Iterated Hopf Ore Extensions over Group Rings

TL;DR

This paper introduces a new class of Hopf algebras constructed as iterated Ore extensions over group rings, unifying and generalizing known families, and provides their module classification and structural properties.

Contribution

It defines and studies a broad class of Hopf algebras via iterated Ore extensions, extending previous constructions and classifying their simple modules.

Findings

Unified framework for various Hopf algebras

Classification of finite-dimensional simple modules

Analysis of ring-theoretic properties

Abstract

We introduce and study a class of Hopf algebras $H(G, \chi, \eta, b, c, \beta)$ which are two-step Ore extensions of a group algebra $\mathbb{K}[G]$. This construction unifies and generalizes some known families of Hopf algebras such as generalized Taft algebras and Hopf algebras related to $\mathfrak{sl}_2$ constructed by Wang, Wu, and Tan. We analyze the ring theoretical properties of these algebras and classify all finite dimensional simple modules over them.