# Pointed Hopf actions on central simple division algebras

## TL;DR

This paper investigates how finite-dimensional pointed Hopf algebras can act faithfully on central simple division algebras in characteristic zero, providing explicit constructions and contrasting with previous results on actions on fields.

## Contribution

It demonstrates that a wide class of pointed Hopf algebras can admit faithful actions on central simple division algebras, expanding understanding beyond actions on fields.

## Key findings

- All considered pointed Hopf algebras admit faithful actions on central simple division algebras.
- Constructed explicit examples of such division algebras for various classes of Hopf algebras.
- Contrasts with earlier work showing most pointed Hopf algebras do not act faithfully on fields.

## Abstract

We examine actions of finite-dimensional pointed Hopf algebras on central simple division algebras in characteristic 0. (By a Hopf action we mean a Hopf module algebra structure.) In all examples considered, we show that the given Hopf algebra does admit a faithful action on a central simple division algebra, and we construct such a division algebra. This is in contrast to earlier work of Etingof and Walton, in which it was shown that most pointed Hopf algebras do not admit faithful actions on fields. We consider all bosonizations of Nichols algebras of finite Cartan type, small quantum groups, generalized Taft algebras with non-nilpotent skew primitive generators, and an example of non-Cartan type.