Green ring of the category of weight modules over the Hopf-Ore extensions of group algebras

TL;DR

This paper analyzes the tensor product structure and Green ring of weight modules over Hopf-Ore extensions of group algebras, providing explicit descriptions of the ring structures under various conditions.

Contribution

It describes the tensor product rules and explicitly characterizes the Green ring of the category of weight modules over Hopf-Ore extensions, extending understanding of their algebraic structure.

Findings

Green ring is isomorphic to a polynomial algebra over the group ring in certain cases.

Green ring is a quotient of a polynomial algebra over the group ring in two variables when specific conditions hold.

Green ring is a quotient of a skew group ring in the finite case.

Abstract

In this paper, we continue our study of the tensor product structure of category $\mathcal W$ of weight modules over the Hopf-Ore extensions $kG(\chi^{-1}, a, 0)$ of group algebras $kG$, where $k$ is an algebraically closed field of characteristic zero. We first describe the tensor product decomposition rules for all indecomposable weight modules under the assumption that the orders of $\chi$ and $\chi(a)$ are different. Then we describe the Green ring $r(\mathcal W)$ of the tensor category $\mathcal W$. It is shown that $r(\mathcal W)$ is isomorphic to the polynomial algebra over the group ring $\mathbb{Z}\hat{G}$ in one variable when $|\chi(a)|=|\chi|=\infty$, and that $r(\mathcal W)$ is isomorphic to the quotient ring of the polynomial algebra over the group ring $\mathbb{Z}\hat{G}$ in two variables modulo a principle ideal when $|\chi(a)|<|\chi|=\infty$. When $|\chi(a)|\le|\chi|<\infty$, $r(\mathcal W)$ is isomorphic to the quotient ring of a skew group ring $\mathbb{Z}[X]\sharp\hat{G}$ modulo some ideal, where $\mathbb{Z}[X]$ is a polynomial algebra over $\mathbb{Z}$ in infinitely many variables.