By weakening the counit and antipode axioms of a C*-Hopf algebra and allowing for the coassociative coproduct to be nonunital, we obtain a quantum group, that we call a weak C*-Hopf algebra, which is sufficiently general to describe the symmetries of essentially arbitrary fusion rules. This amounts to generalizing the Baaj-Skandalis multiplicative unitaries to multipicative partial isometrics. Every finite-dimensional weak C*-Hopf algebra has a dual which is again a weak C*-Hopf algebra. An explicit example is presented with Lee-Yang fusion rules. We briefly discuss applications to amalgamated crossed products, doubles, and quantum chains.
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics