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Twisted homogeneous racks over the alternating groups

Joseph Vulakh

TL;DR

This work advances the classification of finite-dimensional pointed Hopf algebras by examining the type D property of twisted homogeneous racks over alternating groups. By carrying out case analyses for automorphisms given by the identity and by conjugation with a transposition, the authors construct explicit decomposable subracks and prove many twisted homogeneous racks of type $(\mathbb{A}_n, t, \theta)$ are type D, thereby ruling them out as sources of finite-dimensional Nichols algebras. The main theorem narrows the non-type-D possibilities to a small set of explicit configurations, clarifying the landscape of racks that can yield finite-dimensional Nichols algebras and resolving many previously open cases. Consequently, this work significantly tightens the classification program for finite-dimensional pointed Hopf algebras by eliminating a broad class of racks from consideration and providing a clear map of remaining challenges for $n \ge 5$.

Abstract

An important step towards the classification of finite-dimensional pointed Hopf algebras is the classification of finite-dimensional Nichols algebras arising from braided vector spaces of group type. This question is fundamentally linked with the structure of algebraic objects called racks. Of particular interest to this classification is the type D condition on racks, a sufficient condition for a rack to not be the source of a finite-dimensional Nichols algebra. In this paper, we study the type D condition in simple racks arising from the alternating groups. Expanding upon previous work in this direction, we make progress towards a general classification of twisted homogeneous racks of type D by proving that several families of twisted homogeneous racks arising from alternating groups are of type D.

Twisted homogeneous racks over the alternating groups

TL;DR

This work advances the classification of finite-dimensional pointed Hopf algebras by examining the type D property of twisted homogeneous racks over alternating groups. By carrying out case analyses for automorphisms given by the identity and by conjugation with a transposition, the authors construct explicit decomposable subracks and prove many twisted homogeneous racks of type are type D, thereby ruling them out as sources of finite-dimensional Nichols algebras. The main theorem narrows the non-type-D possibilities to a small set of explicit configurations, clarifying the landscape of racks that can yield finite-dimensional Nichols algebras and resolving many previously open cases. Consequently, this work significantly tightens the classification program for finite-dimensional pointed Hopf algebras by eliminating a broad class of racks from consideration and providing a clear map of remaining challenges for .

Abstract

An important step towards the classification of finite-dimensional pointed Hopf algebras is the classification of finite-dimensional Nichols algebras arising from braided vector spaces of group type. This question is fundamentally linked with the structure of algebraic objects called racks. Of particular interest to this classification is the type D condition on racks, a sufficient condition for a rack to not be the source of a finite-dimensional Nichols algebra. In this paper, we study the type D condition in simple racks arising from the alternating groups. Expanding upon previous work in this direction, we make progress towards a general classification of twisted homogeneous racks of type D by proving that several families of twisted homogeneous racks arising from alternating groups are of type D.
Paper Structure (7 sections, 11 theorems, 14 equations, 3 tables)

This paper contains 7 sections, 11 theorems, 14 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Prior Work
  4. Twisted Homogeneous Racks
  5. Identity Automorphism
  6. Automorphism by Conjugation by a Transposition
  7. Proof of Main Result

Key Result

Theorem 1.1

For $n \geq 5$, a permutation $\ell \in \mathbb{A}_n$, an integer $t$ greater than $1$, and an automorphism $\theta$ of $\mathbb{A}_n$ given by conjugation by an element of $\mathbb{S}_n$, if the twisted homogeneous rack $\mathcal{C}_\ell$ of type $(\mathbb{A}_n, t, \theta)$ is not of type D, then $

Theorems & Definitions (22)

  • Theorem 1.1
  • Definition 2.1: andruskiewitsch2011finite*Definition 2.2
  • Definition 2.2
  • Remark 2.3
  • Theorem 2.4: andruskiewitsch2011finite*Theorem 3.6
  • Theorem 3.1: ANDRUSKIEWITSCH2003177*Theorem 3.9--Theorem 3.12
  • Theorem 3.2: andruskiewitsch2010twisted*Proposition 3.3
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • ...and 12 more