in terms of graphs and relations. Quantum Topol. 2 (2011), no. 4, pp. 339–379

DOI 10.4171/QT/24