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Simple Yetter-Drinfeld modules over infinite-dimensional Taft algebras

Xiangjun Zhen, Gongxiang Liu, Jing Yu

TL;DR

This work classifies simple Yetter-Drinfeld modules over the infinite-dimensional Taft algebra $H(n,t,\xi)$ over an algebraically closed field of characteristic zero, distinguishing finite- and infinite-dimensional cases. It constructs a canonical standard-basis framework for simple modules, derives the associated comatrix describing comodule structure, and proves that all finite-dimensional simples are $V(ti,j,\lambda)$ while infinite-dimensional ones arise precisely when $\gcd(t,n)>1$ as $V(i,j)$. The finiteness of Nichols algebras $\mathcal{B}(V)$ is characterized via a reduction to diagonal-type Nichols algebras and Heckenberger diagrams, yielding explicit tables of the finite cases and proving infinite-dimensionality in the remaining regimes. Overall, the results advance the understanding of simple Yetter-Drinfeld modules and Nichols algebras over infinite-dimensional Taft algebras, with implications for the structure and classification of pointed Hopf algebras of GK-dimension one.

Abstract

Let H be an infinite-dimensional Taft algebra over an algebraically closed field k of characteristic 0. We find all the simple Yetter-Drinfeld modules V over H, and classifies those V with B(V) is finite-dimensional.

Simple Yetter-Drinfeld modules over infinite-dimensional Taft algebras

TL;DR

This work classifies simple Yetter-Drinfeld modules over the infinite-dimensional Taft algebra over an algebraically closed field of characteristic zero, distinguishing finite- and infinite-dimensional cases. It constructs a canonical standard-basis framework for simple modules, derives the associated comatrix describing comodule structure, and proves that all finite-dimensional simples are while infinite-dimensional ones arise precisely when as . The finiteness of Nichols algebras is characterized via a reduction to diagonal-type Nichols algebras and Heckenberger diagrams, yielding explicit tables of the finite cases and proving infinite-dimensionality in the remaining regimes. Overall, the results advance the understanding of simple Yetter-Drinfeld modules and Nichols algebras over infinite-dimensional Taft algebras, with implications for the structure and classification of pointed Hopf algebras of GK-dimension one.

Abstract

Let H be an infinite-dimensional Taft algebra over an algebraically closed field k of characteristic 0. We find all the simple Yetter-Drinfeld modules V over H, and classifies those V with B(V) is finite-dimensional.

Paper Structure

This paper contains 18 sections, 41 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Braided vector spaces and Yetter-Drinfeld modules
  4. Nichols Algebras
  5. Infinite-dimensional Taft algebras
  6. Modules and comodules over infinite-dimensional Taft algebras
  7. Modules and comodules over infinite-dimensional Taft algebras
  8. Comodule Structures over Simple Yetter-Drinfeld Modules
  9. Existence of Standard Bases for Simple Yetter-Drinfeld Modules
  10. The Comatrix Relative to a Standard Basis
  11. Further Analysis of the Comatrix
  12. Classification of Simple Yetter-Drinfeld Modules over $H$
  13. The Finite-Dimensional Case
  14. The Infinite-Dimensional Case
  15. Finite-dimensional Nichols algebras over the simple modules $V(ti,j,\lambda)$
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $H=H(n,t,\xi)$. Then

Theorems & Definitions (93)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Definition 2.6
  • Definition 2.7
  • ...and 83 more