The Projective Class Rings of Drinfeld doubles of pointed rank one Hopf algebras

Abstract

Let $\Bbbk$ be an algebraically closed field with char$\Bbbk=0$. In this article, we study the Grothendick ring $G_0(D(H_\mathcal{D}))$ and the Projective Class rings $r_p(D(H_\mathcal{D}))$ of the Drinfeld doubles $D(H_{\mathcal{D}})$ of the rank one pointed Hopf algebra $H_{\mathcal{D}}$. We consider the tensor products of simple modules with simple modules and the tensor products of simple modules with indecomposable projective modules, and the tensor products of indecomposable projective modules with indecomposable projective modules. The decomposition rules are clearly described. Finally, we compute the Grothendick ring $G_0(D(H_\mathcal{D}))$ and Projective Class ring $r_p(D(H_\mathcal{D}))$, we display this two rings by generators with some relations.