Braided racks, Hurwitz actions and Nichols algebras with many cubic relations

TL;DR

This paper classifies certain Nichols algebras over groups with braided racks, focusing on those with specific cubic relations, and introduces a new example using combinatorial invariants related to Hurwitz orbits.

Contribution

It provides a classification of Nichols algebras with braided racks and cubic relations, introducing a new example and a novel combinatorial approach involving Hurwitz orbits.

Findings

Classification of Nichols algebras with braided racks and cubic relations

Introduction of a new Nichols algebra example

Development of a combinatorial invariant based on Hurwitz orbits

Abstract

We classify Nichols algebras of irreducible Yetter-Drinfeld modules over groups such that the underlying rack is braided and the homogeneous component of degree three of the Nichols algebra satisfies a given inequality. This assumption turns out to be equivalent to a factorization assumption on the Hilbert series. Besides the known Nichols algebras we obtain a new example. Our method is based on a combinatorial invariant of the Hurwitz orbits with respect to the action of the braid group on three strands.