Divisibility Relations for the Dimensions and Hilbert series of Nichols Algebras of Non-Abelian Group Type

TL;DR

This paper establishes a divisibility relation for the dimensions and Hilbert series of Nichols algebras of non-abelian group type, extending known results from Coxeter groups with constant cocycle -1.

Contribution

It introduces three groups of isomorphisms acting on Nichols algebras, generalizing the exchange operator concept for Coxeter groups.

Findings

Derived a divisibility relation for Nichols algebra dimensions

Generalized exchange operators to non-abelian group types

Extended Coxeter group results to broader Nichols algebra classes

Abstract

We present a divisibility relation for the dimensions and Hilbert series of certain classes of Nichols algebras of non-abelian group type, which generalizes Nichols algebras over Coxeter groups with constant cocycle -1. For this we introduce three groups of isomorphisms acting on Nichols algebras, which generalize the exchange operator introduced by Milinski and Schneider for Coxeter groups in "Pointed indecomposable Hopf algebras over Coxeter groups".