Fusion Rings and Tensor Categories

Abstract

The definition of a fusion ring F [1], [2], [3] is an abstraction of the properties of the Grothendieck ring K 0(C) of a rigid braided semisimple monoidal category C. For certain issues it is convenient to pass to an algebra (over the complex numbers) thus a fusion algebra F is a unital associative and commutative algebra with a chosen basis I such that the fusion rules N ab c , a, b, c ∈ i.e., the structure constants in this basis, a · b = ∑c N ab c c are in Z + their is an involutive automorphism a → ā such that N> ab 1 = δ a, b The set/corresponds to the sectors, i.e., the equivalence classes of simple objects or irreps, the monoidal structure in C is responsible for the structure of unital associative ring, the braiding for the commutativity, while the rigidity translates in the involutative automorphism.