Representations of Hopf-Ore extensions of group algebras

TL;DR

This paper classifies finite-dimensional simple and indecomposable modules over Hopf-Ore extensions of group algebras, explores tensor product decomposition rules, and describes the Grothendieck and Green rings for specific cases.

Contribution

It provides a comprehensive classification of modules and ring structures for Hopf-Ore extensions of group algebras, including new decomposition rules and ring presentations.

Findings

Classified all finite-dimensional simple modules under different conditions.

Determined indecomposable modules for semisimple cases.

Described Grothendieck and Green rings explicitly.

Abstract

In this paper, we study the representations of the Hopf-Ore extensions $kG(\chi^{-1}, a, 0)$ of group algebra $kG$, where $k$ is an algebraically closed field. We classify all finite dimensional simple $kG(\chi^{-1}, a, 0)$-modules under the assumption $|\chi|=\infty$ and $|\chi|=|\chi(a)|<\infty$ respectively, and all finite dimensional indecomposable $kG(\chi^{-1}, a, 0)$-modules under the assumption that $kG$ is finite dimensional and semisimple, and $|\chi|=|\chi(a)|$. Moreover, we investigate the decomposition rules for the tensor product modules over $kG(\chi^{-1}, a, 0)$ when char$(k)$=0. Finally, we consider the representations of some Hopf-Ore extension of the dihedral group algebra $kD_n$, where $n=2m$, $m>1$ odd, and char$(k)$=0. The Grothendieck ring and the Green ring of the Hopf-Ore extension are described respectively in terms of generators and relations.