On Hopf algebras of dimension $P^{3}$

Abstract

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 also give 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 results from [1] and [4] 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.