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

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 $\mathbb{Z}[X]♯\hat{G}$ modulo some ideal, where $\mathbb{Z}[X]$ is a polynomial algebra over $\mathbb{Z}$ in infinitely many variables.

arXiv admin note: text overlap with arXiv:1806.00753