Fusion operators as generalised Hopf structures

Ross Street – 13 November 1996

Since only one member of the audience admitted to having heard last year's talk on fusion operators, I reviewed my paper Fusion operators and cocycloids in monoidal categories (preprint, August 1995; submitted). A fusion operator V : A(tensor)A -~-> A(tensor)A on A gives rise to a 3-cocycle v : A(tensor)A -~-> A(tensor)A via v = c V where c is (at least) a braiding. A 3-cocycle is a 4-simplex in the nerve of the ambient braided monoidal category regarded as a 1-object 1-arrow tricategory. They also appear in the expression of coherence for associativity constraint of a promonoidal category. Each Hopf algebra H gives a fusion operator V on H which is the composite of (delta)(tensor)1 and 1(tensor)(mu). The unit and counit can be used to recapture the multiplication mu, the comultiplication delta, and the antipode (nu). This is the point of the talk's title. Ideas of Tannaka duality can be used to obtain a Hopf algebra from the monoidal category of modules over a tricocycloid (= object equipped with a 3-cocycle). I gave what I believe to be generators and relations for the free braided monoidal category on a single generating object bearing a tricocycloid structure. It would be nice to have a geometric model and/or something like a faithful representation by free group automorphisms as we do for the Baez-Birman singular (or welded) braids which are relevant to Vassiliev invariants for knots.