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

Abstract

Let $\mathbb{k}$ be an algebraically closed field of characteristic $0$. In this paper, we study the Grothendieck ring $G_0(D(H_{\mathcal{D}}))$ and the projective class ring $r_p(D(H_{\mathcal{D}}))$ of the Drinfeld double $D(H_{\mathcal{D}})$ of the rank one pointed Hopf algebra $H_{\mathcal{D}}$. We analyze the tensor products of simple modules with simple modules, simple modules with indecomposable projective modules, and indecomposable projective modules with indecomposable projective modules, providing explicit decomposition rules in each case. Finally, we compute both the Grothendieck ring $G_0(D(H_{\mathcal{D}}))$ and the projective class ring $r_p(D(H_{\mathcal{D}}))$, and present these two rings in terms of generators and defining relations.