The antipode of a dual quasi-Hopf algebra with nonzero integrals is bijective

TL;DR

This paper proves that for dual quasi-Hopf algebras with nonzero integrals, the antipode is not only injective but also bijective, extending known results from Hopf algebras to this broader class.

Contribution

It establishes the bijectivity of the antipode in dual quasi-Hopf algebras with nonzero integrals, a property previously known only for Hopf algebras.

Findings

Antipode in dual quasi-Hopf algebras with nonzero integrals is bijective.

The space of integrals in such algebras is one-dimensional.

Extends classical results from Hopf algebras to dual quasi-Hopf algebras.

Abstract

For $A$ a Hopf algebra of arbitrary dimension over a field $K$, it is well-known that if $A$ has nonzero integrals, or, in other words, if the coalgebra $A$ is co-Frobenius, then the space of integrals is one-dimensional and the antipode of $A$ is bijective. Bulacu and Caenepeel recently showed that if $H$ is a dual quasi-Hopf algebra with nonzero integrals, then the space of integrals is one-dimensional, and the antipode is injective. In this short note we show that the antipode is bijective.