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

Hua Sun, Hui-Xiang Chen, Libin Li, Yinhuo Zhang

TL;DR

This work computes explicit tensor-product decompositions for the Drinfeld doubles D(H_{\mathcal{D}}) of rank-one pointed Hopf algebras and derives complete presentations for the associated invariants G_0 and r_p. By classifying simple modules V(l,λ) and indecomposable projectives P(l,λ) and analyzing their tensor interactions, the authors obtain closed-form decomposition rules and organize the results into generators-and-relations descriptions. The Grothendieck ring G_0(D(H_{\mathcal{D}})) and the projective class ring r_p(D(H_{\mathcal{D}})) are presented explicitly, with parity-dependent (n even/odd) structures and explicit generating sets, revealing fusion-like commutativity even though the underlying category is not braided. An illustrative example with the generalized Taft algebra H_{4,2} demonstrates limitations of forgetful functors to ordinary module categories, highlighting the nuanced relationship between D(H) and H-mod."

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.

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

TL;DR

This work computes explicit tensor-product decompositions for the Drinfeld doubles D(H_{\mathcal{D}}) of rank-one pointed Hopf algebras and derives complete presentations for the associated invariants G_0 and r_p. By classifying simple modules V(l,λ) and indecomposable projectives P(l,λ) and analyzing their tensor interactions, the authors obtain closed-form decomposition rules and organize the results into generators-and-relations descriptions. The Grothendieck ring G_0(D(H_{\mathcal{D}})) and the projective class ring r_p(D(H_{\mathcal{D}})) are presented explicitly, with parity-dependent (n even/odd) structures and explicit generating sets, revealing fusion-like commutativity even though the underlying category is not braided. An illustrative example with the generalized Taft algebra H_{4,2} demonstrates limitations of forgetful functors to ordinary module categories, highlighting the nuanced relationship between D(H) and H-mod."

Abstract

Let be an algebraically closed field of characteristic . In this paper, we study the Grothendieck ring and the projective class ring of the Drinfeld double of the rank one pointed Hopf algebra . 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 and the projective class ring , and present these two rings in terms of generators and defining relations.

Paper Structure

This paper contains 13 sections, 46 theorems, 163 equations.

Table of Contents

  1. Introduction
  2. Pointed rank one Hopf algebras and their doubles
  3. Simple modules and projective modules
  4. Simple modules $V(l,\lambda)$
  5. Indecomposable projective modules $P(l,\lambda)$
  6. The socle of the tensor product of two simple modules
  7. The decomposition rules for tensor product modules
  8. Tensor products of two simple modules
  9. Tensor products of simple modules with projective modules
  10. The tensor product of two projective modules
  11. The projecitve class rings $r_p(D(H_{\mathcal{D}}))$
  12. The Grothendieck ring $G_0(D(H_{\mathcal{D}}))$
  13. The projective class ring $r_p(D(H_{\mathcal{D}}))$

Key Result

Proposition 2.1

KropRad The Drinfeld double $D(H_{\mathcal{D}})$ is generated as an algebra by $G$, $x$, $\Gamma$ and $\xi$ subject to the relations defining $H_{\mathcal{D}}$ and $H^{*\rm cop}_{\mathcal{D}}$ and the following relations:

Theorems & Definitions (88)

  • Proposition 2.1
  • Proposition 3.1
  • Proposition 3.2
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof
  • Lemma 4.4
  • ...and 78 more