Finite-dimensional Hopf algebras of rank one in characteristic zero

Abstract

Let H be a finite-dimensional pointed Hopf algebra over an algebraically closed field k generated as an algebra by the first term H 1 of its coradical filtration. Generally for a Hopf algebra whose coradical H 0 is a sub-Hopf algebra we assign a measure of complexity to H which we refer to as the rank of H. We obtain a presentation of pointed Hopf algebras H of rank one by generators and relations. In the special case when G ( H ) is an abelian group and k has characteristic 0 we classify the finite-dimensional indecomposable H-modules, determine all simple left modules of the Drinfel'd double D ( H ) of H, their projective covers, and the blocks of D ( H ) .