# Antipodes, preantipodes and Frobenius functors

## TL;DR

This paper establishes equivalences between the existence of a preantipode in a quasi-bialgebra, Frobenius properties of associated functors, and Hopf monads, strengthening the link between Hopf and Frobenius structures.

## Contribution

It proves new equivalences connecting preantipodes, Frobenius functors, and Hopf monads in quasi-bialgebras and bialgebras.

## Key findings

- A quasi-bialgebra admits a preantipode iff its associated functor is Frobenius.
- The results tighten the relationship between Hopf and Frobenius properties.
- Similar equivalences hold specifically for bialgebras.

## Abstract

We prove that a quasi-bialgebra admits a preantipode if and only if the associated free quasi-Hopf bimodule functor is Frobenius, if and only if the relative (opmonoidal) monad is a Hopf monad. The same results hold in particular for a bialgebra, tightening the connection between Hopf and Frobenius properties.