Summary: A fusion category over \(\mathbb{C}\) is a \(\mathbb{C}\)-linear semisimple rigid tensor category with finitely many simple objects and finite-dimensional spaces of morphisms such that the neutral object is simple [see (*) P. Etingof, D. Nikshych and V. Ostrik, “On fusion categories” (to appear in Ann. Math.]. To every fusion category, one can attach a positive number called the Frobenius-Perron (FP) dimension of this category [(*), Section 8]. It is an interesting and challenging problem to classify fusion categories of a given FP dimension \(D\). This problem is easier if \(D\) is an integer, and for integer \(D\), its complexity increases with the number of prime factors in \(D\). Specifically, fusion categories of FP dimension \(p\) or \(p^2\), where \(p\) is a prime, were classified in [(*), Section 8]. The next level of complexity is fusion categories of FP dimension \(pq\), where \(p<q\) are distinct primes. In this case, the classification has been known only in the case when the category admits a fiber functor, that is, it is a representation category of a Hopf algebra [P. Etingof and S. Gelaki, J. Algebra 210, 664–669 (1998; Zbl 0919.16028)].
In this paper, we provide a complete classification of fusion categories of FP dimension \( pq\), thus giving a categorical generalization of Etingof-Galaki (see above). As a corollary, we also obtain the classification of semisimple quasi-Hopf algebras of dimension \(pq\). A concise formulation of our main result is the following theorem: let \({\mathcal C}\) be a fusion category over \(\mathbb{C}\) of FP dimension \(pq\), where \(p<q\) are distinct primes. Then either \(p=2\) and \({\mathcal C}\) is a Tambara-Yamagami category of dimension \(2q\) [D. Tamdeoz and S. Yamagami, J. Algebra 209, 692–707 (1998; Zbl 0923.46052)], or \({\mathcal C}\) is group theoretical in the sense of (*).