Green ring of the category of weight modules over the HopfOre 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 HopfOre 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.
 Publication:

arXiv eprints
 Pub Date:
 June 2018
 arXiv:
 arXiv:1806.01843
 Bibcode:
 2018arXiv180601843S
 Keywords:

 Mathematics  Rings and Algebras;
 16G30;
 16T05;
 19A22
 EPrint:
 arXiv admin note: text overlap with arXiv:1806.00753