Fusion operators and cocycloids in monoidal categories
The Yang-Baxter equation has been studied extensively in the context of monoidal categories. The fusion equation, which appears to be the Yang-Baxter equation with a term missing, has been studied mainly in the context of Hilbert spaces. This paper endeavours to place the fusion equation in an appropriate categorical setting. The author begins by establishing the bijection between fusion operators and 3-cocycles, and shows how these arise from bialgebras. Some constructions with 3-cocycles are described showing, in particular, the relation to the Stasheff-MacLane pentagon. An object is called a tricocycloid if it is equipped with an invertible 3-cocycle. It should be regarded as a generalized Hopf algebra, and hence, as a generalized (quantum) group. By considering representations and using techniques of Tannaka duality, Hopf algebras are associated with tricocycloids. And, the author makes some speculations on higher cocycloids. Finally, the generic tricocycloid is constructed.
scientific article; zbMATH DE number 1195511