Rank 4 finite-dimensional Nichols algebras of diagonal type in positive characteristic

TL;DR

This paper classifies all rank 4 finite-dimensional Nichols algebras of diagonal type over fields of any characteristic, advancing the understanding of their structure and invariants in algebra.

Contribution

It provides a complete classification of rank 4 Nichols algebras with finite arithmetic root systems in arbitrary characteristic, filling a gap in the existing literature.

Findings

Classification of all rank 4 Nichols algebras with finite arithmetic root systems

Extension of classification results to arbitrary characteristic fields

Identification of invariants for these Nichols algebras

Abstract

Nichols algebras are fundamental objects in the construction of quantized enveloping algebras and in the classification of pointed Hopf algebras by lifting method of Andruskiewitsch and Schneider. Arithmetic root systems are invariants of Nichols algebras of diagonal type with a certain finiteness property. In the present paper, all rank 4 Nichols algebras of diagonal type with a finite arithmetic root system over fields of arbitrary characteristic are classified. Our proof uses the classification of the finite arithmetic root systems of rank 4.