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On the representations of a family of pointed Hopf algebras

Agustin Garcia Iglesias, Alfio Antonio Rodriguez

Abstract

For each $\ell\geq 1$ and $λ,μ\in\Bbbk$, we study the representations of a family of pointed Hopf algebras $\mathcal{A}_{λ,μ}$. These arise as Hopf cocycle deformations of the graded algebra $\mathcal{FK}_3\#\Bbbk \mathbb{G}_{3,\ell}$, where $\mathcal{FK}_3$ is the Fomin-Kirillov algebra and $\mathbb{G}_{3,\ell}$ is a given non-abelian finite group. We compute the simple modules, their projective covers and formulate a description of tensor products. We observe that our results are fundamentally different according to the shape of the Hopf cocycle involved in the deformation.

On the representations of a family of pointed Hopf algebras

Abstract

For each and , we study the representations of a family of pointed Hopf algebras . These arise as Hopf cocycle deformations of the graded algebra , where is the Fomin-Kirillov algebra and is a given non-abelian finite group. We compute the simple modules, their projective covers and formulate a description of tensor products. We observe that our results are fundamentally different according to the shape of the Hopf cocycle involved in the deformation.
Paper Structure (37 sections, 41 theorems, 155 equations)

This paper contains 37 sections, 41 theorems, 155 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Our moonshine
  4. General structure
  5. Organization
  6. Preliminaries
  7. Representation of algebras
  8. Pointed Hopf algebras over $\mathbb{G}_{3,\ell}$ -and Hopf cocycles
  9. Irreducible representations of $\mathbb{G}_{3,\ell}$
  10. Computations of $G$-modules
  11. Proof of Proposition \ref{['pro:simplesG']}
  12. On the tensor structure of $\operatorname{Rep} \mathbb{G}_{3,\ell}$
  13. The graded case
  14. Notation
  15. Simple $\mathcal{A}_{\lambda,\mu}$-modules
  16. ...and 22 more sections

Key Result

Proposition 2.2

The irreducible representations of $\mathbb{G}_{3,\ell}$ are given, up to isomorphism by the following modules.

Theorems & Definitions (88)

  • Remark 2.1
  • Proposition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Proposition 2.6
  • proof
  • ...and 78 more