Towards a classification of rational Hopf algebras

Rational Hopf algebras (certain quasitriangular weak quasi-Hopf $^*$-algebras) are expected to describe the quantum symmetry of rational field theories. In this paper methods are developped which allow for a classification of all rational Hopf algebras that are compatible with some prescribed set of fusion rules. The algebras are parametrized by the solutions of the square, pentagon and hexagon identities. As examples, we classify all solutions for fusion rules with two or three sectors, and for the level three affine $A_1$ fusion rules. We also establish several general properties of rational Hopf algebras, and we present a graphical description of the coassociator in terms of labelled tetrahedra which allows to make contact with conformal field theory fusing matrices and with invariants of three-manifolds and topological lattice field theory.