Representations of Hopf-Ore extensions of group algebras

Abstract

In this paper, we study the representations of the Hopf-Ore extensions kG(χ −1 , a, 0) of group algebra kG, where k is an algebraically closed field. We classify all finite dimensional simple kG(χ −1 , a, 0)-modules under the assumption |χ| = ∞ and |χ| = |χ(a)| < ∞ respectively, and all finite dimensional indecomposable kG(χ −1 , a, 0)-modules under the assumption that kG is finite dimensional and semisimple, and |χ| = |χ(a)|. Moreover, we investigate the decomposition rules for the tensor product modules over kG(χ −1 , a, 0) when char(k)=0. Finally, we consider the representations of some Hopf-Ore extension of the dihedral group algebra kDn, 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.