On Hopf algebras of dimension p³

We discuss some general results on finite-dimensional Hopf algebras over an algebraically closed field k of characteristic zero and then apply them to Hopf algebras H of dimension p^{3} over k. There are 10 cases according to the group-like elements of H and H^{*}. We show that in 8 of the 10 cases, it is possible to determine the structure of the Hopf algebra. We give also a partial classification of the quasitriangular Hopf algebras of dimension p^{3} over k, after studying extensions of a group algebra of order p by a Taft algebra of dimension p^{2}. In particular, we prove that every ribbon Hopf algebra of dimension p^{3} over k is either a group algebra or a Frobenius-Lusztig kernel. Finally, using some previous results on bounds for the dimension of the first term H_{1} in the coradical filtration of H, we give the complete classification of the quasitriangular Hopf algebras of dimension 27.