Representations of Hopf-Ore extensions of group algebras

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 $kG(\chi^{-1}, a, 0)$-modulesundertheassumption$|\chi|=\infty$and$|\chi|=|\chi(a)|<\infty$respectively,andallfinitedimensionalindecomposable$kG(\chi^{-1}, a, 0)$-modulesundertheassumptionthat$kG$ is finite dimensional and semisimple, and $\mathrm{\mid }\chi \mathrm{\mid }=\mathrm{\mid }\chi (a)\mathrm{\mid }$. 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 $k{D}_n$, 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.