  By weakening the counit and antipode axioms of a C*-Hopf algebra and allowing
for the coassociative coproduct to be non-unital 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 isometries. Every 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 shortly discuss applications to amalgamated crossed products, doubles, and
quantum chains.
