Fusion categories in terms of graphs and relations

Abstract

Every fusion category $\mathcal{C}$ 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 $\omega\colon\mathcal{C}\to \mathbf{Vect}_k$. We show that $H$ is a quotient $H=H[\mathcal{G}]/I$ of a weak bialgebra $H[\mathcal{G}]$ which has a combinatorial description in terms of a finite directed graph $\mathcal{G}$ that depends on the choice of a generator $M$ of $\mathcal{C}$ and on the fusion coefficients of $\mathcal{C}$. The algebra underlying $H[\mathcal{G}]$ is the path algebra of the quiver $\mathcal{G}\times\mathcal{G}$, and so the composability of paths in $\mathcal{G}$ parameterizes the truncation of the tensor product of $\mathcal{C}$. 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 $\mathcal{C}$. If $\mathcal{C}$ is braided, this includes relations of the form ‘$RTT=TTR$’ where $R$ contains the coefficients of the braiding on $\omega M\otimes\omega M$, 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 $\mathcal{C}$ over all tensor powers of the generator $M$. As examples, we treat the modular categories associated with $U_q(\mathfrak{sl}_2)$.