# Hopf algebras with enough quotients

## TL;DR

This paper investigates the properties of jointly inner faithful algebra maps from Hopf algebras, demonstrating that tensor and free products preserve this faithfulness, thus extending existing results in quantum group theory.

## Contribution

It introduces the concept of joint inner faithfulness for algebra maps and proves that tensor and free products maintain this property in Hopf algebras.

## Key findings

- Tensor products of jointly inner faithful maps are jointly inner faithful.
- Free products of jointly inner faithful maps are jointly inner faithful.
- Generalizes results on torus generation of compact quantum groups.

## Abstract

A family of algebra maps $H\to A_i$ whose common domain is a Hopf algebra is said to be jointly inner faithful if it does not factor simultaneously through a proper Hopf quotient of $H$. We show that tensor and free products of jointly inner faithful maps of Hopf algebras are again jointly inner faithful, generalizing a number of results in the literature on torus generation of compact quantum groups.