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Higher rank Nichols algebras of diagonal type with finite arithmetic root systems in positive characteristic

L. J. Lei, C. Yuan, C. Qian, J. Wang

TL;DR

The paper advances the classification of finite-dimensional Nichols algebras of diagonal type in positive characteristic by leveraging arithmetic root systems and Weyl groupoids. It develops a framework that attaches a semi-Cartan graph to tuples of one-dimensional Yetter–Drinfel'd modules and analyzes finite Cartan graphs, distinguishing non-sporadic and sporadic cases via good neighborhood conditions. The main result is a comprehensive classification theorem for rank r ≥ 5, listing all admissible generalized Dynkin diagrams and providing exchange graphs, thereby enumerating all such Nichols algebras and clarifying the role of positive characteristic in the structure. These findings extend existing low-rank classifications and deepen the understanding of pointed Hopf algebras through the lifting method in positive characteristic settings.

Abstract

The classification of Nichols algebras is an essential step in the classification theory of pointed Hopf algebras by lifting method of N. Andruskiewitsch and H.-J. Schneider. Arithmetic root systems are invariants of Nichols algebras of diagonal type with a certain finiteness property. In this paper, all rank $r\geq 5$ Nichols algebras of diagonal type with a finite irreducible root system over fields of positive characteristic are classified. Weyl groupoids and finite arithmetic root systems are crucial tools for our classification.

Higher rank Nichols algebras of diagonal type with finite arithmetic root systems in positive characteristic

TL;DR

The paper advances the classification of finite-dimensional Nichols algebras of diagonal type in positive characteristic by leveraging arithmetic root systems and Weyl groupoids. It develops a framework that attaches a semi-Cartan graph to tuples of one-dimensional Yetter–Drinfel'd modules and analyzes finite Cartan graphs, distinguishing non-sporadic and sporadic cases via good neighborhood conditions. The main result is a comprehensive classification theorem for rank r ≥ 5, listing all admissible generalized Dynkin diagrams and providing exchange graphs, thereby enumerating all such Nichols algebras and clarifying the role of positive characteristic in the structure. These findings extend existing low-rank classifications and deepen the understanding of pointed Hopf algebras through the lifting method in positive characteristic settings.

Abstract

The classification of Nichols algebras is an essential step in the classification theory of pointed Hopf algebras by lifting method of N. Andruskiewitsch and H.-J. Schneider. Arithmetic root systems are invariants of Nichols algebras of diagonal type with a certain finiteness property. In this paper, all rank Nichols algebras of diagonal type with a finite irreducible root system over fields of positive characteristic are classified. Weyl groupoids and finite arithmetic root systems are crucial tools for our classification.
Paper Structure (12 sections, 69 equations, 2 figures)

This paper contains 12 sections, 69 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Yetter--Drinfel'd modules and braided vector space
  4. Nichols Algebras of Diagonal Type
  5. Cartan graphs, root systems, and Weyl groupoids
  6. The semi-Cartan Graph of a Nichols Algebra of Diagonal Type
  7. Finite Cartan graphs
  8. Non-sporadic finite Cartan graphs
  9. The root systems
  10. Good $D'_r$ neighborhood
  11. Sporadic finite Cartan graphs
  12. The classification theorem

Figures (2)

  • Figure 1: Dynkin diagram of type $D'_r$
  • Figure 2: generalized Dynkin diagram of type $A_r$

Theorems & Definitions (4)

  • proof
  • proof
  • proof
  • proof