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) .