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.Comment: 27 pages, minor changes. Accepted for publication in the Tsukuba
Journal of Mathematic
