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On partial actions of Hopf-Ore extensions

João M. J. Giraldi, Grasiela Martini, Leonardo D. Silva

Abstract

In this work we study how to extend a partial action of a Hopf Algebra $A$ on an algebra $R$ to a partial action of a Hopf-Ore extension of $A$ on $R$. As consequence, we characterize all partial actions of rank one Hopf algebras (in particular, generalized Taft algebras and Radford algebras), under suitable conditions.

On partial actions of Hopf-Ore extensions

Abstract

In this work we study how to extend a partial action of a Hopf Algebra on an algebra to a partial action of a Hopf-Ore extension of on . As consequence, we characterize all partial actions of rank one Hopf algebras (in particular, generalized Taft algebras and Radford algebras), under suitable conditions.

Paper Structure

This paper contains 15 sections, 23 theorems, 69 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. q-combinatorics
  4. Partial actions
  5. Hopf-Ore extensions
  6. Extensions of partial actions via Hopf-Ore extensions
  7. Characterization of partial actions of Hopf-Ore extensions
  8. Sufficient conditions for extending a partial action to a Hopf-Ore extension
  9. Partial actions of Hopf-Ore extensions factorized through quotients
  10. Applications and examples
  11. The particular case when the element $x\cdot 1_R \in Z(R)$
  12. The particular case when $A= \Bbbk G$
  13. Rank one Hopf algebras
  14. The particular case when $A=\mathbb{H}_4$
  15. Nichols Hopf algebras

Key Result

Lemma 2.1

FMS2 Let $q \in \Bbbk^{\times}$ and $i, j, k\in \mathbb{N}_0$. Then,

Theorems & Definitions (46)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Theorem 2.6
  • Remark 2.7
  • ...and 36 more