Fusion categories in terms of graphs and relations

TL;DR

This paper presents a combinatorial approach to describing fusion categories using graphs and relations, connecting them to Weak Bialgebras and providing explicit constructions for categories like U_q(sl_2).

Contribution

It introduces a graph-based description of Weak Bialgebras underlying fusion categories, generalizing known constructions to the weak case and including braided categories.

Findings

Fusion categories are described via quotients of Weak Bialgebras with graph-based combinatorics.

The construction generalizes Faddeev-Reshetikhin-Takhtajan relations to Weak Bialgebras.

Explicit examples for U_q(sl_2) demonstrate the applicability of the approach.

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 \omega: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 GxG, 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 \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 over all tensor powers of the generator M. As examples, we treat the modular categories associated with U_q(sl_2).