Hopf-Ore Extensions and Hopf Algebras of Rank One

TL;DR

This paper classifies rank one pointed Hopf algebras and Hopf-Ore extensions over arbitrary fields, detailing their module structures, tensor product decompositions, and categorical properties.

Contribution

It provides a comprehensive classification of rank one pointed Hopf algebras and modules over Hopf-Ore extensions, including tensor product rules and categorical structures.

Findings

Rank of Hopf-Ore extensions is 1, 2, or infinite.

Finite dimensional simple modules are classified.

Tensor product decompositions of simple modules are described.

Abstract

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 $H'$.