Rational Hopf Algebras: Polynomial Equations, Gauge Fixing, and Low-Dimensional Examples

Rational Hopf algebras, i.e. certain quasitriangular weak quasi-Hopf algebras whose representations form a tortile modular C* category, are expected to describe the quantum symmetry of rational field theories. In this paper the essential structure (hidden by a large gauge freedom) of rational Hopf algebras is revealed. This allows one to construct examples of rational Hopf algebras starting only from the corresponding fusion ring. In particular we classify all solutions for fusion rules with not more than three sectors, as well as for the level 3 affine A$^{(1)}_1$ fusion rules