Tensor product decomposition rules for weight modules over the Hopf–Ore extensions of group algebras

Abstract

In this paper, we investigate the tensor structure of the category of finite dimensional weight modules over the Hopf-Ore extensions kG(χ −1 , a, 0) of group algebras kG. The tensor product decomposition rules for all indecomposable weight modules are explicitly given under the assumptions that k is an algebraically closed field of characteristic zero, and the orders of χ and χ(a) are the same.2010 Mathematics Subject Classification. 16G30, 16T99.

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

“…Proof. Similarly to [11,Theorem 3.6], it can be shown for n s and n > s, respectively. For n s, the proof is similar to Case 1 in the proof of [11,Theorem 3.6].…”

“…In this section, we recall the finite dimensional indecomposable weight H-modules (see [14,11]. We still assume that χ −1 (a) 1, and use the notations of last section.…”

“…Let ǫ denote the identity element of the group G, i.e., ǫ(g) = 1, ∀g ∈ G. The finite dimensional indecomposable weight H-modules are classified in [14]. We state the classification (in case k is an algebraically closed field) as follows (see [11]).…”

“…In [11], we studied the tensor product structure of the category of finite dimensional weight modules over the Hopf-Ore extensions kG(χ −1 , a, 0) of group algebras kG. Hopf-Ore extensions were introduced and studied by Panov in [9].…”

“…Wang, Li and Zhang [12,13] studied the Green rings of finite dimensional rank one pointed Hopf algebras over an algebraically closed field of characteristic zero. In [11], we described the decomposition rules for tensor product of finite dimensional indecomposable weight modules over the Hopf-Ore extensions kG(χ −1 , a, 0) of group algebras kG in case |χ| = |χ(a)|, where the ground field k is an algebraically closed field of characteristic zero.…”