Representations of Hopf Ore extensions of group algebras and pointed Hopf algebras of rank one

Abstract

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 $H=kG(χ, a,\d)$ be a Hopf-Ore extension of $kG$ and $H^{\prime }$ 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 $\chi (a)\ne 1$. We first show that the simple weight modules over $H$ and $H^{\prime }$ are finite dimensional. Then we describe the structures of all simple weight modules over $H$ and $H^{\prime }$, and classify them. We also consider the decomposition of the tensor product of two simple weight modules over $H^{\prime }$ into the direct sum of indecomposable modules. Furthermore, we describe the structures of finite dimensional indecomposable weight modules over $H$ and $H^{\prime }$, and classify them. Finally, when $\chi (a)$ is a primitive $n$-th root of unity for some $n>2$, we determine all finite dimensional indecomposable projective objects in the category of weight modules over $H^{\prime }$.

arXiv admin note: substantial text overlap with arXiv:1206.3943

References in corpus (1)

• Verma and simple modules for quantum groups at non-abelian groups