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

TL;DR

This paper computes the projective class rings of Drinfeld doubles of rank one pointed Hopf algebras, providing explicit decomposition rules and presenting the rings with generators and relations.

Contribution

It offers the first detailed analysis and explicit descriptions of the projective class rings for these specific Hopf algebra doubles.

Findings

Explicit decomposition rules for tensor products of modules.

Complete descriptions of the Grothendieck and projective class rings.

Presentation of the rings via generators and relations.

Abstract

Let $\Bbbk$ 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.