Representations of Hopf-Ore Extensions of Group Algebras

Abstract

In this paper, we study the representations of the Hopf-Ore extensions kG(chi(-1), a, 0) of group algebra kG, where k is an algebraically closed field. We classify all finite dimensional simple kG(chi(-1), a, 0)-modules under the assumption vertical bar chi vertical bar= infinity and vertical bar chi vertical bar = vertical bar chi(a)vertical bar < infinity respectively, and all finite dimensional indecomposable kG (chi(-1), a, 0)-modules under the assumption that kG is finite dimensional and semisimple, and vertical bar chi vertical bar = vertical bar chi(a)vertical bar. Moreover, we investigate the decomposition rules for the tensor product modules over kG (chi(-)(1) , a, 0) when char (k) = 0. Finally, we consider the representations of some Hopf-Ore extension of the dihedral group algebra kD(n), where n = 2m, m > 1 odd, and char(k) = 0. The Grothendieck ring and the Green ring of the Hopf-Ore extension are described respectively in terms of generators and relations.