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 of weight modules over a Hopf–Ore extensions of a group algebra 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 χ and are different. Then we describe the Green ring of the tensor category . It is shown that is isomorphic to the polynomial algebra over the group ring in one variable when , and that is isomorphic to the quotient ring of the polynomial algebra over the group ring in two variables modulo a principle ideal when . When , is isomorphic to the quotient ring of a skew group ring modulo some ideal, where is a polynomial algebra over in infinitely many variables.