Fusion categories in terms of graphs and relations

  • Hendryk Pfeiffer

    The University of British Columbia, Vancouver, Canada


Every fusion category C\mathcal{C} that is kk-linear over a suitable field kk is the category of finite-dimensional comodules of a weak Hopf algebra HH. 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 ⁣:CVectk\omega\colon\mathcal{C}\to \mathbf{Vect}_k. We show that HH is a quotient H=H[G]/IH=H[\mathcal{G}]/I of a weak bialgebra H[G]H[\mathcal{G}] which has a combinatorial description in terms of a finite directed graph G\mathcal{G} that depends on the choice of a generator MM of C\mathcal{C} and on the fusion coefficients of C\mathcal{C}. The algebra underlying H[G]H[\mathcal{G}] is the path algebra of the quiver G×G\mathcal{G}\times\mathcal{G}, and so the composability of paths in G\mathcal{G} parameterizes the truncation of the tensor product of C\mathcal{C}. The ideal II is generated by two types of relations. The first type enforces that the tensor powers of the generator MM have the appropriate endomorphism algebras, thus providing a Schur–Weyl dual description of C\mathcal{C}. If C\mathcal{C} is braided, this includes relations of the form ‘RTT=TTRRTT=TTR’ where RR contains the coefficients of the braiding on ωMωM\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 C\mathcal{C} over all tensor powers of the generator MM. As examples, we treat the modular categories associated with Uq(sl2)U_q(\mathfrak{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