Fusion categories in terms of graphs and relations

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 'RT T = T T R' 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 ).Proposition 4.12. Let C, M and G be as in Theorem 3.8 and E = (E (n) ) n∈AE 0 be an endomorphism system for C with respect to M .

Weak Bialgebras of fractions

“…Let us call them SL q (2), depending on the root of unity q. In [7], we have constructed a WBA H with a central group-like element det q in such a way that SL q (2) agrees with the quotient of H modulo the relation det q = 1. Let us call this WBA H = M q (2) in analogy with Example 4.1.…”

“…This WBA is new and is presented here for the first time. Let us begin by describing M q (2) following [7]. Example 4.6.…”

“…In the broader context of a research programme, Example 4.7 can be seen as the main motivation for the present article. From our work on fusion categories [7], we know that we can obtain a dimension graph G from each fusion category C with a choice of a monoidal generating object M ∈ |C|. There is a 'free' WBA H[G] associated with G and a quotient of H[G] modulo some RT T -relations that plays the role of M q (2).…”

“…Weak Bialgebras are relevant to the study of modular categories [5] and, more generally, fusion categories, see, for example [6,7]. We use our construction of Weak Bialgebras of fractions in order to introduce new Weak Bialgebras whose categories of finite-dimensional comodules relate to SL 2 -fusion categories in the same way as GL(2) relates to SL (2).…”

Algebraic properties of face algebras

“…The main idea of the proof is to consider the Kronecker square Q of the quiver Q [Definition 3.1], and realize H(Q) as the path algebra k Q [Proposition 3.4] (see also [Pfe11,Remark 3.3]). Then, the theorem above follows from (i) prior results on relating algebraic properties of a path algebra with graph-theoretic properties of the underlying quiver [Proposition 2.5] and (ii) a careful analysis of graph-theoretic properties that are shared between a quiver and its Kronecker square [Proposition 3.2].…”