Weak C∗-hopf Algebras: the Coassociative Symmetry of Non-integral Dimensions

By allowing the coproduct to be non-unital and weakening the counit and antipode axioms of a C∗-Hopf algebra too, we obtain a selfdual set of axioms describing a coassociative quantum group, that we call a weak C∗-Hopf algebra, which is sufficiently general to describe the symmetries of essentially arbitrary fusion rules. It is the same structure that can be obtained by replacing the multiplicative unitary of Baaj and Skandalis with a partial isometry. The algebraic properties, the existence of the Haar measure and representation theory are briefly discussed. An algorithm is explained how to construct examples (in particular ones with non-integral dimensions) from non-Abelian cohomology.