Hopf-Ore Extensions and Hopf Algebras of Rank One

In this paper, we study pointed rank one Hopf algebras and Hopf-Ore extensions of group algebras, over an arbitrary field $k$. It is proved that the rank of a Hopf-Ore extension of a group algebra is one or two or infinite. It is also shown that an arbitrary (finite or infinite dimensional) pointed Hopf algebra of rank one is isomorphic to a quotient of a Hopf-Ore extension of its coradical, a group algebra. We classify the finite dimensional simple modules and describe a family of indecomposable modules over a Hopf-Ore extension $H=kG(\chi, a,\delta)$ and its quotient $H'$ of rank one, where $\chi(a)\neq 1$, $G$ is an abelian group and $k$ is an algebraically closed field. The decomposition of the tensor products of two finite dimensional simple modules into a direct sum of indecomposable modules is given too. We also determine all simple objects and a family of indecomposable projective objects in the categories of all weight modules over $H$ and $H'$.