Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Actions of Hopf-Ore Extensions of Group Algebras on Path Algebras of Quivers

Elise Askelsen, Ryan Kinser

Abstract

We classify the (filtered) Hopf actions of Hopf-Ore extensions of group algebras on path algebras of quivers, extending results in several other works from special cases to this general setting. Having done this for general Hopf-Ore extensions of group algebras, we demonstrate application by specializing our main result to certain Hopf-Ore extensions including some Noetherian prime Hopf algebras of GK-dimension one and two.

Actions of Hopf-Ore Extensions of Group Algebras on Path Algebras of Quivers

Abstract

We classify the (filtered) Hopf actions of Hopf-Ore extensions of group algebras on path algebras of quivers, extending results in several other works from special cases to this general setting. Having done this for general Hopf-Ore extensions of group algebras, we demonstrate application by specializing our main result to certain Hopf-Ore extensions including some Noetherian prime Hopf algebras of GK-dimension one and two.

Paper Structure

This paper contains 15 sections, 14 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Context and Motivation
  3. Summary of Main Results
  4. Relation to existing literature
  5. Background
  6. Hopf Algebras and Hopf Actions
  7. Quivers and Path Algebras
  8. Preliminary results on actions of skew-primitive elements
  9. Hopf-Ore Extensions of Group Algebras
  10. Main results
  11. Applications in small GK-dimension
  12. Gelfand-Kirillov Dimension One
  13. Lg
  14. Generalized Liu Algebra: Lg
  15. Gelfand-Kirillov Dimension Two

Key Result

Theorem 1

The following data determines a (filtered) Hopf action of the Hopf-Ore extension $R=\Bbbk G(\chi, h, \delta)$ on $\Bbbk Q$, and all such actions are of this form:

Theorems & Definitions (29)

  • Theorem
  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.6
  • Proposition 2.7
  • proof
  • Lemma 2.12
  • Proposition 2.13
  • proof
  • ...and 19 more