Tensor categories with fusion rules of self-duality for finite abelian groups
Split semisimple tensor categories are a special class of monoidal categories. They have associated an algebra which is called the fusion algebra of the given category. Let \(A\) be a finite group, \(S\) the disjoint union of \(A\) with the element \(m\) and \({\mathbb Z}[S]\) the algebra obtained by extension of the group algebra \({\mathbb Z}[G]\) whose product satisfies \(m^2 = \sum_{a\in A}a\) and \(am = ma = m\) for all \(a\in A\). The main result of this paper classifies split semisimple tensor categories over a field \(K\) whose fusion algebra is \({\mathbb Z}[S]\). As an application is obtained the existence of four classes of semisimple tensor categories over \({\mathbb C}\) when \(A = {\mathbb Z}_2 \oplus {\mathbb Z}_2 \). Three of these are realized as the class of the category of representations of Hopf algebras of dimension \(8\) (dihedral group algebra, quaternion group algebra and the Kac and Paljutkin algebra). So these Hopf algebras have the same fusion algebra but different classes of categories of representations.
scientific article; zbMATH DE number 1247622