Fusion in the Extended Verlinde Algebra

The features of a usual conformal field theory were captured by the formalism of a tensor or fusion category. The notion of a Verlinde algebra which arises from a fusion category, the action of a modular group on the algebra, and the Verlinde formula gave us beautiful means to study these theories. Such they were that in [Ki] Kirillov put forth the notion of a G-equivariant fusion category so that it generalized the formalism of a fusion category. Roughly, a G-equivariant fusion category C additionally possesses a G-grading and -action satisfying similar functorality properties which respect the group grading and action. Unlike the fusion category there are not braiding isomorphisms but rather, as Turaev [Tu] called them, G-crossed braidings: Denote the action of g P G on an object V P Ch by V . Then for V P Cg and W P Ch the isomorphism corresponding to the crossed braiding is σ : V bW€ÑgW b V . Then Kirillov generalized the Verlinde algebra, giving us the so-called extended Verlinde algebra, which as he demonstrated can, like the Verlinde algebra, too have the action of the a modular group defined on it. Figure 1 summarizes this development.