Representations of Hopf-Ore Extensions of Group Algebras and Pointed Hopf Algebras of Rank One

In this paper, we study the representation theory of Hopf-Ore extensions of group algebras and pointed Hopf algebras of rank one over an arbitrary field k. Let H=k G(chi,a,delta) be a Hopf-Ore extension of k G and H (') a rank one quotient Hopf algebra of H, where k is a field, G is a group, a is a central element of G and chi is a k-valued character for G with chi(a)not equal 1. We first show that the simple weight modules over H and H (') are finite dimensional. Then we describe the structures of all simple weight modules over H and H ('), and classify them. We also consider the decomposition of the tensor product of two simple weight modules over H (') into the direct sum of indecomposable modules. Furthermore, we describe the structures of finite dimensional indecomposable weight modules over H and H ('), and classify them. Finally, when chi(a) is a primitive n-th root of unity for some na parts per thousand yen2, we determine all finite dimensional indecomposable projective objects in the category of weight modules over H (').