# Hopf modules, Frobenius functors and (one-sided) Hopf algebras

## TL;DR

This paper explores the Frobenius property of functors related to Hopf modules over bialgebras, characterizing certain one-sided Hopf algebras through this property and connecting FH-algebras with Frobenius functors.

## Contribution

It provides a characterization of one-sided Hopf algebras with anti-(co)multiplicative antipodes via Frobenius functors and relates FH-algebras to Frobenius properties of associated functors.

## Key findings

- Characterization of one-sided Hopf algebras with Frobenius free Hopf module functor
- Relation between FH-algebras and Frobenius functors for bialgebras
- Insight into the structure of Hopf modules and their functorial properties

## Abstract

We investigate the property of being Frobenius for some functors strictly related with Hopf modules over a bialgebra and how this property reflects on the latter. In particular, we characterize one-sided Hopf algebras with anti-(co)multiplicative one-sided antipode as those for which the free Hopf module functor is Frobenius. As a by-product, this leads us to relate the property of being an FH-algebra (in the sense of Pareigis) for a given bialgebra with the property of being Frobenius for certain naturally associated functors.