On group theoretical Hopf algebras and exact factorizations of finite groups

Abstract

We show that a semisimple Hopf algebra A is group theoretical if and only if its Drinfeld double is a twisting of the Dijkgraaf–Pasquier–Roche quasi-Hopf algebra D ω ( Σ), for some finite group Σ and some ω∈ Z 3( Σ, k ×). We show that semisimple Hopf algebras obtained as bicrossed products from an exact factorization of a finite group Σ are group theoretical. We also describe their Drinfeld double as a twisting of D ω ( Σ), for an appropriate 3-cocycle ω coming from the Kac exact sequence.