Tensor Categories with Fusion Rules of Self-Duality for Finite Abelian Groups

Semisimple tensor categories with fusion rules of self-duality for finite abelian groups are classified. As an application, we prove that the Tannaka duals of the dihedral and the quaternion groups of order 8 and the eight-dimensional Hopf algebra of Kac and Paljutkin are not isomorphic as abstract tensor categories whereas they have the same fusion rule.