On Nichols algebras over PGL(2,q) and PSL(2,q)

TL;DR

This paper investigates conditions under which Yetter-Drinfeld modules over PGL(2,q) and PSL(2,q) produce finite-dimensional Nichols algebras, advancing classification of related pointed Hopf algebras and excluding certain cases over Mathieu groups.

Contribution

It provides necessary conditions for finite-dimensional Nichols algebras over PGL(2,q) and PSL(2,q), and shows no non-trivial finite-dimensional pointed Hopf algebras exist over M20 and M21.

Findings

Necessary conditions for Nichols algebras over PGL(2,q) and PSL(2,q)

No non-trivial finite-dimensional pointed Hopf algebras over M20 and M21

Progress towards classifying pointed Hopf algebras with these groups

Abstract

We compute necessary conditions on Yetter-Drinfeld modules over the groups $\mathbf{PGL}(2,q)=\mathbf{PGL}(2,\FF_q)$ and $\mathbf{PSL}(2,q)=\mathbf{PSL}(2,\FF_q)$ to generate finite dimensional Nichols algebras. This is a first step towards a classification of pointed Hopf algebras with group of group-likes isomorphic to one of these groups. As a by-product of the techniques developed in this work, we prove that there is no non-trivial finite-dimensional pointed Hopf algebra over the Mathieu groups $M_{20}$ and $M_{21}=\mathbf{PSL}(3,4)$.