To put the results of this paper in a perspective, this reviewer would like to recall the present situation of the classification of complex semisimple Hopf algebras. This classification revealed to be very hard, because of the lack of powerful general results. There is, however, a series of conjectures whose solution would shed some light into the fine structure of semisimple Hopf algebras. First, it would be very useful to have a positive answer to the Kaplansky conjecture: The dimension of an irreducible module of a semisimple Hopf algebra divides the dimension of the Hopf algebra itself. For finite groups, this is a classical result of Frobenius. Several important instances where this conjecture is true are known, but the general statement is still open. Next, the first folkloric conjecture asks whether any complex semisimple Hopf algebra has a form over a ring of algebraic integers; this conjecture implies Kaplansky’s conjecture by a result of Larson in 1972. The second folkloric conjecture asks whether any complex semisimple Hopf algebra can be build up from finite groups via some standard operations like duals, extensions, twisting, etc. A form of this conjecture is Question 2.6 in [N. Andruskiewitsch, Contemp. Math. 294, 1–57 (2002)]; some adjustments to this statement might be needed. Both conjectures are natural and appeared to many algebraists, hence the adjective “folkloric”. Although the second folkloric conjecture is somewhat imprecise, it is likely that it implies the first, and then also Kaplansky’s. Recently, a fourth conjecture on the structure of complex semisimple Hopf algebras was proposed by P. Etingof, D. Nikshych and V. Ostrik [math.QA/0203060 (2002)]; let us call it the ENO conjecture. In V. Ostrik [math.QA/0202130 (2002)], a semisimple Hopf algebra was constructed out from a collection \((\Sigma,F,G,\omega,\sigma,\tau)\), where \(\Sigma\) is a finite group, \(F\) and \(G\) are subgroups of \(\Sigma\), \(\omega\), resp. \(\sigma\), \(\tau\), is a 3-cocycle on \(\Sigma\), resp. a 2-cocycle on \(F\), \(G\), with values in \(\mathbb{C}^\times\); such semisimple Hopf algebras are called group-theoretical (an analogous construction was announced by A. Ocneanu in several talks, naming them “baby examples”). The ENO conjecture asks whether any complex semisimple Hopf algebra is group-theoretical; actually, they just ask whether there exists a semisimple Hopf algebra which is not group-theoretical.
The first result of the present paper is that a semisimple Hopf algebra is group-theoretical if and only if its Drinfeld double is a twisting of a Dijkgraaf-Pasquier-Roche quasi-Hopf algebra. Roughly speaking, this says also that the ENO conjecture implies the second folkloric conjecture. (It can be shown directly that the ENO conjecture implies Kaplansky’s conjecture.) It is also shown that an Abelian extension of Hopf algebras, that is a Hopf algebra of the form \(\mathbb{C}^G{^\tau\#_\sigma}\mathbb{C} F\) where \(\Sigma=FG\) is an exact factorization of a finite group and \((\sigma,\tau)\) is a compatible pair of cocycles, is group-theoretical; the 3-cocycle \(\omega\) is read off from the Kac exact sequence. The proof follows the lines of a more general result of P. Schauenburg [Adv. Math. 165, No. 2, 194–63 (2002; Zbl 1006.16054)].